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Error correction and quantumness testing of quantum annealing devices

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Error correction and quantumness testing of quantum annealing devices

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Content ERROR CORRECTION AND QUANTUMNESS TESTING OF QUANTUM ANNEALING DEVICES by Anurag Mishra A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) August 2018 Copyright 2018 Anurag Mishra To my mother ii Acknowledgments If I have seen further, it is by standing on the shoulders of Giants. This dissertation would not have been possible without the support of friends, family and mentors around me. Let me first start by expressing my gratitude to my academic advisor, Prof. Daniel Lidar, who supported me throughout the years and taught me how to write, organize, and think about my research. I am also indebted to Dr. Tameem Albash, who acted as my second mentor. All the work described in this dissertation was done in collaboration with Tameem and Daniel. Where Daniel let me fly high with my imagination, Tameem would ground me and make me come up with better physics in support of my ideas. They are the Yin and Yang of this work. I also thank my collaborators Dr. Walter Vinci and Prof. Paul Warburton. Many thanks go out to all the faculty members at USC and IIT Kanpur that prepared me for this arduous journey. The knowledge and wisdom I gained from them will follow me throughout my life. I thank Prof. Deshdeep Sahdev of IIT Kanpur for his amazing ability to impart the joy of quantum mechanics in me and my fellow students. I especially thank Prof. Itzak Bars, Prof. Ben Reichardt, Prof. Richard Thompson and Prof. Paulo Zanardi. I thank my friends here and around the world, Abhay, Abhishek, Anurag, Vaibhaw, Neelabh, Geetak and Anchal. I will fondly remember the time spent iii with Ryan, Ignacio and Siddharth solving many homework exercises in first year of the PhD program. Many thanks for Josh for my endless discussion of statistical methods with him. Many thanks to Kristen and Zhihui for helping me settle down in Pasadena. I thank all the health professional who make my life better. I thank Micheal for his unending patience with my rants and excuses. I thank Genevieve for her love and support. Lastly, I will thank my mother and my sisters for their unending love and support and their patience. iv Contents Acknowledgments iii List of Figures vii List of Tables xxxiv Abstract xxxvi 1 Introduction 1 1.1 Adiabatic quantum computing . . . . . . . . . . . . . . . . . . . . . 2 1.2 The D-Wave device . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Quantum annealing correction . . . . . . . . . . . . . . . . . . . . . 6 1.4 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Consistency Tests of Classical and Quantum Models for a Quan- tum Annealer 12 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 D-Wave Control Noise Sources . . . . . . . . . . . . . . . . . . . . . 22 2.4 Introducing an energy scale . . . . . . . . . . . . . . . . . . . . . . 24 2.5 Numerical simulations without cross-talk . . . . . . . . . . . . . . . 26 2.6 Experimental results and numerical simulations including cross-talk 33 2.7 Ground state entanglement during the course of the annealing evolution 38 2.8 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . 40 2.9 The D-Wave Two device . . . . . . . . . . . . . . . . . . . . . . . . 44 2.10 Enhancement vs suppression of the isolated state in SA vs QA . . . 44 2.11 Experimental data collection methodology . . . . . . . . . . . . . . 51 2.12 Kinks in the time dependence of the gap . . . . . . . . . . . . . . . 60 2.13 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.14 Comparing the models in the noiseless case . . . . . . . . . . . . . . 68 2.15 ME vs Modified SSSV models with “decoherence” from O(2) rotors to Ising spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 v 2.16 An alternative model for breaking the symmetry of the cluster states 79 2.17 Effect of varying the annealing time or the total number of spins . . 87 2.18 Derivation of the O(3) spin-dynamics model . . . . . . . . . . . . . 93 3 Performance of two different quantum annealing correction codes101 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.2 QAC using the [4, 1, 4] 0 and [3, 1, 3] 1 codes . . . . . . . . . . . . . . 106 3.3 Benchmarking using antiferromagnetic chains . . . . . . . . . . . . 113 3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.5 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.7 Optimizing γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 3.8 Comparing decoding strategies . . . . . . . . . . . . . . . . . . . . . 132 4 Finitetemperaturequantumannealingsolvingexponentiallysmall gap problem with non-monotonic success probability 140 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.2 The alternating sectors chain model . . . . . . . . . . . . . . . . . . 143 4.3 Empirical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.4 Fermionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 4.5 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.6 Master equation model . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.7 Comparison to the classical SVMC model . . . . . . . . . . . . . . . 157 4.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5 Conclusion 163 5.1 Open questions and future work . . . . . . . . . . . . . . . . . . . . 165 A Appendix 167 A.1 Chain generation algorithm . . . . . . . . . . . . . . . . . . . . . . 167 A.2 Jordan-Wigner transform . . . . . . . . . . . . . . . . . . . . . . . . 167 A.3 Fermionic domain-wall states . . . . . . . . . . . . . . . . . . . . . 172 A.4 Spin Vector Monte Carlo (SVMC) algorithm . . . . . . . . . . . . . 174 A.5 Test of the adiabatic condition for ASC . . . . . . . . . . . . . . . . 176 A.6 ASC at different annealing times . . . . . . . . . . . . . . . . . . . 177 A.7 Computing master equation matrix elements for ASC . . . . . . . . 178 A.8 Exponential dependence of the success probability on k ∗ . . . . . . 183 Reference List 185 vi List of Figures 2.1 DW2 annealing schedules A(t) and B(t) along with the operating temperature of T = 17mK (black dashed horizontal line). The large A(0)/(k B T) value ensures that the initial state is the ground state of the transverse field Hamiltonian. The largeB(t f )/(k B T) value ensures that thermal excitations are suppressed and the final state reached is stable. Also shown are the attenuated αB (t) curves for (a) the value of α at which the intersection between A(t) and αB (t) coincides with the operating temperature (blue dot-dashed curve), and (b) the largestα such thatαB (t) remains below the temperature line for the entire evolution (blue dotted curve). . . . . . . . . . . . 19 2.2 The eight-spin Ising Hamiltonian. The inner “core” spins (green circles) have local fieldsh i = +1 [using the convention in Eq. (2.2)] while the outer spins (red circles) have h i =− 1. All couplings are ferromagnetic: J ij = 1 (black lines). . . . . . . . . . . . . . . . . . . 20 2.3 Schematic representation of the 12, 16, and 20 spin Hamiltonians used in our tests. Extensions to larger N follow the same pattern, with N/2 qubits in the inner ring and N/2 qubits in the outer ring. Notation conventions are as in Fig. 2.2. . . . . . . . . . . . . . . . . 20 vii 2.4 Numerically calculated evolution of the gap between the instan- taneous ground state and the 17-th excited state (which becomes the first excited state att = t f ), for the eight-spin Hamiltonian in Eq. (2.1), following the annealing schedule of the DW2 device (Fig. 2.1). The gap value is shown for some interesting values of α (see Fig. 2.1). The kinks are due to energy level crossings, as explained in Section 2.12. A reduction in α results in a reduction of the size of the minimal gap and delays its appearance. . . . . . . . . 24 2.5 ME simulation for the time-dependence of the probability of being in the lowest 17 energy eigenstates, for different values ofα . Simulation parameters are t f = 20μs (the minimal annealing time of the DW2) and κ = 1.27× 10 − 4 , where κ is an effective, dimensionless system- bath coupling strength defined in Section 2.13.3. The chosen value of κ allows us to reliably probe the small α regime. . . . . . . . . . 27 viii 2.6 Distribution of the ground states for N = 8 for (a) ME with no noise on{h i , J ij }, (b) SSSV with no noise on{h i , J ij }, and (c) SSSV with{h i , J ij } noise using σ = 0.085. The cluster states are labeled by their Hamming distance H from the isolated state, and by their multiplicity M for a given value of H. The vertical axis is the final probabilityp of a given (H,M) set, divided by its multiplicity and the total ground state probability. The data symbols (◦ , etc.) are the mean values of the bootstrapped [103] distributions, and the error bars are two standard deviations below and above the mean representing the 95% confidence interval. Note that the SSSV model prefers the|1111 0000i cluster state, whereas the ME gives a uniform distribution over all cluster states. SSSV parameters are T = 10.56mK and 1× 10 5 Monte Carlo step updates per spin (“sweeps”). The same parameters are used in all subsequent SSSV figures. These results do not include the cross-talk correction. . . . 28 ix 2.7 Time evolution (according to the ME) of energy eigenstate pop- ulations for α = 0.1 and κ = 8.9× 10 − 4 (this relatively large value was chosen here since it results in increased thermal exci- tation/relaxation). P i denotes the population of the ith eigenstate, with i = 1 being the instantaneous ground state. The energy eigen- state that eventually becomes the isolated ground state is i = 6 (dashed red line). This state acquires more population at the end of the evolution than the other 16 eigenstates that eventually be- come the cluster (solid purple line). (Inset) The difference of the population ratio between the open system and the closed system evolution, Δ(P I /P C ) = (P I /P C ) Open − (P I /P C ) Closed . The deviation from closed system dynamics starts at t/t f ≈ 0.4, when the i = 6 eigenstate becomes thermally populated at the expense of the lowest five eigenstates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 x 2.8 Statistical box plot of the probability divided by the multiplicity of being in a given state with Hamming distanceH and multiplicityM. Shown are the ME isolated state and cluster states for N = 8, with 512 noise realizations applied to the h’s and J’s [with distribution N (0, 0.06)] at α = 1. The isolated state (H = 0,M = 1) is sup- pressed while the cluster states are, on average, equally populated. The red bar is the median, the blue box corresponds to the lower and upper quartiles, respectively, the segment contains most of the samples, and the +’s are outliers [104]. The horizontal axis label indicates the Hamming distance from the isolated state and the multiplicity of the cluster-states at each value of H. States that are equivalent up to 90 ◦ rotations are grouped together. For example, there are four rotationally equivalent cluster-states that have two adjacent outer qubits pointing down, while the other two are pointing up. Only the H = 6 case splits into two rotationally inequivalent sets. 31 xi 2.9 (a) Ground state populations for DW2. Legend: (H,M), correspond- ing to Hamming distance from the isolated state and multiplicity respectively. Error bars represent the 95% confidence interval. (b) Cluster state populations for the ME. Solid lines correspond to the results with no noise on the{h i ,J ij }’s, while the data points include Gaussian noise with mean 0 and standard deviation σ = 0.025 for 100 noise realizations. The error bars represent the 95% confidence interval. The DW2 data from (a) is also plotted as the shaded region representing a 95% confidence interval with the dashed lines corre- sponding to the mean. The inset shows the behavior for the noiseless ME for small α . (c) Cluster state population for SSSV for N = 8 with the DW2 data plotted as in (b). In contrast to Fig. 2.6, both the ME and the noisy SSSV model include the cross-talk correction, Eq. (2.6), with χ chosen to optimize the fit for the cluster state populations at α = 1. (d) Only the cluster states with Hamming distance 4 and 8 from the isolated state are shown for DW2, the ME, and SSSV from panels (b) and (c) in order to highlight their differ- ences. Panel (e) displays the same for N = 20 (excluding the ME, which is too costly to simulate at this scale). (f) The isolated state populations for DW2, SSSV, ME, and noisy ME, which highlights the qualitative agreement between the models and DW2. Experimental data were collected using the in-cell embeddings strategy described in Section 2.11. The embedding and gauge-averaging strategies are also discussed in Section 2.11. The color coding of states is consistent across all panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 xii 2.10 Subset of the first excited state populations for (a) DW2 forN = 8; (b) SSSV for N = 8; and (c) ME for N = 8. In (b) and (c), the simulations include qubit cross-talk correction with χ chosen as in Fig. 2.9 to optimize the fit for the cluster state populations at α = 1. Panels (d) and (e) are for N = 20. The Π symbol denotes all permutations. The SSSV model does not reproduce the correct ordering. The error bars represent the 95% confidence interval. . . . 37 2.11 Time-dependence of the negativity [Eq. (2.8)] for (a) a closed system evolution and (b) an open system evolution ofN = 8 qubits (modeled via the ME withκ = 1.27×10 − 4 ), as a function ofα . The rapid decay of negativity for small α in the open system case signals a transition to classicality. However, for large α the closed and open system negativities are similar, suggesting that the system is quantum in this regime. The apparent jaggedness of the closed system plot near α = 0 is due to our discretization of α in steps of 0.01. . . . . . . . 39 2.12 Qubits and couplers in the DW2 device. The DW2 “Vesuvius” chip consists of an 8× 8 two-dimensional square lattice of eight-qubit unit cells, with open boundary conditions. The qubits are each denoted by circles, connected by programmable inductive couplers as shown by the lines between the qubits. Of the 512 qubits of the device located at the University of Southern California used in this chapter, the 503 qubits marked in green and the couplers connecting them are functional. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 xiii 2.13 Embedding according to the random parallel embeddings strategy for eight spins. An example of 15 randomly generated different parallel copies of the eight-spin Hamiltonian. Our data collection used a similar embedding with 50 different copies. . . . . . . . . . . . . . . 52 2.14 Embedding according to the random parallel embeddings strategy for 16 spins. An example of 10 randomly generated different parallel copies of the 16-spin Hamiltonian. Our data collection used a similar embedding with 93 different copies. . . . . . . . . . . . . . . . . . . 53 2.15 Embedding according to the “in-cell embeddings” strategy. An example of a randomly generated in-cell embedding. . . . . . . . . . 54 2.16 Some representative autocorrelation tests at t f = 20μs showing the standard error of the mean Δx as a function of binning size, for different values of α : (a) α = 0.1, (b) α = 0.35, (c) α = 1. Each curve is the result of the binning test for a different state. The relatively flat lines for all states suggest that there are no significant autocorrelations in the data. . . . . . . . . . . . . . . . . . . . . . . 58 2.17 (a) Time-dependence of the gap between the 18th excited state and the instantaneous ground state, for different values of the energy scale factor α . (b)-(d) Time-dependence of the lowest 56 energy eigenvalues for different values of α [(b) α = 1/7, (c) α = 2/7, (d) α = 3/7]. Note that the identity of the lowest 17 energy eigenvalues changes over the course of the evolution. . . . . . . . . . . . . . . . 61 xiv 2.18 Simulated annealing shows quantitatively similar behavior for var- ious annealing schedule. The schedules are: exponential T(k) = T(0)(T (K)/T (0)) k/K , linear T(k) = T(0) + k K (T(K)− T(0)), and constant T(k) = T(K), with K = 1000, with k B T(0)/~ = 8 GHz and k B T (K)/~ = 0.5 GHz. . . . . . . . . . . . . . . . . . . . . . . . 62 2.19 Numerical results distinguishing the quantum ME and classical SA, SD, and SSSV models. (a) Results for the ratio of the isolated state population to the average population in the cluster-states (P I /P C ), and(b)thegroundstateprobability(P GS ), asafunctionoftheenergy scale factor α , at a fixed annealing time oft f = 20μs . The error bars represent the 95% confidence interval. Two striking features are the “ground state population inversion” between the isolated state and the cluster (the ratio of their populations crosses unity), and the manifestly non-monotonic behavior of the population ratio, which displays a maximum. At the specific value of the system-bath coupling used in our simulations (κ = 1.27× 10 − 4 ), it is interesting that the ME underestimates the magnitude and position of the peak inP I /P C but qualitatively matches the experimental results shown in Fig. 2.9(a), capturing both the population inversion and the presence of a maximum even in the absence of noise. In contrast to the ME results shown, the SA, SSSV, and SD results for the population ratio are not in qualitative agreement with the ME. Specifically, all three classical models miss the population inversion and maximum seen for the ME. Simulation parameters can be found in Section 2.13. . 67 xv 2.20 Statistical box plot of the average z component for all core qubits (main plot) and all outer qubits (inset) att =t f = 20μs . (a) The SD model. The data are taken for 1000 runs with Langevin parameters k B T/~ = 2.226 GHz (i.e., 17mK, to match the operating temperature of the DW2) andζ = 10 − 3 . In Section 2.18.2 we show that the results do not depend strongly on the choice ofζ . (b) The SSSV model. The data are taken for 1000 runs with parameters k B T/~ = 1.382 GHz (i.e., T = 10.56mK, as in Ref. [48]) and 5× 10 5 sweeps. . . . . . . . 72 2.21 “Forced” and strongly or weakly “decohering” SSSV models. Shown are the results for the ratio of the isolated state population to the average population in the cluster-states (P I /P C ) as a function of the energy scale factor α , for N = 8 and at a fixed annealing time of t f = 20μs . The error bars represent the 95% confidence interval. For reference the plot also includes the curves for the ME from Fig. 2.19(a). Additional parameters for the modified SSSV models: g 2 η = 10 − 6 for the strongly decohered model and g 2 η = 2.5× 10 − 7 for the weakly decohered and forced models. . . . . . . . . . . . . 73 2.22 Trace-norm distance of the ME, SA, SD, SSSV, weakly and strongly decohered SSSV, and forced SSSV states from the T = 17mK Gibbs state at t f = 20μs and N = 8. The error bars represent the 95% confidence interval. Three regions are clearly distinguishable for the ME: (1) 1≥ α & 0.3, whereD is decreasing as α decreases; (2) 0.3&α & 0.1, whereD is increasing asα decreases; (3) 0.1&α ≥ 0, whereD is again decreasing as α decreases. Both SA and SD lack the minimum at α ≈ 0.3. . . . . . . . . . . . . . . . . . . . . . . . 76 xvi 2.23 Master equation results for the populations of the 17 Ising ground states, with α = 1,|h| = 0.981|J|, t f = 20μs , and κ = 1.27× 10 − 4 . The cluster-states split by Hamming distance from the isolated state (bottom curve), in agreement with the experimental results shown in Fig. 2.24(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.24 Statistical box plot of the gauge-averaged ground states population (a,c) before and (b), (d) after optimization of J ij as per Table 2.1, for (a),(b) N = 8 and (c),(d) N = 16, α = 1 and t f = 20μs . Only the N = 8,H = 6 case splits into two rotationally inequivalent sets. Note the clear step structure in the cluster-states (H > 0) in (a),(c), while in (b),(d) the population of the cluster-states is fairly equalized (less so in the N = 16 case since Table 2.1 is optimized for N = 8). (a) Data taken with the random parallel embeddings strategy. (b) Data taken using the in-cell embeddings strategy, with the optimized values of the couplings given in Table 2.1. The same optimization removes the step structure from data taken with the random parallel embeddings strategy (not shown). . . . . . . . . . . . . . . . . . . . 82 2.25 Ratio of isolated state population to average cluster-state population as a function of the energy scale factor α , for t f = 20μs and N = 8. Shown are the ratios calculated with both uncorrected and corrected values of J (as per Table 2.1), the latter tuned to flatten the steps seen in the population of the cluster states. Error bars represent the standard error of the mean value of the ratio estimated using bootstrapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 xvii 2.26 (a) Noisy SSSV; (b) ME, (c) perturbation theory [Eq. (2.46)] with a population ordering correction (offset of h vsJ) atα = 1. The error bars represent the 95% confidence interval. The SSSV model now has the right ordering of the cluster states but clearly disagrees with the DW2 result [Fig. 2.9(a)] for α . 0.3, near where the isolated state has its maximum. The ME result is in qualitative agreement with the DW2 result except that the cluster state populations do not equalize for small α , which is a consequence of not including the α -dependence of the offset. . . . . . . . . . . . . . . . . . . . . . . . 84 2.27 (a) DW2 and (b) noisy SSSV model with a population equalizing correction at α = 1. (c) Noisy SSSV model with offset chosen to equalize the cluster state populations at α = 0.2. The error bars represent the 95% confidence interval. . . . . . . . . . . . . . . . . . 85 2.28 Subset of the first excited state populations for (a) perturbation theory [as in Eq. (2.46)], (b), (c) and (d) SSSV with offsets matching Figs. 2.26(a), 2.27(b) and 2.27(c) respectively. Panel (f) shows the case with qubit cross-talk discussed in Sec. 2.3. The Π symbol denotes all permutations. Whereas the perturbation theory result for a QA Hamiltonian [Eq. (2.46)] reproduces the correct ordering, none of the three SSSV cases shown does. These three SSSV cases were chosen to optimize the fit for the cluster state populations. The error bars represent the 95% confidence interval. . . . . . . . . . . . 88 xviii 2.29 Ratio of the isolated state population to the average population in the cluster (P I /P C ) as a function of the energy scale factorα , for two different values of t f , and N = 8. The inset shows the ME results. Data were collected using the “in-cell embeddings” strategy (see Section 2.11.1 for details). Error bars are one standard deviation above and below the mean. . . . . . . . . . . . . . . . . . . . . . . . 89 2.30 Numerically calculated instantaneous energy gap between the ground and first excited state for the 8, 12 and 16 spin Hamiltonians. The gap vanishes since the first excited state becomes part of the2 N/2 +1- fold degenerate ground state manifold at t =t f . . . . . . . . . . . . 89 2.31 Ratio of the isolated state population to the average population in the cluster-states (P I /P C ) as a function of the energy scale factor α , for different values ofN, at a fixed annealing time oft f = 20μs . The non-monotonic dependence of the population ratio on α is observed for all values of N. The growth of the P I /P C peak with increasing N is consistent with the discussion presented in Section 2.17.2. The increasingly large error bars are due to the smaller amount of data collected as N grows. For N = 12, 16, 20 data were collected using the “random parallel embeddings” strategy and for N = 40 using the “designed parallel embedding” strategy (see Section 2.11.1 for details). Error bars are one standard deviation above and below the mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 xix 2.32 Evolution of a core (blue) and outer (green) spin with t f = 20μs , subject to the O(3) SD model with α = 1. All spins start with M x = 1, M y = M z = 0, i.e., point in the x direction. (a) Closed system case given by Eq. (2.63). (b) Open system case given by Eq. (2.65). Rapid oscillations at the beginning of the evolution in (a) are because the initial conditions used are not exactly the ground state of the system [because of the finiteB(0)]. In (b), Langevin parameters are k B T/~ = 2.226 GHz and ζ = 10 − 6 . . . . . . . . . . . 96 2.33 Distribution of M z at the end of the evolution for all (a) core spins and (b) all outer spins for α = 1. Langevin parameters are k B T/~ = 2.226 GHz and ζ = 10 − 3 . Data collected using 1000 runs of Eq. (2.65). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.34 Statistical box plot of the z component for all core qubits (main plot) and all outer qubits (inset) at t = t f = 20μs . The data is taken for 1000 runs of Eq. (2.65) with Langevin parametersk B T/~ = 2.226 GHz (to match the operating temperature of the DW2) and (a) ζ = 10 − 1 and (b) ζ = 10 − 5 . The ζ = 10 − 3 is shown in Fig. 2.20(a). This illustrates that the results do not depend strongly on the choice of ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.35 Distribution of M z at the end of the evolution for (a) all core spins and (b) all outer spins (i.e., the values of all core spins and all outer spins are included in each respective box plot) forα = 2/7. Langevin parameters are k B T/~ = 2.226 GHz and ζ = 10 − 3 . Data collected using 1000 runs of Eq. (2.65). Note that the t f -axis scale is not linear.100 xx 3.1 The DW2 processors have unit cells arranged in an 8× 8 grid, each containing 8 qubits forming a K 4,4 bipartite graph. The active (inactive) qubits are shown in green (red), and active couplers are shown as solid black lines. Out of the 512 qubits on the full DW2 Chimera graph, 504 and 476 were functional on the DW2-ISI and S6 devices, respectively. A comparison of the physical parameters of the two devices is given in Table 3.1. . . . . . . . . . . . . . . . . . 104 3.2 Construction of the [4, 1, 4] 0 code. Two encoded qubits are constructed using the four physical qubits in the upper and the lower halves of the unit cell. In (a), the dotted lines represent the penalty terms. The solid lines form the encoded Hamiltonian couplings. Since each encoded coupling is formed by 2 physical couplings, the energy scale of the Ising problem is boosted by factor of 2. In (b) we show the section of the encoded graph formed by (a), with the same color scheme for the couplings. Roman letters labels the same encoded qubits in (a) and (b). . . . . . . . . . . . . . . . . . . . . . 106 xxi 3.3 Construction of the [3, 1, 3] 1 code. An encoded qubit is con- structed using three data qubits from each vertical half of the unit cell and a penalty qubit from the opposite half. In (a), the four physical qubits forming the encoded group are shown in the same color, and the dashed lines represent the stabilizer couplings. The solid lines form the encoded Hamiltonian coupling. Since each en- coded coupling comprises 3 physical couplings, the energy scale of the encoded problem is boosted by a factor of 3. In (b) we show the section of the encoded graph formed by (a), with the same color scheme for the couplings. In both (a) and (b), the Roman alphabet labels the respective encoded qubits. . . . . . . . . . . . . . . . . . 107 3.4 The [4, 1, 4] 0 encoded graph. Each encoded qubit is composed of four physical qubits, and the encoded couplings are formed from two physical couplers. The green (red) circles denote functional (inactive) qubits. Out of 128 possible encoded qubits on the complete graph, 120 were functional on the DW2-ISI device (a) and 99 on the S6 device (b). The encoded graph is a 2-level grid [124] and has degree 5.108 xxii 3.5 The [3, 1, 3] 1 code encoded graph. Each encoded qubit is com- posed of four physical qubits, and the encoded couplings are formed from three physical couplers. The green (red) circles denote func- tional (inactive) qubits. Orange circles indicate encoded qubits that have all three data qubits but are missing their penalty qubits. Out of 128 possible encoded qubits on the complete graph, 120 were fully functional while 3 were missing penalty qubits on the DW2-ISI device (a); 95 were fully functional and 18 were missing penalty qubit on the S6 device (b). We only used fully functional encoded qubits in our experiments. The encoded graph has degree 3. . . . . 108 3.6 ResultsfortheDW2-ISIdevice. Panels(a),(b)and(c)compare the results for chains using the U, C, [3, 1, 3] 1 code and [4, 1, 4] code at scaling parameters α = 1, 0.5 and 0.4 respectively. Panel (d) shows a comparison at a fixed chain length ofN = 98 as α is varied. For α . 0.5 and sufficiently long chain lengths, the [4, 1, 4] 0 code starts to outperform the C strategy. The [3, 1, 3] 1 code outperforms all other strategies at all values ofα , for sufficiently long chain lengths.116 3.7 Results for the S6 device. Panels (a), (b) and (c) compare the results for chains using the U, C, [3, 1, 3] 1 code and [4, 1, 4] code at scaling parameters α = 1, 0.5 and 0.4 respectively. Panel (d) shows a comparison at a fixed chain length ofN = 80 as α is varied. For α . 0.90 and for sufficiently long chains, the [4, 1, 4] 0 code starts to outperform the C strategy. The [3, 1, 3] 1 code outperforms all other strategies at all values of α , for sufficiently long chain lengths. . . . 117 xxiii 3.8 Independence test. Using the data from the DW2-ISI device, we compare the performance of the U strategy to the C strategy at α = 0.45. The C strategy agrees with the prediction from binomial theory that assumes independent chains. . . . . . . . . . . . . . . . 118 3.9 Decoding strategies for the [4, 1, 4] 0 code. DW2-ISI device at α = 0.70. The EP strategy is marginally improved upon by the use of decoding. Ties are broken by either coin tossing or energy minimization; the latter performs slightly better at all chain lengths. 119 3.10 Code comparison at equalized effective energy scales for the DW2-ISI device. Panel (a) compares the EP performance of the two codes at equivalent effective energy scales: 3× 0.3 and 2× 0.45 for the [3, 1, 3] 1 and [4, 1, 4] 0 code, respectively. Panel (b) compares the two codes after decoding. Code performance is essen- tially indistinguishable in the EP case, indicating that the effective energy scale is the dominant performance-determining factor. The [3, 1, 3] 1 code exhibits a slight advantage across all chain lengths after decoding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 xxiv 3.11 Decodable states. Panels (a) and (b) display the performance of the [3, 1, 3] 1 and [4, 1, 4] 0 codes, respectively, for the longest chain of length 100, α = 1, and optimal γ observed on the DW2-ISI device. We show the energies of all the states observed relative to the encoded ground state along the vertical axis, while the horizontal axis is the Hamming distance from the encoded ground state. We did not find any decodable states of Hamming distance higher than 50. Color indicates the fraction of decodable states at each observed energy and Hamming distance. States with a small Hamming distance are mostly decodable. The [3, 1, 3] 1 code is decoded via majority vote, while the [4, 1, 4] 0 code uses the EM strategy. . . . . . . . . . . . . . 121 3.12 Comparison of the performance of the ISI and S6 devices. Correlation plot of the success probability on the S6 and ISI devices for instances that have the same γ opt (for the same α and chain length) using the EP strategy [panel (a), with 264 instances] and the QAC with EM strategy [panel (b), with 376 instances]. Instances to the left (right) of the diagonal line have a higher success probability on the S6 (DW2-ISI) device. Virtually all instances were solved with a higher success probability on the S6 device. . . . . . . . . . . . . 122 xxv 3.13 Optimal γ for the [4, 1, 4] 0 code. Panels (a) and (b) show the optimalγ values for the DW2-ISI device. Panels (c) and (d) show the same for the S6 device. For three representative values of the scaling parameter α , we note that the EP strategy consistently requires a higher value for the optimal γ . There is a slight tendency for longer chains to have a larger optimal penalty. Additionally, since the DW2-ISI device operates at a higher temperature and hence is more prone to errors, it requires a higher value for the optimal γ than the S6 device. The difference is most prominent in the QAC case, i.e., when comparing panels (b) and (d). . . . . . . . . . . . . 126 3.14 Thermodynamic comparison of codes. Panel (a) shows the thermal error rates of the two codes at the same encoded energy scale h. Panel (b) compares the two codes at equivalent effective energy scales, i.e., h for the [4, 1, 4] 0 code and 2h/3 for the [3, 1, 3] 1 code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.15 Two coupled encoded [4, 1, 4] 0 code qubits. The two encoded [4, 1, 4] 0 code qubits consists of 8 physical qubits. We show one of the two degenerate ground states, and all inequivalent ways in which bit-flip errors might accumulate on one of the encoded qubits. Each flip leads to an excited state. The number in parentheses denotes the multiplicity of such states, followed by the energy separation from the shown ground state. The↑ symbol denotes the correct state of the qubit while the⇑ symbol denotes the occurrence of a flip. . . . 128 xxvi 3.16 Decodability analysis for α = 0.3. (a) The first few excited states at the final time for two coupled [4, 1, 4] 0 code qubits. The legend labels refer to the states in Fig. 3.15, indicating their decod- ability. The optimalγ value occurs right before the first excited state goes from being decodable to undecodable. Logical error states are represented by solid lines, decodable states by dashed lines. Thick lines indicate degenerate excited states, some of which are decodable, and some of which are encoded errors. (b) Success probability calcu- lated using the adiabatic master equation [45]. Inset: a zoomed out version. The success probability of the QAC strategy is maximized near γ ≈ 0.3, which agrees with the value of γ in (a) where an undecodable state becomes the first excited state. Population lost to this state in the simulations cannot be recovered after decoding. This explains the comment made in Sec. 3.2.2 that the optimal γ keeps the decodable states lower in the energy spectrum. The EP strategy is optimized at a larger value of γ . . . . . . . . . . . . . . . 130 3.17 Optimalγ forthe [4, 1, 4] 0 codeontheDW2-ISIdevice. The color scale represents the success probability, while the white circles indicate the optimal penalty value for a given chain length. The top and bottom three panels show the EP and QAC with EM strategies, respectively. The optimal γ increases proportionally to the problem scale α . A higher γ opt is required for the EP case, where we perform no decoding. The success probability depends strongly on N,γ , and α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 xxvii 3.18 Optimalγ forthe [3, 1, 3] 1 codeontheDW2-ISIdevice. The optimal γ increases proportionally to the problem scale α in the EP case, but remains fairly constant for the QAC case, in agreement with Ref. [86]. The success probability depends strongly on N and α , but not as strongly on γ . . . . . . . . . . . . . . . . . . . . . . . 134 3.19 Optimalγ forthe [4, 1, 4] 0 codeontheS6device. Theoptimal γ increases proportionally to the problem scale α . A higher γ opt is required for the EP case, where we perform no decoding. The success probability depends strongly on N, γ , and α . . . . . . . . . 135 3.20 Optimalγ ontheS6deviceforthe [3, 1, 3] 1 code. Theoptimal γ increases proportionally to the problem scale α in the EP case, but remains fairly constant for the QAC case, again in agreement with Ref. [86] (though note that the latter used the DW2-ISI device). The success probability depends strongly on N and α , but not as strongly on γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.21 PerformancecomparisonfortheDW2-ISIdevice. The [4, 1, 4] 0 code starts to outperform the C strategy below α ≈ 0.5 and for suf- ficiently long chains. . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3.22 Performance comparison for the S6 device. The [4, 1, 4] 0 code starts to outperform the C strategy belowα ≈ 0.9 and for sufficiently long chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.23 Ties in the [4, 1, 4] 0 code. The number of ties per qubit is shown for each value of α ∈ [0.1, 1.0], minimized over the penalty strength γ . The number of ties generally increases with the chain length N and decreases with α . The largest number of ties per qubit is ∼ 5× 10 − 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 xxviii 4.1 Illustration of an alternating sector chain (ASC). This exam- ple has sector size n = 3, length N = 10 and number of sectors 2b + 1 = 3. Red lines denote the heavy sector with coupling W 1 , blue lines denote the light sector with coupling W 2 <W 1 . . . . . . . 143 4.2 Empirical success probability vsk ∗ for the ASC problem on the DW2X processor. k ∗ denotes the number of single-fermion energies that fall below the thermal energy gap at the point of the minimum gap s ∗ . The legend entries indicate the chain parameters: (W 1 ,W 2 ,N). The error bars everywhere indicate 95% confidence intervals calculated using a bootstrap [103, 142] over different gauges and embeddings. (a)-(c) Contrary to closed-system theory expec- tations, the success probability P G is non-monotonic in the sector sizen, first decreasing and then increasing, exponentially.Inset (a): The minimum gap (in GHz) of the chains as a function of the sector size n∈{ 1,..., 20}. The solid black line denotes the operating temperature energy scale of the DW2X. (d)-(f) For all chains we studied the ground state success probability has a minimum at the sector size n ∗ where the peak in the number of single-fermion states k ∗ occurs [compare with (a)-(c)]. The rise and fall pattern, as well as the location of n ∗ , are in agreement with the behavior of P G within the error bars. Inset (d): The total number of energy eigenstates that fall below the thermal energy gap as a function of the sector sizen. In this case the peak position does not agree with the ground state success probability minimum. . . . . . . . . . . . . . . . . . . 145 xxix 4.3 Ratio of the gap to the thermal density of states, as a func- tion of sector size. Two alternating sector chain cases are shown. The position of the minimum is determined by d rather than Δ, as can be seen by comparing to Fig. 4.2(d) where the plot of d = 2 k ∗ alone correlates well with the position of minima in the empirical success probability curves. . . . . . . . . . . . . . . . . . . . . . . . 150 4.4 Master equation results for the state populations when re- stricting the excited states to single-fermion states. (a) The population in each single-fermion state at t = t f in a one-fermion simulation. The chain parameters are N = 176, W 1 = 1, W 2 = 0.5, t f = 5μ s, and n = 5. With the annealing schedule given in Methods, the quantum minimum gap is at s ∗ =t ∗ /t f ≈ 0.424. At this point we findk ∗ = 18 single-fermion states below the thermal energy T = 12mK (D-Wave processor operating temperature). As expected, inone-fermionsimulations, mostofthepopulationisfoundinthefirst k ∗ states. A long tail of more energetic single particle states beyond the firstk ∗ retain some population. (b) Evolution of the instanta- neous ground state populations for ASCs with the same parameters as in (a), but for different sector sizes n and with two-fermion states. The ground state loses the majority of its population as it approaches the minimum gap point at t/t f =s ∗ . The largest drop is found for n =n ∗ = 5. Inset: Magnification of the region around the minimum gap. Relaxation plays essentially no role. Instead, the population freezes almost immediately. . . . . . . . . . . . . . . . . . . . . . . 154 xxx 4.5 Master equation results for the ground state population whenrestrictingtheexcitedstatestosingleandtwofermion states. (a) The result of simulating the ASC problem with pa- rameters (1, 0.5, 175) via the adiabatic Pauli master equation (4.8), restricted to the vacuum + single-fermion states, and vacuum + single-fermion + two-fermion states. Also shown is the dependence on the system-bath coupling parameter g in the two-fermion case; doubling it has little impact, whereas halving it increases the success probability somewhat for n < 14. The position of the minimum at n ∗ = 5 matches the empirical result seen in Fig. 4.2(a), except when g = 1/2, i.e., the position is robust to doubling g but not to halving it. Panels (b) and (c) show additional 2-fermion master equation results with g = 1. Note that for the (1, 0.5, 200) chain, these simulations exhibit better agreement with the DW2X data than the simple k ∗ analysis plotted in Figs. 4.2(d)-4.2(f). This is because the simulations also keep track of the Boltzmann factor. . . 155 xxxi 4.6 Comparison of the SMVC model to the empirical DW2X results. The error bars everywhere indicate 95% confidence intervals calculated using a binomial bootstrap over the different runs of the simulation. (a) The SVMC parameters were optimized to match the empirical DW2X success probability results for the chain with parameters (W 1 ,W 2 ,N) = (1, 0.5, 175). The optimal values found are: N s = 120× 10 3 ,β = 0.75 (GHz) − 1 ,σ = 0.05 [compare to the DW2X’s t f = 5 μ s,β = 0.637 (GHz) − 1 ,σ ∼ 0.03]. (b) With the same optimal SVMC parameter values, but with chain parameters (0.8, 0.4, 175), theSVMCmodelpredictsincreasedsuccessprobability, in contrast to the empirical results. The same trend continues but is more pronounced in (c), with additionally the position of the minimum shifting to the wrong location (n = 7 vs n ∗ = 4). Panel (d) shows that increasing the chain length causes a large deviation in the SVMC results [compare to panel (a)], and also shifts the location of the minimum to the wrong value, but (e) shows showing that reducing the chain length does not degrade the agreement much. (f) Results for another chain parameter set, exhibiting a similar discrepancy to that seen in (c). . . . . . . . . . . . . . . . . . . . . 158 4.7 Spin boundary correlation function computed using the SVMC model. The spin boundary correlation function for the same chain and SVMC parameters as in Fig. 4.6(a). The SVMC model does not correctly capture the empirical results despite pro- viding a close match to the success probability in Fig. 4.6(a). . . . . 160 xxxii A.1 Test of the adiabatic condition. The solid line is the anneal- ing time used in our experiments. Symbols represent the quantity appearing in two versions of the adiabatic condition [for ASC param- eters (1, 0.5, 175)] that should be smaller than the annealing time in order for the adiabatic condition to hold. . . . . . . . . . . . . . . . 176 A.2 Dependence of the success probability on the annealing times. WeshowtheresultsfortheASCwithparameters (1.0, 0.5, 175). The error bars everywhere indicate 95% confidence intervals calcu- lated using a bootstrap over different gauges and embeddings. The location of the minimum is unchanged as the annealing time is varied.177 xxxiii List of Tables 2.1 Optimized|J ij | values for a given|h i | value, yielding the flat popula- tion structure shown in Fig. 2.24(b). The systematic corrections are of the order of 1%, smaller than the random control errors of 5% at α = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.1 Physical parameter of the two quantum annealing devices used in our study. Both devices belong to the same Vesuvius generation, with the major difference being a lower temperature and lower noise on the S6 device, and a higher qubit yield on the DW2-ISI device. M AFM is the inter-qubit coupling energy when a coupler is set to provide the maximum antiferromagnetic (AFM) coupling. N f is the number of functional qubits on the annealer. 1/f is the low frequency flux noise in units of flux quanta,Φ 0 =h/2e, where h is the Planck constant and e is the electron charge. Details about these physical parameters can be found in Ref. [127]. . . . . . . . . . . . . . . . . 105 xxxiv 3.2 Comparisonofthetwocodesfora 1-qubitencodedproblem with a local field. (a) [3, 1, 3] 1 code. The first three columns are the values of the data qubits; the fourth column is the penalty qubit; the fifth is the energy penalty counted as twice the number of violated couplings v; the sixth is the magnetization m =− P i s i ; the seventh is the multiplicity, which is the number of states that are topologically equivalent to the state shown, up to relabeling of qubits; and the last is the decodability of the state. (b) [4, 1, 4] 0 code. The first four columns are the values of the data qubits; the fifth is the energy penalty; the sixth is the magnetization m =− P i s i ; the seventh is the multiplicity, which is number of states that are topologically equivalent to the state shown, up to relabeling of qubits; and the last is the decodability, where “t” denotes a tie. Rows 3–5 denote three distinct ways of placing the two plus and two minus states, where the qubits are numbered 1–4 clockwise, starting from top-left [as in Fig. 3.2(a)]. . . . . . . . . . . . . . . . . . . . . . . . 123 4.1 Chain length (N) and sector size (n) for N∼ 175. . . . . . . . . . 144 4.2 Locations n ∗ of the DW2X success probability minima vs those found by the fermionic model based on the peak of k ∗ , the master equation model (ME), and the SVMC model. When the location of the minimum is ambiguous within our 95% confidence interval we list all values ofn ∗ that overlap to within oneσ . The best agreement is obtained by the master equation model. . . . . . . . . . . . . . . 159 xxxv Abstract Quantum computing is rapidly progressing from an academic field to one with commercially available quantum computers. We now have access to quantum annealers with thousands of qubits. The end user only has a very limited access to the inner working of such devices. Often, they present themselves as a black box: the user has control over some dials to program her problem, and the machine returns a classical solution at the end of its computation. How can one say that the black box presented is truly solving the problem with a quantum process? In this dissertation, we try to characterize the behavior of commercially available quantum annealers in different regimes and we show that the currently accepted quantum master equation models adequately explain the behavior of these annealers on problems with hundreds of qubits. Not surprisingly, these black box quantum annealers suffer from decoherence and errors due to their interaction with the external environment. As an example, we will discuss an extreme case where the behavior of the device deviates significantly from the noise-free closed system prediction. Quantum error correction techniques must be used to recover the theoretical closed system scaling advantage. We discuss the performance of the proposed error correction technique for quantum annealing and discuss the relative advantages of the two proposed codes. xxxvi Chapter 1 Introduction What are the fundamental physical limitations of computing [1]? In past few decades, computers have become smaller and faster. This is succinctly captured by the celebrated Moore’s law [2, 3]: the transistor density doubles about every two years. This increasing density has been achieved by miniaturization of computing components. The laws of physics at small sizes are described by quantum mechanics, and as the computing components become small, the need to understand the implications of quantum mechanics on computing becomes important. Quantum computing [4] is a new approach towards computing where quantum-mechanical phenomena, such as super-position and entanglement, are utilized to do useful calculations. A quantum computer [5] is a device that implements quantum computation. Already, quantum computing algorithms have started showing demonstrable advantages over their classical analogs. The algorithm from Shor [6] offers an exponential speedup in finding the prime factors of a large number over any known classical algorithm and Grover’s algorithm [7] can search through an unsorted list with quadratically fewer steps than any classical method. Another problem that has proved intractable without quantum computers is the task of simulating quantum interactions. A quantum system with n variables has an associated space of 2 n configurations which makes the task of brute-force search impossible. Feynman [8] was first to advocate for using quantum computers to simulate quantum systems. 1 As the field progresses, commercially available quantum computing devices are being made available online to the general public [9–12]. In this dissertation, we will work on few such devices manufactured by the D-Wave Inc [13]. These devices implement a limited version of the adiabatic quantum computing model. This is a different model of quantum computation which is nevertheless equivalent to the better known circuit model of quantum computation. In Chapter 2, we will discuss how to verify the quantum behavior of these devices in a “black-box” model of computing. Next, in Chapter 3, we will try to analyze the performance of two different error correction codes proposed for these devices. In Chapter 4, we will discuss how thermal noise can have a great impact on these devices, negating the predictions made from “noise-free” analysis. This dissertation will focus on the adiabatic quantum computing model as implemented in a limited fashion by the D-Wave devices and how error correction can enhance performance lost due to the noisy nature of these commercial devices. We turn to these topics next. 1.1 Adiabatic quantum computing Adiabatic quantum computation (AQC) is a model of quantum computing that utilizes the quantum adiabatic theorem to implement quantum computing. AQC is polynomially equivalent to the standard circuit model of quantum computation [14]. In other words, AQC can be simulated on the circuit model and vice-versa with an overhead that grows only polynomially in the system size. In AQC, we first start by preparingthegroundstateofan“easy”toimplementHamiltonian. TheHamiltonian of the system is then slowly changed to reach a desired final Hamiltonian, the ground state of which encodes the result of the computation. If the total evolution 2 time of the algorithm is large enough, the adiabatic theorem guarantees that the system will find the correct ground state of the final Hamiltonian with a high probability. Mathematically, AQC can be implemented by the following Hamiltonian: H(t) =A(t)H init +B(s)H final (1.1) where where t f is the total evolution time, s = t/t f ∈ [0, 1], and A(s) and B(s) are the annealing schedules, monotonically decreasing and increasing, respectively, satisfying B(0) = 0 and A(1) = 0. The quantum adiabatic theorem then places a lower bound on the evolution time t f [15]. For a detailed review of adiabatic quantum computing, see Ref. [16]. When used as an optimization algorithm, AQC is frequently referred as adiabatic quantum algorithm (AQO) [17, 18] or Quantum annealing (QA) [19–26]. While the adiabatic theorem guarantees a solution to a problem, it may not be the optimal way for implementing quantum annealing for the problem. Often, evolving the system faster than that predicted by the adiabatic theorem leads to higher success probability [27, 28]. Frequently, as in this dissertation, quantum annealing is used as an all-encompassing term where the Hamiltonian (1.1) is implemented without any restriction on t f . 1.2 The D-Wave device The D-Wave device [13, 29–32] is a practical realization of a quantum annealing device. The work discussed in this dissertation has used two generations of these 3 processors, the D-Wave 2 device (DW2) and the D-Wave 2X (DW2X) device. Both generations have been designed to implement the transverse Ising model, H(s) =A(s) X i σ x i +B(s)   X i h i σ z i + X (i,j) J ij σ z i σ z j   , (1.2) with dimensionless time s =t/t f . The coupling strengths J ij between qubits i and j can be set in the range [− 1, 1] and the local fieldsh i can be set in the range [− 2, 2]. The qubits on these processors are arranged on the so-called Chimera graph [33]. The qubits are arranged in cells of completely bipartite K 4,4 where the cells themselves are arranged on a 2-D grid. Every newer generation of the device has a bigger graph compared to the earlier generations. The DW2 generation devices feature a cell-grid of size 8× 8 and the DW2X generation devices have a cell grid of size 12× 12. Theoretically, these are graphs that contain 512 and 1152 qubits respectively. In practice, calibration and manufacturing errors result in some non-functional qubits on the graph and hence reduce the number of active qubits. In order to be useful across a large range of problems, a quantum device should allow interactions between any arbitrary pair of qubits. On the other hand, the Chimera graph has a fixed structure with limited connectivity. This limitation can be worked around by minor-embedding [33, 34], where we find a minor of the Chimera graph that can implement the connectivity required for a specific problem. In this manner, these processors can implement many different NP-hard problem [35]. Since their introduction, considerable debate has happened in the scientific community regarding the nature and usefulness of these devices. Do the D-Wave devices exhibit quantum behavior, and if so, can they utilize this quantum behavior to provide a speed-up over classical computers [36]? 4 In theory, quantum systems can be differentiated from classical systems by using Bell’s inequality [37, 38] or other tests [39, 40] based on similar ideas. These tests of quantum behavior require preparation of entangled states on the quantum device, and frequently the presence of such entanglement is thought to be a sufficient demonstration of the quantum behavior of a device. Such entanglement has been alreadyshownforsmallsizesontheD-Wavedevice[41,42]. Whiletheseexperiments establish the existence of quantum effect at small sizes, showing the existence of such entanglement measure with hundreds of qubits remains challenging. In lieu of such experiments, the next most relevant task can be elimination of all known reasonable model of the device. Frequently, simulated annealing is thought to be the classical analog of quantum annealing [25]. Early results with random instances [43] and small, specially designed instances [44] ruled out simulated annealing as a proper model for the device and showed that a time-dependent adiabatic quantum master equation [45] can adequately explain the behavior of the device. However, ruling out one classical model does not preclude the existence of other such phenomenological classical models that can explain the behavior of the device. Indeed, objections were raised about the results presented in aforementioned work, and a new classical model was able to explain the ground state success probability of these experiments [36, 46– 49]. Later experiments on the D-Wave devices have conclusively ruled out any existing classical model [42, 50, 51] as the proper description of the D-Wave device, while various forms of quantum master equations continue to show a good match with the experimental results. Whether these devices show a conclusive speed-up over the classical algorithms remains an open question. In 2014, Rønnow et al. [52] gave a proper definition of a quantum speed-up in this context. In their work, they found no evidence of a quantum speed-up on the D-Wave annealer against a random set of problems. Other 5 such work also found a lack of speed-up against the best classical algorithms [53, 54]. Recently, the D-Wave device was shown to demonstrably outperform some classical algorithms on a synthetic set of instances [55]. Further work is needed to find a definitive set of problems where quantum annealing on the D-Wave device can beat well known classical algorithms. Even without demonstrable speed-up, the D-Wave device has found utility in other areas of computation. Specifically, quantum annealing machine learning (QAML) [56, 57] has shown promising results when applied to novel problems, such as identifying a Higgs boson event against the noise background [58] and in classification and ranking of transcription factor binding [59]. Many more applications of quantum annealing [57, 60–62] await proper connectivity and size of such devices. 1.3 Quantum annealing correction Although QA is robust against certain forms of decoherence appearing in the more realistic open system setting [63–69], it remains susceptible to thermal noise and specification errors [70], which can jeopardize the efficiency of the quantum computation. Therefore, any scalable QA architecture will require quantum error correction [71]. Unfortunately, theoretical progress in quantum error correction for adiabatic quantum computing and quantum annealing has not enjoyed the same success as that of other quantum computing paradigms, in spite of recent advances [72–78] . Physical constraints, such as locality of the interaction terms in the Hamiltonian [79, 80], and a no-go theorem constraining what can be achieved 6 with commuting two-local interactions [81], remain stubborn hurdles. An accuracy- threshold theorem rivaling that of the circuit model(e.g., [82, 83]) remains elusive despite recent attempts [84]. Commercial quantum annealers, such as the D-Wave device, present their own challenges in implementation of quantum error correction. The standard scheme of error-suppression via energy penalties for QA requires encoding of the entire Hamiltonian [72, 76]. Moreover, such techniques frequently require k-local interactions with k≥ 3 [79], though codes with 2-local interactions also exist [75]. On the other hand, commercial annealers only allow for control over the Ising part of the Hamiltonian 1.2 and only implement two-body couplers (k = 2). Quantum annealing correction (QAC) [85–87] presents an implementation of quantum error correction that can operate under these compromises. In its simplest definition, QAC can be thought of as a quantum repetition code which implements error suppression and a decoding procedure on the classical output of an anneal under Hamiltonian (1.1). For k logical qubits, the data of the problem Hamiltonian (h i and J ij of Hamiltonian 1.2) are encoded over n D - data qubits and additional n P -“penalty” qubits add terms to Hamiltonian which energetically suppress errors during the anneal. In total, k qubits are encoded in n =n D +n P . If the distance of the code is d, we represent the code by notation [n,k,d] n P [87]. The practical use of these codes requires three steps. First, the problem Hamiltonian is converted to the encoded Hamiltonian. This is achieved by replacing the Ising operators σ z i in Eq. (1.2) by its encoded counterpart σ z i such that the encoded Hamiltonian is H I = X i h i σ z i + X ij J ij σ z i σ z j , (1.3) 7 where theh i andJ ij values are inherited from the original problem Hamiltonian. In our second step, we add the penalty termH P composed of the penalty qubits. This term is the sum of the stabilizer generators [71] of the code. It ties together the data qubits of the Hamiltonian and energetically penalizes disagreement between the data qubits. With this term included, the final Hamiltonian implemented on the annealer is given by H(t) =A(t) X i σ x i +B(t)(H I +γH P ) , (1.4) where γ is a term that control the relative strength of the encoded problem Hamiltonian and the penalty Hamiltonian. In this formulation, we are not encoding the transverse field term and the penalty Hamiltonian evolves with the encoded Hamiltonian. This reflects the limitations of the D-Wave device and makes QAC different from regular scheme of error suppression, such as the one found in Ref. [72]. The transverse field term ( P i σ x i ) prepares the initial ground state, but it also acts as a logical error on the encoded Hamiltonian. An optimal value of the term γ ensures that the decodable states remain lower in the energy spectrum. We let this Hamiltonian evolve on the annealer, and in step three, the logical qubits are decoded using the spin state of the data qubits. For codes where the number of data qubits n D is even, a simple majority vote will suffice to properly decode the logical qubits. When the number of data qubits is even, a majority vote alone cannot always tell us the state of the logical qubits. In this case, more complex decoding schemes must be implemented [87] to break any ties that occur between the data qubits. We will present a more detailed review of these scheme in Chapter 3. The first QAC code was introduced by Pudenzet al. [86] which encodes 1 logical qubits using 3 data qubits and 1 penalty qubits. In our notation, we refer to 8 this as the [3, 1, 3] 1 code. Another such code, the [4, 1, 4] 0 code, was introduced by Vinci et al. [87]. In order to properly compare the performance of these codes with the unencoded Hamiltonian, one must allow for the unencoded case to have same amount of resources as the encoded case. The aforementioned codes use 4 physical qubits to every encoded qubits, and hence use four time as many resources as an unencoded problem. We can remove this handicap on the unencoded problem by running it in parallel. A run is considered a success if any of these copies find the correct ground state. This scheme is called the classical repetition technique. For a range of different instances [50, 87, 88], both codes show an improvement over the classical repetition strategy. What explains the success of this error correction scheme? Intuitively, this is achieved by suppressing single bit flips error via the penalty term, and decoding corrects for any remaining bit flip errors. Mean field analysis of QAC withp-body Ising model [89, 90] shows that QAC softens the closing of the quantum minimum gap and pushes the critical point towards larger transverse field values. If this value is larger than the initial transverse field implemented on the annealer, QAC can completely avoid the minimum gap of the problem. 1.4 Summary of the results We now summarize the results contained in this dissertation. The question of whether the D-Wave processors exhibit large-scale quantum behavior or can be described by a classical model has attracted significant interest. In Chapter 2, we address this question by studying a 503 qubit D-Wave Two device in the “black box” model, i.e., by studying its input-output behavior. Our work generalizes an approach introduced in Boixo et al. [44], and uses groups of up to 20 qubits to 9 realize a transverse Ising model evolution with a ground state degeneracy whose distribution acts as a sensitive probe that distinguishes classical and quantum models for the D-Wave device. Our findings rule out all classical models proposed to date for the device and provide evidence that an open system quantum dynamical description of the device that starts from a quantized energy level structure is well justified, even in the presence of relevant thermal excitations and a small value of the ratio of the single-qubit decoherence time to the annealing time. The work contained in this chapter was published in Ref. [91]. In Chapter 3, we propose and benchmark the [4, 1, 4] 0 error correcting code developed to work on the D-Wave quantum annealing processors. This code offers a slightly higher connectivity for the encoded graph than the previous [3, 1, 3] 1 code proposed by Pudenz et al. [86] for the same number of physical qubits per logical group. However, the code uses fewer physical couplings per logical coupling, which reduces the energy scale amplification that encoding provides. We test the performance of the code using anti-ferromagnetic chains, and while the code does not beat the Pudenz et al.’s code for the same energy scales, we show that it can provide an advantage over a classical repetition strategy when the noise level is sufficiently high. The work contained in this chapter was published in Ref. [88]. Closed-system quantum annealing is expected to sometimes fail spectacularly in solving simple problems for which the gap becomes exponentially small in the problem size. Much less is known about whether this gap scaling also impedes open- system quantum annealing. In Chapter 4, we study the performance of a quantum annealing processor in solving the problem of a ferromagnetic chain with sectors of alternating coupling strength. Despite being trivial classically, this is a problem for which the transverse field Ising model is known to exhibit an exponentially decreasing gap in the sector size, suggesting an exponentially decreasing success 10 probability in the sector size if the annealing time is kept fixed. We find that the behavior of the quantum annealing processor departs significantly from this expectation, with the success probability rising and recovering for sufficiently large sector sizes. Rather than correlating with the size of the minimum energy gap, the success probability exhibits a strong correlation with the number of thermally accessible excited states at the critical point. We demonstrate that this behavior is consistent with a quantum open-system description of the process and is unrelated to thermal relaxation. Our results indicate that even when the minimum gap is several orders of magnitude smaller than the processor temperature, the system properties at the critical point still determine the performance of the quantum annealer. The work contained in this chapter has been accepted for publication [92]. We conclude in Chapter 5. Appendix A contains technical arguments and other supplementary material. 11 Chapter 2 Consistency Tests of Classical and Quantum Models for a Quantum Annealer 2.1 Introduction How can one determine whether a given “black box” is quantum or classical [93]? A case in point are the devices built by D-Wave [29, 30, 32]. These devices are commercial computers that the user can only access via an input-output interface. Reports [43, 44, 94] that the D-Wave devices implement quantum annealing (QA) with hundreds of qubits have attracted much attention recently, and have also generated considerable debate [36, 46–49]. At stake is the question of whether the experimental evidence suffices to rule out classical models, and whether a quantum model can be found that is in full agreement with the evidence. It is our goal in this chapter to distinguish several classical models (simulated annealing, spin dynamics [46], and hybrid spin-dynamics Monte Carlo [48]) and a quantum adiabatic master model [45] of the D-Wave device, and to decide which of the models survives a comparison with the experimental input-output data of a “quantum signature” test. This test is not entanglement-based and does not provide a Bell’s-inequality-like [38] no-go result for classical models. Instead, our approach is premised on the standard notion of what defines a “good theory:” it should 12 have strong predictive power. That is, if the theory has free parameters then these can be fit once, and future predictions cannot require that the free parameters be adjusted anew. It is in this sense that we show that we can rule out the classical models, while at the same time we find that the adiabatic quantum master equation passes the “good theory” test. The D-Wave devices operate at a non-zero temperature that can be compara- ble to the energy gap from the ground state, so one might expect that thermal excitations act to drive the system out of its ground state, potentially causing the annealing process to be dominated by thermal fluctuations rather than by quantum tunneling. Furthermore, the coupling to the environment should cause decoherence, potentially resulting in the loss of any quantum speedup. This is- sue was recently studied in Refs. [43, 44], where data from a 108-qubit D-Wave One (DW1) device was compared to numerical simulations implementing classical simulated annealing (SA), simulated quantum annealing (SQA) using quantum Monte Carlo, and a quantum adiabatic master equation (ME) derived in Ref. [45]. These studies demonstrated that SA correlates poorly with the experimental data, while the ME (in Ref. [44]) and SQA (in Ref. [43]) are in good agreement with the same data. Specifically, the eight-qubit “quantum signature” Hamiltonian introduced in Ref. [44] has a 17-fold degenerate ground state that splits into a single “isolated” state and a 16-fold degenerate “cluster,” with the population in the former suppressed relative to the latter according to the ME but enhanced according to SA; the experiment agreed with the ME prediction [44]. Subsequently, Ref. [43] rejected SA on much larger problem sizes by showing that the ground state population (“success probability”) distribution it predicts for random Ising instances on up to 108 spin variables is unimodal, while the experimental data and 13 SQA both give rise to a bimodal distribution. This was interpreted as positive evidence for the hypothesis that the device implements QA. However, interesting objections to the latter interpretation were raised in Refs. [46, 48], where it was argued that there are other classical models that also agree with the experimental data of Refs. [43, 44]. First, Smolin and Smith [46] pointed out that a classical spin-dynamics (SD) model of O(2) rotors could be tuned to mimic the suppression of the isolated ground state found in Ref. [44] and the bimodal success probability histograms for random Ising instances found in Ref. [43]. Shortly thereafter this classical model was rejected in Ref. [47] by demon- strating that the classical SD model correlates poorly with the success probabilities measured for random Ising instances, while SQA correlates very well. In response, a new hybrid model where the spin dynamics are governed by Monte Carlo updates that correlates at least as well with the DW1 success probabilities for random Ising instances as SQA was very recently proposed by Shin, Smolin, Smith, and Vazirani (SSSV) [48] 1 . In this model the qubits are replaced by O(2) rotors with classical Monte Carlo updates along the annealing schedule of the D-Wave device. This can also be interpreted as a model of qubits without any entanglement, updated at each time step to the classical thermal equilibrium state determined by the instantaneous Hamiltonian. Moreover, the hybrid model correlates almost perfectly with SQA, suggesting that the SSSV model is a classical analog of a mean-field approximation to SQA [95], and that this approximation is very accurate for the set of problems solved by the DW1 in Ref. [43]. 2 1 In literature, SSSV algorithm is also often called the spin vector Monte Carlo (SVMC) [51] algorithm. 2 We note that phases of quantum models often have an accurate mean-field description, and that SQA is a classical simulation method obtained by mapping a quantum spin model to a classical one after the addition of an extra spatial dimension of extent β (the inverse temperature). 14 At this point it is important to note that recent work already established that eight-qubit entangled ground states are formed during the course of the annealing evolution in experiments using a D-Wave Two device [41]. This demonstration of entanglement was done outside of the “black box” paradigm we are considering here 3 and is, of course, a crucial demonstration of non-classicality. However, it does not necessarily imply that non-classical effects play a role in deciding the final outcome of a computation performed by the D-Wave devices. It is the latter that we are concerned with in this chapter, and it is the fundamental reason we are interested in the “black-box” paradigm. Using a physically motivated noise model, we show that none of the three classical models introduced to date matches new data we obtained from the D-Wave Two (DW2) device using a generalized “quantum signature” Hamiltonian on up to 20 qubits. At the same time the ME matches the new data well. Thus, our results confirm the earlier rejection of the SA and SD models—this is of independent interest since the “quantum signature” provided in Ref. [44] for the DW1 had remained in question in light of the SD-based critique of Ref. [46]—and also serve to reject the new SSSV model [48] for system sizes of up to 20 qubits. Of course, this still leaves open the possibility that a classical model can be found that will match the experimental data while satisfying the “good theory” criteria. Since, as mentioned above, our quantum signature-based test does not provide a no-go result for classical models, the distinction we demonstrate between a natural quantum Moreover, SQA scales polynomially in problem size, which is the reason that Ref. [43] was able to use SQA to predict the experimental outcomes of 108 qubit problem instances. 3 The experiment had access to the internal workings of the D-Wave device, in particular the ability to perform qubit tunneling spectroscopy and thus obtain the instantaneous energy spectrum. 15 model and fine-tuned classical models is perhaps the best that can be hoped for within our approach. The “quantum signature” Hamiltonian we consider here is defined in Sec. 2.2 and is a direct generalization of the Hamiltonian introduced in Ref. [44]. We introduce a controllable overall energy scale, or an effective (inverse temperature) “noise control knob”. Decreasing the energy scale amounts to increasing thermal excitations, enabling us to drive the D-Wave processor between qualitatively distinct regimes. At the largest energy scale available, the annealing process appears to be dominated by coherent quantum effects, and thermal fluctuations are negligible. As the energy scale is decreased, thermal excitations become more relevant, and for a sufficiently small energy scale, the system behaves more like a classical annealer based on incoherent Ising spins. Nevertheless, at all energy scales the system is very well described by the ME. This suggests that an open system quantum dynamical description of the D-Wave device is well justified, even in the presence of relevant thermal excitations and a small single-qubit decoherence time to annealing time ratio, at least for the class of Hamiltonians studied here. The structure of this chapter is as follows. We provide theoretical background on the quantum signature Hamiltonian—our workhorse in this study—in Sec. 2.2. We describe a noise model for the D-Wave device in Sec. 2.3, which includes both stochastic and systematic components, the latter being dominated by spurious qubit cross-talk. We analyze the effect of tuning the thermal noise via the magnitude of the final Hamiltonian in Sec. 2.4. Our first set of main results is presented in Sec. 2.5, where we demonstrate a clear difference between the behavior of the classical model and that of the quantum model in the absence of cross-talk. We then include the cross-talk and establish in Sec. 2.6 the input-output characteristics of the D-Wave device that allow us to critically assess the classical models, and 16 confirm the agreement with an open quantum system description via the adiabatic quantum ME. We achieve a close match between the ME and the experimental data, while rejecting the SSSV model, the strongest of the classical models. We demonstrate that the ME predicts that an entangled ground state is formed during the course of the annealing evolution in Sec. 2.7. We provide a discussion and conclusions in Sec. 2.8. Sections 2.9 to 2.18 provide further technical details and experimental and numerical results. 2.2 Theoretical background Quantum and classical annealing are powerful techniques for solving hard optimiza- tion problems, whether they are implemented as numerical algorithms or on analog (physical) devices. The general simulation strategy is to implement an “escape” rule from local minima of an energy or penalty function to reach the global minimum, representing a solution of the optimization problem [24, 25, 96]. The physical strategy is to use a natural system or build a device whose physical ground state represents the sought-after solution [13, 63, 97, 98]. In both cases, by progressively reducing the escape probability, the system is allowed to explore its configuration space and eventually “freeze” in the global minimum with some probability. 2.2.1 Quantum annealing Hamiltonian The QA Hamiltonian is given by H(t) =A(t)H X +B(t)H I , (2.1) 17 where H X =− P i σ x i (with σ x i being the Pauli matrix acting on qubit i) is the transverse field,H I is the classical Ising Hamiltonian, H I =− X i∈V h i σ z i − X (i,j)∈E J ij σ z i σ z j , (2.2) and the time-dependent functions A(t) and B(t) control the annealing schedule. Typically A(t f ) =B(0) = 0, where t f is the total annealing time, and A(t) [B(t)] decreases (increases) monotonically. The local fields{h i } and couplings{J ij } are fixed. The qubits occupy the verticesV of a graph G ={V,E} with edge setE. A spin configuration is one of the 2 N elements of a set of±1 eigenvalues of all the Pauli matrices{σ z i } N i=1 , which we denote without risk of confusion by ~ σ z = (σ z 1 ,...,σ z N ). The goal is to find the minimal energy spin configuration ofH I , i.e., argmin ~ σ zH I . In QA, the non-commuting fieldH X [24–26, 99] allows quantum tunneling out of local minima. This “escape probability” is reduced by turning off this non-commuting field adiabatically, i.e., the time-scale of the variation of the A(t) and B(t) functions must be slow compared to the inverse of the minimal energy gap of H(t). In a physical device implementation of QA there is always a finite temperature effect, and hence one should consider both tunneling and thermal barrier crossing [64, 65, 100, 101]. Such physical QA devices, operating at∼ 20 mK using superconducting flux technology, have been built by D-Wave [29, 30, 32]. The qubits occupy the vertices of the “Chimera” graph (shown in the Section 2.9). Excluding the coupling to the thermal bath, the Hamiltonian driving the device is well-described by Eq. (2.1), with the functions A(t) and B(t) depicted in Fig. 2.1. 18 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 35 t/t f Annealing schedule (GHz) A(t) B(t) 0.2834B(t) 0.1099B(t) Figure 2.1: DW2 annealing schedules A(t) and B(t) along with the operating temperature of T = 17mK (black dashed horizontal line). The large A(0)/(k B T) value ensures that the initial state is the ground state of the transverse field Hamiltonian. The large B(t f )/(k B T) value ensures that thermal excitations are suppressed and the final state reached is stable. Also shown are the attenuated αB (t) curves for (a) the value of α at which the intersection between A(t) and αB (t) coincides with the operating temperature (blue dot-dashed curve), and (b) the largest α such that αB (t) remains below the temperature line for the entire evolution (blue dotted curve). 2.2.2 The quantum signature Hamiltonian Reference[44]introducedaneight-qubit“quantumsignatureHamiltonian,” schemat- ically depicted in Fig. 2.2, designed to distinguish between SA and QA. The eight- spin problem comprises four spins connected in a ring, which we refer to as core spins, and four additional spins connected to each core spin, which we refer to as outer spins. One special property of this Hamiltonian is that it has a 17-fold degenerate ground state. Of these, 16 states form a subspace of spin configurations connected via single flips of the outer spins, hence we refer to them as theclustered (C) ground states, or just the “cluster-states,” or “cluster.” There is one additional 19 + + + + − − − − Figure 2.2: The eight-spin Ising Hamiltonian. The inner “core” spins (green circles) have local fieldsh i = +1 [using the convention in Eq. (2.2)] while the outer spins (red circles) have h i =− 1. All couplings are ferromagnetic: J ij = 1 (black lines). + + + + + + − − − − − − + + + + + + + + − − − − − − − − + + + + + + + + + + − − − − − − − − − − Figure 2.3: Schematic representation of the 12, 16, and 20 spin Hamiltonians used in our tests. Extensions to larger N follow the same pattern, with N/2 qubits in the inner ring and N/2 qubits in the outer ring. Notation conventions are as in Fig. 2.2. state, which we call the isolated (I) ground state, connected to the cluster-states via four core spin flips: C :{|0000 0000i,|0001 0000i,...,|1111 0000i} , (2.3a) I :{|1111 |{z} outer 1111 |{z} core i} , (2.3b) where|0i and|1i are, respectively, the +1 and− 1 eigenstates ofσ z . This structure of the ground state manifold is easily verified by inspection of the Hamiltonian of Fig. 2.2. 20 The clustered ground states arise from the frustration of the outer spins, due to the competing effects of the ferromagnetic coupling and local fields. This frustration arises only when the core spins have eigenvalue +1, which is why there is only a single additional (isolated) ground state where all spins have eigenvalue− 1 (for a more detailed discussion of the energy landscape of this eight-spin Hamiltonian, see Ref. [44].) Here we also consider quantum signature Hamiltonians with larger numbers of spins N, as depicted in Fig. 2.3. These Hamiltonians share the same qualitative features, but the degeneracy of the ground state grows exponentially with N: Degeneracy of C : N C = 2 N/2 Degeneracy of I : N I = 1 . (2.4) At the end of any evolution, be it quantum or classical, at t = t f , there is a certain probability of finding each ground state. Let us denote the observed population of the isolated state att =t f byP I , and the average observed population in the cluster at t =t f by P C = 1 N C N C X c=1 P c , (2.5) where N C = 16 for the eight-spin case, and where P c is the population of cluster- state number c. As shown in Ref. [44] for the eight-spin case, SA and QA can be distinguished because they give opposite predictions for the population ratio P I /P C . For SA, the isolated state population is enhanced relative to any cluster state’s population, i.e.,P I /P C ≥ 1, whereas for QA, the isolated state population is suppressed relative to any cluster state’s population, i.eP I /P C ≈ 0. This conclusion also holds for the N > 8 cases, as we show in detail in Section 2.10. These two starkly different predictions for QA and SA allowed Ref. [44] to rule out SA as an explanation of the experimental results obtained from the DW1 using the eight-spin Hamiltonian. In addition, Ref. [44] demonstrated that the ME [45] 21 correctly predicts the suppression of the isolated state, including the dependence on the annealing time t f , thus providing evidence that the DW1 results correlate well with the predictions of open system quantum evolution. However, as we discuss in detail and demonstrate with data from the DW2, the suppression of the isolated state can change as a function of the thermal noise, and suppression can turn into enhancement at sufficiently high noise levels. Yet, this does not imply that the system admits a classical description. One limitation of the analysis so far is that the quantityP C as defined in Eq.(2.5) is an average over the cluster state populations, so it does not account for variations in individual cluster state populations. In the absence of any non-idealities in the quantum Hamiltonian of Eq. (2.1), SA and QA (under a closed system evolution) predict that all the cluster states end up with equal populations. Therefore, the cluster state populations alone cannot be used to distinguish between SA and QA, yet, as we will show, not all classical models preserve the cluster state symmetry, making it a useful feature to take into account. However, when we make|J| and|h| unequal or introduce additional spurious couplings between qubits, the symmetry between the cluster states is broken in all models, and we must be careful to model such noise sources accurately. 2.3 D-Wave Control Noise Sources So far we have not considered the effect of control noise on the local fields and couplings in H I , which are important effects on the D-Wave processor. For each annealing run of a given problem Hamiltonian, the values of{h i , J ij } are set with a Gaussian distribution centered on the intended value, and with standard error of about 5% [102]. In our experiments we used averaging techniques described in 22 Section 2.11 to minimize the effects of this noise. However, it is important to test its effects on the quantum and classical models as well. Besides this random noise source on the{h i , J ij }, there is a spurious cross-talk between qubits. The model accounting for this cross-talk is h i 7→ h i − χ X k6=i J ik h k (2.6a) J ij 7→ J ij +χ X k6=i,j J ik J jk , (2.6b) whereχ is the qubit background susceptibility multiplied by the mutual inductance [102]. To understand the correction to J ij consider first the case wherei and j are nearest-neighbor qubits (i.e., are perpendicular, intersecting superconducting loops in the same unit cell of the Chimera hardware graph), and k is an index of qubits that are next nearest neighbors of i (i.e., are parallel, non-intersecting loops in the same unit cell of the Chimera hardware graph) [32]. Then both J ij and J jk are existing “legitimate” couplings in the unit cell, but χJ jk is a spurious next nearest neighbor coupling. Next consider the case where qubits i and j are next-nearest neighbors (i.e., parallel loops), so that nominally J ij = 0. Now the correction is due to qubits k that are nearest neighbors of (i.e., perpendicular to) both i and j. The sign is determined by the following rule of thumb: a ferromagnetic chain or ring is strengthened by the background χ , i.e., the intermediate qubit is mediating an effective ferromagnetic interaction between next-nearest neighbors. While in reality χ is time and distance dependent, for simplicity we model it as constant and separately fitχ for the ME and SSSV to the DW2 cluster-state populations. The sum in Eq. (2.6) extends over the unit cell of the Chimera graph, i.e., in the sum over k we include all the couplings between physically parallel 23 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 t/tf Energy gap (GHz) α = 0.11 α = 0.28 α = 1 Temperature Figure 2.4: Numerically calculated evolution of the gap between the instantaneous ground state and the 17-th excited state (which becomes the first excited state at t =t f ), for the eight-spin Hamiltonian in Eq. (2.1), following the annealing schedule of the DW2 device (Fig. 2.1). The gap value is shown for some interesting values of α (see Fig. 2.1). The kinks are due to energy level crossings, as explained in Section 2.12. A reduction in α results in a reduction of the size of the minimal gap and delays its appearance. qubits. We set J ik and J jk equal to the unperturbed J ij . To account in addition for the effect of Gaussian noise on h and J, we apply the cross-talk perturbation terms after the Gaussian perturbation. 2.4 Introducing an energy scale The QA and SA protocols represent two opposite extremes: in the former quantum fluctuations dominate while in the latter thermal fluctuations dominate. Can we interpolate between these regimes on a physical annealer? Since we are unable to directly change the temperature on the DW2 device 4 , our strategy to answer this 4 On-chip variability of the SQUID critical currents leads to uncertainty in both the qubit biases h and the qubit coupling strengths J. These uncertainties are calibrated out each time 24 question is to indirectly modify the relative strength of thermal effects during the annealing process. As we now discuss, this can be done by modifying the overall energy scale. A straightforward way to tune the thermal noise indirectly is to change the overall energy scale of the problem Hamiltonian H I by rescaling the local fields and couplings by an overall dimensionless factor denoted by α : (J ij ,h i )7→ α (J ij ,h i ) (2.7) In the notation above, α = 1 corresponds to implementing the largest allowed value of the physical couplings on the device (assuming|h max | =|J max | = 1 in dimensionless units). The scale of the transverse fieldH X is not changed. Due to the form of the cross-talk corrections this scales the cross-talk corrections by α 2 . For SA, reducing α is tantamount to increasing the temperature. Since this does not change the energy spectrum, the earlier arguments for SA remain in effect, and we expect to have P I ≥ P C for all α > 0 values. This is confirmed in our numerical simulations as shown in Section 2.13. For QA, decreasing α from the value 1 has two main effects, as can be clearly seen in Fig. 2.4. First, the minimal gap between the instantaneous ground state and the 17th excited state is reduced (the lowest 17 states become degenerate at the end of the evolution as explained previously). Since thermal excitations are suppressed by a factor of e − β Δ (with β = 1/k B T the inverse temperature and Δ the energy gap), a reduction in the gap will increase the thermal excitation rate [45]. One might expect that, by sufficiently reducing α , it is possible to make the gap small the chip is thermally cycled. The conditions required for optimum calibration are however temperature-dependent. We therefore conduct our experiments at a fixed operating temperature of 17 mK. 25 enough that the competition between non-adiabaticity and thermalization becomes important. However, simulations we have performed for the closed system case with α ∈ [0.01, 1] show that P I /P C is essentially 0 over the entire range. Therefore we do not expect non-adiabatic transitions to play a role over the entire range of α ’s we studied. Second, reducing α delays the appearance of the minimal gap. This effectively prolongs the time over which thermal excitations can occur, thus also increasing the overall loss of the ground state population. Hence we see that by changing α we expect to move from a regime where thermal fluctuations are negligible (α ' 1), to a regime where they are actually dominant (α . 0.1), when the minimal gap is comparable to or even smaller than the physical temperature of the device. In agreement with these considerations, the effect that the position and size of the minimal gap have on the probability of being in a given energy eigenstate is shown in Fig. 2.5. This figure shows the total population of the 17 lowest energy eigenstates (i.e., the subspace that eventually becomes the ground state manifold), computed using the ME. As is clear from Fig. 2.5, as α decreases, the increasingly delayed and smaller minimum gap causes this subspace to lose more population due to the increased rate and duration of thermal excitations. This behavior is interrupted by kinks caused by level crossing, whose position is a function of α , with the kinks occurring later for smaller α (see Section 2.12). 2.5 Numerical simulations without cross-talk We have performed extensive numerical simulations using SA (described in Sec- tion 2.13.1), SD (Section 2.13.2), the ME (Section 2.13.3), and the SSSV model (Appendix A.4). Since experimental evidence for rejection of SA and the SD models 26 0 0.2 0.4 0.6 0.8 1 0.7 0.75 0.8 0.85 0.9 0.95 1 t/tf Population in 17 lowest eigenstates α = 0.11 α = 0.28 α = 1 Figure 2.5: ME simulation for the time-dependence of the probability of being in the lowest 17 energy eigenstates, for different values of α . Simulation parameters are t f = 20μs (the minimal annealing time of the DW2) and κ = 1.27× 10 − 4 , where κ is an effective, dimensionless system-bath coupling strength defined in Section 2.13.3. The chosen value ofκ allows us to reliably probe the smallα regime. has already been presented in Refs. [43, 44], while the SSSV model presents a particularly interesting challenge since it nicely reproduces the success probability correlations that were used in Ref. [43] to reject both SA and SD, we focus on the SSSV model here, and present our discussion of SA and SD in Section 2.14. We note that our ME simulations have only one adjustable parameter, κ , an effective, dimensionless system-bath coupling strength (defined in Section 2.13.3). The SSSV model has two: the temperature and the number of Monte Carlo update steps. In addition, as we discuss in detail below, it requires the addition of stochastic noisetothelocalfieldsandcouplingsinordertomatchtheMEandtheexperimental results, which introduces a third free parameter in the form of the noise standard deviation. When we discuss the effect of cross-talk in the next section, both the SSSV model and the ME will require the susceptibility χ as an additional free parameter. 27 , 0 0.2 0.4 0.6 0.8 1 p =( M p G S ) 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 α p/(MpGS) (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 α p/(MpGS) (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 α p/(MpGS) (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 α p/(MpGS) (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) (a) ME, no noise , 0 0.2 0.4 0.6 0.8 1 p =( M p G S ) 0 0.02 0.04 0.06 0.08 0.1 0.12 (b) SSSV, no noise , 0 0.2 0.4 0.6 0.8 1 p =( M p G S ) 0 0.02 0.04 0.06 0.08 0.1 0.12 (c) SSSV, noisy Figure 2.6: Distribution of the ground states for N = 8 for (a) ME with no noise on{h i , J ij }, (b) SSSV with no noise on{h i , J ij }, and (c) SSSV with{h i , J ij } noise using σ = 0.085. The cluster states are labeled by their Hamming distance H from the isolated state, and by their multiplicity M for a given value of H. The vertical axis is the final probabilityp of a given (H,M) set, divided by its multiplicity and the total ground state probability. The data symbols (◦ , etc.) are the mean values of the bootstrapped [103] distributions, and the error bars are two standard deviations below and above the mean representing the 95% confidence interval. Note that the SSSV model prefers the|1111 0000i cluster state, whereas the ME gives a uniform distribution over all cluster states. SSSV parameters are T = 10.56mK and 1× 10 5 Monte Carlo step updates per spin (“sweeps”). The same parameters are used in all subsequent SSSV figures. These results do not include the cross-talk correction. We now present our first set of numerical findings, where we do not include the cross-talk correction discussed in Sec. 2.3, but focus instead on the role of the stochastic noise on the local fields and couplings. The results in this section will help to clarify the roles played by these various sources of imperfection in the experiment. 2.5.1 Ratio of the populations of the isolated state to the cluster states Figure 2.6 shows the distribution of cluster states and isolated state for the entire range of α values. First, we observe that in the absence of noise on{h i , J ij } the behavior of the isolated state is strikingly different between SSSV and the ME. 28 Whereas SSSV shows a monotonic increase with decreasing α , the ME result for the isolated state is non-monotonic in α ; see Fig. 2.6(a). Initially, as α is decreased from its largest value of 1, the ratio of isolated to cluster state population increases and eventually becomes larger than 1; i.e., the population of the isolated state becomes enhanced rather than suppressed. For sufficiently smallα , the ME isolated state population turns around and decreases towards 1. Thus, the SSSV model captures the suppression of the isolated state at high α but does not capture the ground state population inversion at low α (as we discuss below, this conclusion changes after noise on h and J is included). On the other hand, we note that SA correctly predicts an enhanced isolated state at low α but does not predict the suppression at high α (see Section 2.14). This observation led us to consider classical models that interpolate between SSSV at high α and SA at low α . These models exploit the fact that in SA the qubits are replaced by fully incoherent, classical Ising spins, while in SSSV each qubit is replaced by a “coherent” O(2) rotor. 5 Therefore a natural way to interpolate between SSSV and SA is to “decohere” the O(2) rotors over an α -dependent timescale τ α , and two natural decoherence models we considered are discussed in Section 2.15. However, these models do not reproduce the behavior of the ME. In order to understand what contributes to the increase in the isolated state population asα is lowered, it is useful to study the time evolution of the population in the lowest 17 energy eigenstates according to the ME. An example is shown in Fig. 2.7, for α = 0.1, i.e., close to the peak of the isolated state population. This figure clearly shows how the relative ratio of the isolated state population to the 5 Recall that SU(2) is (locally) isomorphic to SO(3), so a qubit can always be mapped to an SO(3) rotor. [Strictly, SO(3) is isomorphic to SU(2)/Z 2 .] The restriction to O(2) rotors is heuristically justified by SSSV via the observation that the QA Hamiltonian contains only x and z components. 29 t = t f 0 0.2 0.4 0.6 0.8 1 P r ob ab i l i t y 0 0.05 0.1 0.15 0.2 0.25 P 6 1 5 ( P 5 i = 1 P i ) 1 1 1 P 1 7 i = 7 P i 1 1 6 ( P 1 7 i = 1 P i ! P 6 ) t = t f 0 0.2 0.4 0.6 0.8 1 " ( P I =P C ) 0 0.5 1 Figure 2.7: Time evolution (according to the ME) of energy eigenstate populations for α = 0.1 and κ = 8.9× 10 − 4 (this relatively large value was chosen here since it results in increased thermal excitation/relaxation). P i denotes the population of the ith eigenstate, with i = 1 being the instantaneous ground state. The energy eigenstate that eventually becomes the isolated ground state is i = 6 (dashed red line). This state acquires more population at the end of the evolution than the other 16 eigenstates that eventually become the cluster (solid purple line). (Inset) The difference of the population ratio between the open system and the closed system evolution, Δ(P I /P C ) = (P I /P C ) Open − (P I /P C ) Closed . The deviation from closed system dynamics starts at t/t f ≈ 0.4, when the i = 6 eigenstate becomes thermally populated at the expense of the lowest five eigenstates. mean cluster state populationP I /P C becomes> 1. The sixth energy eigenstate (red line) evolves to become the isolated ground state, while the other 16 eigenstates evolve tobecome thecluster(purple line). During thetime evolution, the population in the sixth eigenstate grows slightly larger than that of the cluster (red curve ends up above the purple one), which explains why P I /P C > 1. In more detail, we observe that (around t/t f = 0.4) the sixth eigenstate acquires population (via thermal excitations) from the lowest five eigenstates (blue line). Somewhat later (around t/t f = 0.6) the sixth eigenstate loses some population due to thermal excitations, which is picked up in part by the highest 11 eigenstates (green). Finally, thermal relaxation returns some population to the 17 eigenstates, but the sixth eigenstate gains more population than the other 16 eigenstates since it is connected to a larger number of excited states. During this relaxation phase, the system 30 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 12 3456 7 Probability divided by multiplicity (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) (H,M) Figure 2.8: Statistical box plot of the probability divided by the multiplicity of being in a given state with Hamming distance H and multiplicity M. Shown are the ME isolated state and cluster states for N = 8, with 512 noise realizations applied to the h’s and J’s [with distributionN (0, 0.06)] at α = 1. The isolated state (H = 0,M = 1) is suppressed while the cluster states are, on average, equally populated. The red bar is the median, the blue box corresponds to the lower and upper quartiles, respectively, the segment contains most of the samples, and the +’s are outliers [104]. The horizontal axis label indicates the Hamming distance from the isolated state and the multiplicity of the cluster-states at each value of H. States that are equivalent up to 90 ◦ rotations are grouped together. For example, there are four rotationally equivalent cluster-states that have two adjacent outer qubits pointing down, while the other two are pointing up. Only the H = 6 case splits into two rotationally inequivalent sets. 31 behaves like classical SA. The inset shows that deviations from the closed system behavior occur around t/t f = 0.4, i.e., when the population of the sixth eigenstate first starts to grow (along with the highest 11 eigenstates) due to excitations from the lowest five eigenstates. 2.5.2 Cluster state populations The other important feature to note from Fig. 2.6 is that the population degeneracy of the cluster states is broken in the SSSV model, giving rise to a staircase pattern organized according to Hamming distance (HD) from the isolated state [Fig. 2.6(b)], while the ME exhibits a uniform distribution over the cluster states [Fig. 2.6(a)]. Except for very small α , the SSSV pattern remains fixed asα is decreased. The preference for the|1111 0000i (HD= 4) state and the insensitivity of this feature to α in the SSSV model can be understood from the following simple argument. Consider first the closed system (no thermal noise) case and note that for t> 0.6t f , the transverse field is almost completely turned off. Therefore the cluster states’ outer spins are free to rotate with no energy cost when the core spins are pinned at M z c = cosθ c = 1, leading to M z o = cosθ 0 = 0 (the c and o subscripts stand for “core” and “outer”, respectively). However, in the open system case the core spins are not fixed atM z c = 1 due to thermal noise, and M z o =− 1 becomes energetically favorable for the outer spins. To see why, consider the case of a single pair of core and outer spins. In this case, the Ising potential is simply V =− α (h o M z o +h c M z c +J oc M z o M z c ) =α (M z o − M z c − M z o M z c ). WhenM z c = 1, the dependence onM z o vanishes so the outer spin is free to rotate, however whenM z c 6= 1 (as happens when thermal noise is present), this Ising potential is minimized when M z o =− 1. This explains why the SSSV model prefers the|1111 0000i cluster state for all α . 32 Note that this argument depends on|h| =|J|; i.e., it will not necessarily apply when there is noise on h and J. As an example, if we add Gaussian noise Δh i , ΔJ ij ∼N (0, 0.085) so that h i 7→ h i + Δh i and J ij 7→ J ij + ΔJ ij in Eq. (A.22), as first shown in Ref. [49], the resulting “noisy SSSV” model is able to reproduce the non-monotonic behavior of the isolated state observed for the ME, as shown in Fig. 2.6(c). However it maintains its preference and ordering of cluster state populations. 2.6 Experimental results and numerical simula- tions including cross-talk 2.6.1 The distribution of cluster states Having developed an understanding of the role of noise on the local fields and couplings on the ground state distribution, we now present experimental results for the DW2 in Fig 2.9(a), which we believe to include cross-talk. We immediately observe a strong discrepancy between the DW2 and both the ME and SSSV results shown in Fig. 2.6. This discrepancy implies that both models require an adjustment. We next introduce the cross-talk correction. (An alternative model which gives rise to the breaking of the cluster state symmetry by detuning|h| relative to|J| is discussed in Section 2.16; this gives a less satisfactory fit to the experimental data.) We fit the cross-talk magnitudeχ at α = 1 in order to force both the noisy SSSV model and the ME to reproduce the correct ordering of the cluster states at this value of α . However, we refrain from excessively fine-tuning the models for additional values ofα . That is, we adhere to the idea that a good theoretical model should have predictive power after its free parameters are fit to the data once. 33 , 0 0.2 0.4 0.6 0.8 1 p =( M p G S ) 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 α p/(MpGS) (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 α p/(MpGS) (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 α p/(MpGS) (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 α p/(MpGS) (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) (a) DW2, N =8 , 0 0.2 0.4 0.6 0.8 1 p =( M p G S ) 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.05 0.06 0.07 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 (b) DW2 & ME, χ = 0.015, N =8 , 0 0.2 0.4 0.6 0.8 1 p =( M p G S ) 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 (c) DW2 & SSSV,χ =0.035, σ =0.085, N =8 , 0 0.2 0.4 0.6 0.8 1 p = ( M p G S ) 0 0.02 0.04 0.06 0.08 0.1 0.12 , 0 0.2 0.4 0.6 0.8 1 p =( M p G S ) 0 0.02 0.04 0.06 0.08 0.1 0.12 D W 2 , ( 4 ; 1 ) D W 2 , ( 8 ; 1 ) S S S V , ( 4 ; 1 ) S S S V , ( 8 ; 1 ) M E , ( 4 ; 1 ) M E , ( 8 ; 1 ) (d) DW2, ME (χ = 0.015), SSSV, (χ = 0.035, σ = 0.085), N =8 , 0 0.2 0.4 0.6 0.8 1 p = ( M p G S ) #10 -3 0 0.5 1 1.5 2 2.5 3 3.5 D W 2 , ( 1 0 ; 1 ) D W 2 , ( 2 0 ; 1 ) S S S V , ( 1 0 ; 1 ) S S S V , ( 2 0 ; 1 ) (e) DW2 & SSSV (χ =0.035, σ =0.085), N =20 , 0 0.2 0.4 0.6 0.8 1 p = ( M p G S ) 0 0.02 0.04 0.06 0.08 0.1 0.12 D W 2 ( 0 , 1 ) S S S V ( 0 , 1 ) M E ( 0 , 1 ) N o i s y M E ( 0 , 1 ) (f) DW2, SSSV, (χ =0.035, σ =0.085), ME (χ =0.015), and noisy ME (χ = 0.015, σ =0.025), N =8 Figure 2.9: (a) Ground state populations for DW2. Legend: (H,M), corresponding to Hamming distance from the isolated state and multiplicity respectively. Error bars represent the 95% confidence interval. (b) Cluster state populations for the ME. Solid lines correspond to the results with no noise on the{h i ,J ij }’s, while the data points include Gaussian noise with mean 0 and standard deviation σ = 0.025 for 100 noise realizations. The error bars represent the 95% confidence interval. The DW2 data from (a) is also plotted as the shaded region representing a 95% confidence interval with the dashed lines corresponding to the mean. The inset shows the behavior for the noiseless ME for small α . (c) Cluster state population for SSSV for N = 8 with the DW2 data plotted as in (b). In contrast to Fig. 2.6, both the ME and the noisy SSSV model include the cross-talk correction, Eq. (2.6), with χ chosen to optimize the fit for the cluster state populations atα = 1. (d) Only the cluster states with Hamming distance 4 and 8 from the isolated state are shown for DW2, the ME, and SSSV from panels (b) and (c) in order to highlight their differences. Panel (e) displays the same for N = 20 (excluding the ME, which is too costly to simulate at this scale). (f) The isolated state populations for DW2, SSSV, ME, and noisy ME, which highlights the qualitative agreement between the models and DW2. Experimental data were collected using the in-cell embeddings strategy described in Section 2.11. The embedding and gauge-averaging strategies are also discussed in Section 2.11. The color coding of states is consistent across all panels. 34 Noise on the local fields and couplings has no such effect on the ME. Indeed, we have checked that introducing noise on the couplings and the local fields does not, on average, break the population degeneracy of the ME cluster states, while it does break the degeneracy for any given noise realization. This can be seen in Fig. 2.8. We have also checked that introducing different system-bath couplings for each qubit (by adding Gaussian noise to each coupling) does not break the population degeneracy of the cluster states. The ME result [Fig. 2.9(b)] is now a significantly closer match to the DW2 cluster populations than before [Fig. 2.6(a)], over the entire range of α values. The ME captures quantitatively the cluster state populations, while the noisy SSSV model [Fig. 2.9(c)], with χ and the noise variance optimized to match the DW2 results at α = 1, does not capture the cluster state populations correctly. To highlight this difference, the same data are plotted in Fig. 2.9(d) for only two cluster states. The same conclusions apply for N = 20 spins, as seen in Fig. 2.9(e), where we show only the two extremal of the 2 10 cluster ground states. The ME’s main discrepancy is in not capturing the full isolated state population, especially the strength of the peak at small α , and it can only capture qualitatively the behavior of the isolated state as shown in Fig. 2.9(f). As illustrated by the SSSV results in Figs. 2.6(b) and 2.6(c), the inclusion of noise on the local fields and couplings can have a dramatic effect on the small α behavior, while keeping the large α behavior mostly untouched. Indeed, we have shown in Fig. 2.8 that noise of a certain magnitude on the local fields and couplings does not significantly alter the cluster state distribution at α = 1. To study this effect over the entire range of α would require performing simulations for a large number of noise samples, which is unfeasible given the high computational cost of running the ME. However, even for a moderate number of small noise samples, we observe [Fig. 2.9(f)] an increase 35 in the strength of the isolated state peak. To increase the population at larger α , increasing the system-bath coupling would increase the strength of thermal excitations, which would allow for the isolated state to be further populated. We believe that an optimization over these parameters, albeit at a huge computational cost, could significantly improve the quantitative agreement between the ME and DW2. However, our focus here has been to illustrate that the ME captures the behavior of the DW2 data remarkably well with no significant parameter fitting. 2.6.2 The distribution of first excited states While the results presented in the previous subsection provide a clear quantitative discrepancy between the noisy SSSV model and the DW2 results, and demonstrate that the agreement with the ME is quantitatively better, it is important to provide a clearcut example of a qualitative discrepancy. To address this we now go beyond the ground subspace and consider an eight-dimensional subspace of the subspace of first excited states. We arrange these according to permutations of the core or outer qubits, i.e., we group the states as|1111 Π(0001)i and|Π(1110) 1111i, where Π denotes a permutation. As shown in Fig. 2.10(a), the DW2 prefers the set|Π(1110) 1111i. However, the noisy SSSV model prefers the set|1111 Π(0001)i, as seen in Fig. 2.10(b). This discrepancy becomes observable for α . 0.2, where thermal excitations start to significantly populate the excited states. This also helps explain why α ≈ 0.2 played a threshold role in our ground state analysis. This conclusion persists for N = 20, as shown in Figs. 2.10(d) and 2.10(e). In Fig. 2.10(c), we show similar results for the ME for N = 8. The results qualitatively match the DW2 ordering forα & 0.15. The error bars are large since it is computationally prohibitive to run a large number of noise instances, which also restricted us to a relatively low noise level (σ = 0.025). It is difficult to conclude 36 , 0 0.2 0.4 0.6 0.8 1 p 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 1 1 1 1 & ( 0 0 0 1 ) & ( 1 1 1 0 ) 1 1 1 1 (a) DW2, N =8 , 0 0.2 0.4 0.6 0.8 1 p 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 1 1 1 1 & ( 0 0 0 1 ) & ( 1 1 1 0 ) 1 1 1 1 (b) SSSV, χ =0.035, σ =0.085, N =8 , 0 0.2 0.4 0.6 0.8 1 p 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 1 1 1 1 & ( 0 0 0 1 ) & ( 1 1 1 0 ) 1 1 1 1 (c) ME, χ =0.015, σ =0.025, N =8 , 0 0.2 0.4 0.6 0.8 1 p =( M p G S ) #10 -3 0 1 2 3 4 5 1 1 1 1 1 1 1 1 1 1 & ( 0 0 0 0 0 0 0 0 0 1 ) & ( 1 1 1 1 1 1 1 1 1 0 ) 1 1 1 1 1 1 1 1 1 1 (d) DW2, N =20 , 0 0.2 0.4 0.6 0.8 1 p =( M p G S ) #10 -3 0 1 2 3 4 5 1 1 1 1 1 1 1 1 1 1 & ( 0 0 0 0 0 0 0 0 0 1 ) & ( 1 1 1 1 1 1 1 1 1 0 ) 1 1 1 1 1 1 1 1 1 1 (e) SSSV, χ =0.035, σ =0.085, N =20 Figure 2.10: Subset of the first excited state populations for (a) DW2 forN = 8; (b) SSSV forN = 8; and (c) ME for N = 8. In (b) and (c), the simulations include qubit cross-talk correction with χ chosen as in Fig. 2.9 to optimize the fit for the cluster state populations at α = 1. Panels (d) and (e) are for N = 20. The Π symbol denotes all permutations. The SSSV model does not reproduce the correct ordering. The error bars represent the 95% confidence interval. 37 much for α . 0.15 because the high computational cost forces us to truncate the energy spectrum in our ME simulations, which predominantly degrades our ability to compute the excited state populations at low α . To summarize, we showed in the previous section that with the inclusion of the cross-talk terms in the Hamiltonian the ME captures the convergence of the cluster state populations for small α as well, while the noisy SSSV model predictions do not improve relative to the case without the cross-talk correction. The discrepancy between the noisy SSSV model and the experimental data is amplified when we consider the excited states, for which the former predicts the opposite population ordering from the one observed, as shown in Fig. 2.10. We note that varying α is not the only way in which a control parameter for thermal excitations can be introduced. In Section 2.17 we discuss the similar effect of increasing the total annealing time or the number of spins, along with experimental results. 2.7 Ground state entanglement during the course of the annealing evolution Having established that the ME is, at this point, the only model consistent with the DW2 data, we are naturally led to ask whether the ME displays other quantifiable measures of quantum mechanical behavior. We thus use the ME to compute an entanglement measure for the time-evolved state. Ground state entanglement was already demonstrated experimentally in Ref. [41] for a different Hamiltonian; here we are concerned with the time-dependent entanglement as a function of α , and are relying on the good qualitative match between the ME and our experimental 38 0 0.5 1 0 0.5 1 −0.05 0 0.05 0.1 0.15 t/tf α Negativity (a) Closed system 0 0.5 1 0 0.5 1 −0.05 0 0.05 0.1 0.15 t/tf α Negativity (b) Open system Figure 2.11: Time-dependence of the negativity [Eq. (2.8)] for (a) a closed system evolution and (b) an open system evolution of N = 8 qubits (modeled via the ME with κ = 1.27× 10 − 4 ), as a function of α . The rapid decay of negativity for small α in the open system case signals a transition to classicality. However, for large α the closed and open system negativities are similar, suggesting that the system is quantum in this regime. The apparent jaggedness of the closed system plot near α = 0 is due to our discretization of α in steps of 0.01. results to justify this as a proxy for the actual entanglement. To this end we use the negativity (a standard measure of entanglement [105]) N (ρ ) = 1 2 ||ρ Γ A || 1 − 1 , (2.8) where ρ Γ A denotes the partial transpose of ρ with respect to a partition A. Fig- ure 2.11 shows the numerically calculated negativity as a function of α along the time evolution for a “vertical” partition of the eight-qubit system, i.e., with an equal number of core and outer qubits on each side. Both the closed and open system evolution cases are shown. We observe that in the case of the closed system evolution there is always a peak in the negativity for all values of α ≥ 0.01 studied, with an α -dependent position. This is not surprising since as we change α , we change the relative position of the fixed ratio value ofA(t)/(αB (t)), and we expect the negativity peak to correspond to the position of the minimum gap of H(t) 39 [106]. 6 For the open system case, in contrast, the negativity peak drops when α is sufficiently small. This can be said to signal a transition to classicality. The reason for this drop is that as α decreases the system thermalizes more rapidly towards the Gibbs state, but the Gibbs state is also approaching the maximally mixed state, which has vanishing entanglement. However, for large α the peak position and value is similar to that of the closed system case, so that the simulated system exhibits quantum features and has not decohered into a classical evolution. This can be interpreted as another reason for the failure of classical models to reproduce the experimental data. 2.8 Discussion and Conclusions Motivated by the need to discern classical from quantum models of the D-Wave processor, in this chapter we examined three previously published classical models of the D-Wave device (SA, SD [46], SSSV [48]). We studied the dependence of the annealing process on the energy scale of the final “quantum signature” Hamiltonian. Lowering this energy scale acts as an effective temperature increase and thus enhances the effects of thermal fluctuations. While this strategy might appear counterproductive as a means to rule out classical models since it promotes a transition to the classical regime, it in fact presents a challenge for classical models that must now accurately describe not only the ground subspace but also the excited state spectrum of a quantized system. We found that all of the classical models we studied are inconsistent with the experimental data for our quantum signature Hamiltonian, covering the range of 8 6 While this peak position does not precisely match the position of the minimum gap, the result in Ref. [106] holds in the thermodynamic limit and predicts a strict singularity; a discrepancy is therefore excepted in the case of a finite system size. 40 to 20 qubits (thus extending beyond the 8-qubit unit cell of the DW2 device), in a “black-box” setting of a study of the input-output distribution of the device. The SA and SD models were already rejected based on such inconsistency in earlier work [43, 44, 47] and the present evidence supports and strengthens these conclusions. The SSSV model was of particular interest since it matches the ground state success probabilities of random Ising model experiments on the DW1 device [48]. While it is possible that with additional fine-tuning a better match can be achieved with a classical model, an adiabatic quantum ME [45] which we have examined is capable of reproducing most of the key experimental features with only one free parameter (the effective system-bath coupling κ ). Our most complete and accurate model for the D-Wave device accounts for qubit cross-talk and local field and coupling noise, where we demonstrated that the ME captures all the features in the experimental data, in contrast to the noisy SSSV model (Fig. 2.9). We have thoroughly analyzed and explained these findings. It is important to stress that the ME exhibits decoherence not in the compu- tational basis but in the energy eigenbasis. Such decoherence is not necessarily a detriment to QA since it is consistent with maintaining computational basis coherence in the ground state. How can the rejection of the classical SSSV model by our experimental data on quantum signature Hamiltonian problem instances of up to 20 qubits be reconciled with the conclusions of Ref. [48], which demonstrated a strong correlation between success probabilities of the SSSV model and the DW1 device for random Ising problem instances of 108 qubits? One obvious consideration is problem size, though we have found no evidence to suggest that the agreement with experiment improves for the SSSV model as the number of qubits increases. More pertinent seems to be the fact that the quantum signature Hamiltonian experiment probes different 41 aspects of the QA dynamics than the random Ising problem instances experiment. The former is, by design, highly sensitive to the detailed structure of the ground state degeneracy and the manner in which this degeneracy is dynamically generated, and these aspects are different for quantum and classical models. In this sense, it is a more sensitive probe than the random Ising experiment [43], which did not attempt to resolve the ground state degeneracy structure. While Ref. [48] established that the SSSV model correlates very well with the experimental success probability distribution for random Ising instances, and even better with SQA, our results suggest the possibility that a closer examination would reveal important differences between the SSSV model and QA also for the random Ising experiment. For example, Ref. [43] presented additional evidence for QA by also considering excited states and correlations between hardness and avoided level crossings with small gaps. Specifically, we conjecture that a detailed study of the ground state degeneracy for random Ising instances would determine the suitability of the SSSV model as a classical model for QA in this setting as well. Such a study might also circumvent an important limitation of our quantum signature Hamiltonian approach: the exponential degeneracy of the cluster states (2 N/2 ) makes gathering statistically significant data prohibitively time-consuming forN& 20. Clearly, ruling out any finite number of classical models still leaves open the possibility that a new classical model can be found that explains the experimental data. Nevertheless, in the absence of a strict no-go test such as a Bell inequality violation, ruling out physically reasonable classical models while establishing close agreement with a quantum model (the adiabatic ME), is a strategy that should bolster our confidence in the role played by quantum effects, even if it falls short of a proof that all classical models are inconsistent with the experiment. 42 Finally, we stress that the results reported here do not address the scaling of the performance of the D-Wave devices against state-of-the-art classical solvers, or whether this scaling benefits from a quantum speedup [52]. Recent work has highlightedtheimportanceofthechoiceofthebenchmarkproblems[107]. Moreover, a careful estimate of the scaling performance of the D-Wave devices must take into account the effects of limited connectivity and precision in setting the intended problem [108, 109]. The presence of quantum speedup is possible only if the device displays relevant quantum features and defies a classical description. Our work rules out plausible classical models while at the same time showing consistency with an open quantum system description. For small values ofα our ME predicts that entanglement rapidly vanishes, signaling a transition to classicality as the effective temperature becomes high enough, though a quantized energy spectrum persists. This observation can be of practical importance in the case of optimization problems where one expects that classical annealing can be more efficient than QA [52, 107]. In this case one might obtain a performance improvement by allowing the device to work in the classical, thermal region. The possibility of such “thermally assisted” QA has been indeed demonstrated experimentally in [94], in the case of a specific toy problem. It is interesting to more generally characterize the potentially beneficial role played by thermal effects in affecting the performance of QA and adiabatic quantum computing [101, 110]. Apart from being a practical issue for the D-Wave device, thermal excitations present a fundamental obstacle for any adiabatic algorithm [64, 65, 68]. This issue must be addressed by adding error correction to QA [70, 86], or by exploiting thermal noise as a computational resource [111]. Future work shall revisit these questions using new tests and larger system sizes. 43 2.9 The D-Wave Two device All our experiments were performed on the DW2 “Vesuvius” processor located at the Information Sciences Institute of the University of Southern California. Details of the device have been given in Section 1.2, and we only provide a brief overview here. As shown Fig. 2.12, the device is organized into an 8×8 grid of unit cells, each comprising eight qubits arranged in a K 4,4 bipartite graph, which together form the “Chimera” connectivity graph [33] of the entire device. Of the 512 qubits 503 were calibrated to within acceptable working margins in the DW2 processor used in our experiments. Figure 2.12 also gives a schematic representation of the most general problem Hamiltonian [as specified in Eq.(2.2)] that can be implemented in the device. 2.10 Enhancement vs suppression of the isolated state in SA vs QA Here we review and generalize the detailed argument given in Ref. [44] for the enhancement of the isolated state in SA vs its suppression in QA. 2.10.1 Classical ME explanation for the enhancement of the isolated state for general N We first explain why SA predicts an enhancement of the isolated state for general (even) N. To do so we closely follow the arguments from Ref. [44] concerning the N = 8 case. Consider a signature Hamiltonian with N = 2n qubits. As depicted in Fig. 2.3, n of these are the ferromagnetically coupled “core” qubits (J ij = 1), while the othern “outer” qubits are each ferromagnetically coupled to a single core qubit 44 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 Figure 2.12: Qubits and couplers in the DW2 device. The DW2 “Vesuvius” chip consists of an 8× 8 two-dimensional square lattice of eight-qubit unit cells, with open boundary conditions. The qubits are each denoted by circles, connected by programmable inductive couplers as shown by the lines between the qubits. Of the 512 qubits of the device located at the University of Southern California used in this chapter, the 503 qubits marked in green and the couplers connecting them are functional. 45 (J ij = 1). The local fields applied to the core qubits areh i = 1, while h i =− 1 for the outer qubits. Under our classical annealing protocol the system evolves via single spin flips. That is, at each step of the evolution a state can transfer its population only to those states which are connected to it by single spin flip. Thus the rate of population change in a state depends only on the number of states it is connected to via single spin flips. Let the indexj run over all the states connected to state a; the Pauli master equation for the populations can then be written as ˙ p a = X j f(E a − E j )p j − f(E j − E a )p a , (2.9) where we have assumed that the transfer function f(ΔE) does not depend on j and satisfies the detailed balance condition. For a derivation starting from the quantum ME, see Ref. [44] (Supplementary Information). We now derive a classical rate equation for the generalization to N spins of the clustered and isolated states given in Eq. (2.3). Let P I denote the population in the isolated state|11··· 1 | {z } n outer 11··· 1 | {z } n core i, and let P C = 2 − n P c P c denote the average population of the cluster-states{|00··· 0 00··· 0i,...,|11··· 1 00··· 0i}. For the isolated state, flipping either a core spin or an outer spin creates an excited state. An outer-spin flip changes the core-outer spin-pair from|11i to|10i. This flip has an associated cost of 4 units of energy, and there are n such cases. A flip of one of the core qubits changes the core-outer spin-pair from|11i to|01i. This results in two unsatisfied links in the core ring, raising the energy by 4 units. However, the 46 flip leaves the energy of the given core-outer spin-pair unchanged. There are again n such cases. Thus, for the isolated state, the rate equation is ˙ P I = 2n[f(− 4)P 4 − f(4)P I ] (2.10) where P 4 is the population in the excited states which are 4 units of energy higher than the ground states. The derivation of the rate equation for the cluster-states is somewhat more involved. We note first that a flip of any of the outer spins involves no energy cost, so all the excited states created from the cluster-states arise from flipping a core spin. Depending on the state of the outer spin when the core qubit is flipped, we have two different cases. • If the outer is spin|0i, the configuration of the core-outer pair changes from |00i to|10i. This transition involves a change of 4 units of energy. Moreover, this creates a pair of unsatisfied links in the core ring, at a cost of another 4 energy units. Overall, it takes 8 units of energy to accomplish this flip. To count the total number of such excited states connected to all cluster-states, let us consider a cluster-state withl outer spins in|0i andn− l outer spins in |1i. There are n l such cluster-states. In each of these states, we can choose any of the l core spins to flip. Thus, the overall number of all such possible excited states connected to cluster-states is P n l=0 l n l =n2 n− 1 . • If the outer is spin|1i, the configuration of the core-outer pair changes from |01i to|11i. This core-outer transition involves no change of energy. However, this creates a pair of unsatisfied links in the core-ring, at a cost of 4 energy units. The counting argument for number of these excited states is same as 47 in the previous case. Thus, the number of all such possible excited states connected to cluster-states is again n2 n− 1 . We assume that all cluster-states have the same population, equal to the average population. The rate equation of the average cluster-states population is then ˙ P C = n2 n− 1 2 n ([f(− 8)P 8 − f(8)P C ] + [f(− 4)P 4 − f(4)P C ]) (2.11a) = n 2 [f(− 8)P 8 − f(8)P C +f(− 4)P 4 − f(4)P C ], (2.11b) where P 8 is the population in the excited states that are 8 units of energy above the ground states. For most temperatures of interest, relative to the energy scale of the Ising Hamiltonian, the dominant transitions are those between the cluster and states with energy− 4. Transitions to energy 0 states are suppressed by the high energy cost, and transitions from energy 0 states to the cluster-states are suppressed by the low occupancy of the 0 energy states. ˙ P C ≈ n 2 [f(− 4)P 4 − f(4)P C ] (2.12) In classical annealing at constant low temperature starting from arbitrary states (that is, the high energy distribution), probability flows approximately ˙ P I / ˙ P C ≈ 4 times faster into the isolated state initially, and it gets trapped there by the high energy barrier. To show that ˙ P I ≥ ˙ P C for slow cooling schedules, assume that this is indeed the case initially. Then, in order forP C to become larger thanP I , they must first become equal at some inverse annealing temperatureβ 0 : P I (β 0 ) =P C (β 0 )≡ P g , 48 and it suffices to check that this implies that P I grows faster than P C . Subtracting the two rate equations at this temperature yields ˙ P I − ˙ P C = 3n 2 (f(− 4)P 4 − f(4)P g ) (2.13a) = 3n 2 f(− 4)P g P 4 P g − P (g→ 4) P (4→ g) ! , (2.13b) where in the second line we used the detailed balance condition, and P(4→ g) denotes the probability of a transition from the excited states with energy 4 units above the ground state to the ground state g. Now, because the dynamical SA process we are considering proceeds via cooling, the ratio between the non- equilibrium excited state and the ground state probabilities will not be lower than the corresponding thermal equilibrium transition ratio, i.e., P 4 Pg ≥ P (g→ 4) P (4→ g) =e − 4β 0 . Therefore, as we set out to show, ˙ P I − ˙ P C ≥ 0, (2.14) implying that at all times P I ≥ P C . 2.10.2 Perturbation theory argument for the suppression of the isolated state in QA for general N We consider the breaking of the degeneracy of the ground state of our N spin benchmark Ising Hamiltonian by treating the transverse fieldH X =− P N i=1 σ x i as a perturbation of the Ising Hamiltonian (thus treating the QA evolution as that of a closed system evolving backward in time). As pointed out in the main text, the ground state is 2 N/2 + 1-fold degenerate. According to standard first order degenerate perturbation theory, the perturbation ˆ P g of the ground subspace is 49 given by the spectrum of the projection of the perturbation H X onto the ground subspace. Π 0 = (|1ih1|) ⊗ N + (|0ih0|) ⊗ N/2 (|+ih+|) ⊗ N/2 , (2.15) where the first term projects onto the isolated state, and we have written the state of the outer qubits of the cluster in terms of|+i = (|0i +|1i)/ √ 2. We therefore wish to understand the spectrum of the operator ˆ P g = Π 0   − N X j=1 σ x j   Π 0 . (2.16) The isolated state is unconnected via single spin flips to any other state in the ground subspace, so we can write ˆ P g as a direct sum of the 0 operator acting on the isolated state and the projection onto the space Π 0 0 = Π 0 − (|1ih1|) ⊗ N = (|0ih0|) ⊗ N/2 (|+ih+|) ⊗ N/2 of the cluster ˆ P g =− 0⊕ Π 0 0   − N X j=1 σ x j   Π 0 0 (2.17a) =− 0⊕   − N X j=N/2+1 σ x j   , (2.17b) where the sum is over the outer qubits. This perturbation splits the ground space of H I , lowering the energy of |00··· 0 + +··· +i, and theN/2 permutations of|−i = (|0i−| 1i)/ √ 2 in the outer qubits of|00··· 0 + +··· +−i . None of these states overlaps with the isolated ground state, which is therefore not a ground state of the perturbed Hamiltonian. Furthermore, after the perturbation, only a higher (the sixth for N = 8) excited state overlaps with the isolated state. The isolated state becomes a ground state 50 only at the very end of the evolution (with time going forward), when the per- turbation has vanished. This explains why the isolated state is suppressed in a closed system model. A numerical solution of the ME agrees with this prediction for sufficiently large values of the problem energy scale α . 2.11 Experimental data collection methodology Our data collection strategy was designed to reduce the effects of various control errors. In this section we explain the main sources of such errors and our methods for reducing them. These methods are distinct from, and complementary to other error correction methods [71], inspired by stabilizer codes, that have been previously proposed and implemented [70, 86]. Each time (programming cycle) a problem Hamiltonian is implemented on the DW2 device, the values of the local fields and couplings{h i , J ij } are set with a Gaussian distribution centered on the intended value, and with standard error of about 5% [102]. To average out these random errors we ran several different programming cycles for the same problem Hamiltonian as described in Sec. 2.11.1 on this section. Differences among the individual superconducting flux qubits can contribute to systematic errors. To average out these local biases, we embedded our Hamiltonian multiple times in parallel on the device using different flux qubits, as also explained in Sec. 2.11.1 of this section. Furthermore, we implemented different “gauges”, a technique introduced in Ref. [44]. A gauge is a given choice of {h i , J ij }; a new gauge is realized by randomly selectinga i =±1 and performing the substitutionh i 7→ a i h i andJ ij 7→ a i a j J ij . Provided we also perform the substitution σ z i 7→ a i σ z i , we map the original Hamiltonian to a gauge-transformed Hamiltonian with the same energy spectrum but where the identity of each energy eigenstates is 51 Figure 2.13: Embedding according to the random parallel embeddings strategy for eight spins. An example of 15 randomly generated different parallel copies of the eight-spin Hamiltonian. Our data collection used a similar embedding with 50 different copies. 52 Figure 2.14: Embedding according to the random parallel embeddings strategy for 16 spins. An example of 10 randomly generated different parallel copies of the 16-spin Hamiltonian. Our data collection used a similar embedding with 93 different copies. 53 1 5 2 6 3 7 4 8 Figure 2.15: Embedding according to the “in-cell embeddings” strategy. An example of a randomly generated in-cell embedding. 54 relabeled accordingly. In total, there are 2 N different gauges for anN-spin problem. We averaged our data using different programming cycles, embeddings, and gauges, as explained in Sec. 2.11.2 of this section. In addition we checked for errors due to correlations between successive runs (Sec. 2.11.3 of this section) and found these to be negligible. Finally, we describe a new method for correcting control errors that assumes that the degenerate cluster states should ideally have the same population ( Section 2.16.1). 2.11.1 Data collection strategies We used two different data collection strategies that resulted in perfectly consistent results. Strategy A: Random parallel embedding. As illustrated in Fig. 2.13, 50 different parallel embeddings of the 8-spin problem Hamiltonian were generated randomly in suchawaythatanembeddingisnotnecessarilylimitedtoaunitcell. Wethussolved 50 different copies of the same 8-spin problem in parallel during each programming cycle. We generated two such parallel embeddings containing 50 copies each. For each embedding, we performed 100 programming cycles and 1000 readouts for the runs with annealing time t f = 20μ s, 200μ s; 200 programming cycles and 498 readouts for the runs with annealing time t f = 2000μ s; 500 programming cycles and 48 readouts for the runs with annealing time t f = 20000μ s. An example set of randomly generated embeddings for the 16 spin Hamiltonian is shown in Fig. 2.14. Strategy B: In-cell embeddings. We utilized 448 qubits to program 56 parallel copies of the eight-spin problem Hamiltonian, with an identical gauge for all copies, with one copy per unit cell. All possible 256 gauges were applied sequentially. For a given annealing time t f , the number of readouts was min(1000,b5× 10 5 /t f c). For example, 1000 readouts were done for t f = 20μ s and 100 readouts for t f = 5000μ s. 55 One such copy of an in-cell embedding is shown in Fig. 2.15. No in-cell embeddings are possible for problems involving N > 8 spins. Strategy C: Designed parallel embedding. For N = 40, we utilized 320 qubits to program 8 parallel copies of the 40-spin problem Hamiltonian. The 40-qubit Hamil- tonian was designed with three different embeddings spread across the Chimera graph. Each embedding occupied six different unit cells. One hundred random gauges were chosen out of the 2 40 possible gauges and were identically applied to all eight copies. To collect significant statistics, we performed 200 programming cycles with 10000 readouts for every gauge. The annealing time was t f = 20μ s. 2.11.2 Data analysis method The following method was used to analyze the data. Let us denote the number of gauges by N G and the number of embeddings by N E . For a given embedding a and gauge g, the number of total readouts (number of readouts times the number of programming cycles) for the ith computational state is used to determine the probability p a,g (i) of that computational state. The gauge-averaged probabilities for the ith computational state p GA a (i) of the ath embedding are determined by averaging over the gauges for a fixed embedding: p GA a (i) = 1 N G N G X g=1 p a,g (i) , a = 1,...,N E . (2.18) Let us now consider a function of interestF, for example P I /P C or the trace-norm distanceD(ρ DW2 ,ρ Gibbs ). Using the raw probabilities p a,g (i), we can calculateF a,g . For example, ifF =P I /P C , we have: F a,g = P I P C a,g = 16p a,g (I) P 16 i=1,i∈ C p a,g (i) . (2.19) 56 For a fixed embeddinga, we calculate the standard deviation σ G a associated with the distribution ofF using the raw probabilities values over the N G gauges, i.e. σ G a = std h {F a,g } N G g=1 i . (2.20) For each embedding, we also calculateF GA a using the gauge-averaged probabilities, e.g., F GA a = P I P C GA a = 16p GA a (I) P 16 i=1,i∈ C p GA a (i) . (2.21) Therefore, for each embedding, we now have the following sets of data {(F GA a ,σ G a )} N E a=1 . We refer to this as the gauge-averaged data, of which we have N E data points. We then drew 1000 bootstrap [103] data samples from the gauge- averaged data (giving us a total of 1000×N E data points), which we denote byF GA a,b where a = 1,...,N E , b = 1,..., 1000. In order to account for the fluctuations in the gauge data, for a fixed bootstrap sampleb, we add noise (normally distributed with the standard deviation σ G a ) to everyF GA a,b in the bootstrap sample. For each of the 1000 bootstrap data samples, we calculated the mean: ¯ F GA b = 1 N E N E X a=1 F GA a,b , b = 1,..., 1000 . (2.22) Therefore we now have a distribution of 1000 means. The mean of the 1000 means corresponds to the data points in our plots, and twice the standard deviation of the 1000 means is the error bar used in the plots in the main text. 2.11.3 Autocorrelation Tests Correlations between the outputs of different runs on the device could be a re- sult of errors on the device (such correlations were reported in [43]) in that the 57 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 x 10 −3 Bin Sizes Δx (a) α =0.1 1 2 3 4 5 6 7 8 9 10 0 0.002 0.004 0.006 0.008 0.01 0.012 Bin Sizes Δx (b) α =0.35 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 −3 Bin Sizes Δx (c) α =1 Figure 2.16: Some representative autocorrelation tests at t f = 20μs showing the standard error of the mean Δx as a function of binning size, for different values of α : (a)α = 0.1, (b)α = 0.35, (c)α = 1. Each curve is the result of the binning test for a different state. The relatively flat lines for all states suggest that there are no significant autocorrelations in the data. results of each run are not completely independent. This can in turn affect the ground state populations by preferentially picking the first state observed. To test for this possibility, we use a binning test, which is a simple method to test for autocorrelations in statistical data [112]. Consider a list of n uncorrelated binary numbers{x i } with P (x i = 1) =p. The standard error of the mean for this dataset is Δx≡ q Var[x]/n = q p(1− p)/n. We bin together the average of consecutive pairs in this list to produce a new list y i of n/2 numbers such that y i ∈{ 0, 0.5, 1}. 58 Since P(y i = 1) = p 2 , P(y i = 0.5) = 2p(1− p) and P(y i = 0) = (1− p) 2 , the error in the mean of this derived list is Δy = q p(1− p)/n = Δx. If how- ever, the list were correlated such that P(x i+1 = 1| x i = 1) = q 6= p, then Δy = q (p +q− p 2 − q 2 )/(2n)6= Δx. The idea of the binning test easily follows: keep on binning data with larger bin sizes until the error in the means converges to a constant value. The minimal bin size where this occurs is the autocorrelation length, ξ . We used the binning test on all 256 different states for the N = 8 problem. For each state, we generated a list of 1000 binary numbers{x i } such that x i = 1 when that state was read from the D-Wave device and x i = 0 otherwise. The probability of occurrence of the state is denoted by ¯ x and Δx is its error. We found that the error in the mean does not change appreciably with the size of bins used. This indicates that the autocorrelation length for any state in our system is zero, and there are no significant autocorrelations in our data. Figure 2.16 shows a few representative cases from the data collected using the “in-cell embeddings” strategy for various choices of α and random gauge choices. The Wald-Wolfowitz runs test is a standard statistical test for autocorrelations. We tested the null hypothesis,H 0 , that the sequence in consideration was generated in an unbiased manner. The Wald-Wolfowitz test relies on comparing the number of “runs” in the dataset to a normal distribution of runs. A run is defined as consecutive appearance of same state. In our dataset of binary valued sequences, a run occurs every time there is a series of either 0’s or 1’s. For example, the sequence 011100100111000110100 contains 6 runs of 0 and 5 runs of 1. The total number of runs is 11. Let R be the number of runs in the sequence, N 1 be the number of times value 1 occurs and N 0 be the number of times value 0 occurs. (In our example, R = 11, N 1 = 10 and N 0 = 11.) It can be shown that if the sequence 59 were unbiased, the average and the standard deviation of the number of runs would be given by [113] ¯ R = 2N 1 N 0 N 1 +N 0 + 1, (2.23) σ 2 R = 2N 0 N 1 (2N 1 N 0 − N 1 − N 0 ) (N 1 +N 0 ) 2 (N 1 +N 0 − 1) (2.24) The test statistic is Z = R− ¯ R σ R . At 5% significance level, the test would reject the null hypothesis if|Z|> 1.96 (in this case the obtained value of the number of runs differs significantly from the number of runs predicted by null hypothesis). We applied the Wald-Wolfowitz test to the binary sequences used in the binning test. We found that for each value of α , fewer than 0.01% of the sequences failed the Wald-Wolfowitz test. For example, 112 sequences for α = 1, 343 sequences for α = 0.35 and 195 sequences for α = 0.1 failed the test. The total number of such sequences tested for each value ofα were 256× 56× 256≈ 3.6× 10 6 . This suggests once more that autocorrelations do not affect our dataset significantly. 2.12 Kinks in the time dependence of the gap Here we explain the origin of the kinks in the time dependence of the gap seen in Fig. 2.4. First, just as in Fig. 2.4 but for different values of α , we show in Fig. 2.17(a) how as α is decreased, the minimal gap occurs at a later time in the evolution and decreases in magnitude. The kinks that appear in both Fig. 2.4 and Fig. 2.17(a) are a consequence of energy level crossings apparent in the evolution of the spectrum, as shown in Figs. 2.17(b)-2.17(d) for the same values of α as in Fig. 2.17(a). There are energy eigenstates that become part of the 17 degenerate ground states that “cut” through other energy eigenstates. 60 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 t/t f Energy gap (GHz) α = 1/7 α = 2/7 α = 3/7 (a) 0 0.2 0.4 0.6 0.8 1 −35 −30 −25 −20 −15 −10 −5 t/t f Energy (GHz) (b) α =1/7 0 0.2 0.4 0.6 0.8 1 −35 −30 −25 −20 −15 −10 −5 t/t f Energy (GHz) (c) α =2/7 0 0.2 0.4 0.6 0.8 1 −35 −30 −25 −20 −15 −10 −5 t/t f Energy (GHz) (d) α =3/7 Figure 2.17: (a) Time-dependence of the gap between the 18th excited state and the instantaneous ground state, for different values of the energy scale factor α . (b)-(d) Time-dependence of the lowest 56 energy eigenvalues for different values of α [(b) α = 1/7, (c) α = 2/7, (d) α = 3/7]. Note that the identity of the lowest 17 energy eigenvalues changes over the course of the evolution. 2.13 Simulation details 2.13.1 Simulated Annealing We describe here our implementation of classical SA. As the state of system at any given step is a classical probability distribution, we can represent it by a state vector ~ p, where the component p i of the vector denotes the probability of finding the system in the ith state with energy E i . We initialize in the maximally mixed state (infinite temperature distribution), i.e.,p i = 1/2 N ∀ i. Note that the initial 61 0 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 3.5 4 α P I /P C 0 0.5 1 0 0.2 0.4 0.6 0.8 1 PGS α Exponential Linear Constant Exponential Linear Constant Figure 2.18: Simulated annealing shows quantitatively similar behavior for various annealing schedule. The schedules are: exponential T(k) =T(0)(T (K)/T (0)) k/K , linearT (k) =T (0) + k K (T (K)− T (0)), and constantT (k) =T (K), withK = 1000, with k B T (0)/~ = 8 GHz and k B T (K)/~ = 0.5 GHz. Gibbs state of the quantum annealer also has a uniform probability distribution over all computational states, so this choice for the classical initial state is well motivated. The system then evolves via single spin flips. The transition probability between two states with energy difference ΔE is given by 1 N min(1, exp(− β ΔE)), the Metropolis update rule [114]. The transition matrix has elements T(i→ j) = 1 N min(1, exp(− β (E j − E i ))) . (2.25) The system is evolved for 1000 steps by acting with the transition matrix on the state vector ~ p. At each step, the temperature is adjusted so as to reduce thermal excitations. If the temperature is reduced slowly enough and to low enough energies, SA can find an optimal solution. The choice of temperature schedule to follow for SA is often motivated by experimental circumstances, and in the main text we used β − 1 B(t) (as shown in Fig. 2.1) as the schedule with β − 1 /~ = 2.226 GHz. Here we tested three other different temperature schedules. As shown in Fig. 2.18, 62 we find that the qualitative features of the simulation results do not depend on a particular choice of temperature schedule. While the numerical values of the ratio of the isolated state and cluster populations changes, the ratio is always greater than unity. The ground state population curves are indiscernible regardless of the temperature schedule used. 2.13.2 Spin Dynamics In the O(3) SD model, qubits are replaced by classical spins ~ M i = (sinθ i cosφ i , sinθ i sinφ i , cosθ i ). This is a natural semi-classical model since it amounts to the saddle-point approximation of the path integral for the spin system (the derivation is presented in Section 2.18) and can be interpreted as describing coherent single qubits interacting classically. This model is closely related to one that was proposed and analyzed by Smolin and Smith [46] in its planar, O(2) version, i.e., ~ M i = (sinθ i , 0, cosθ i ) (a spin in thex− z plane). While the SD model was already shown to be inconsistent with the experimental data in the context of correlations with DW1 spin glass benchmarks in Refs. [43, 47], the SD model was shown in Ref. [46] to give the same suppression of the isolated state prediction as QA for α = 1, and hence the evidence in Ref. [44] alone does not suffice to rule out the SD model as a classical description of the D-Wave device. In this section we demonstrate that similarly to SA, the SD model is also inconsistent with the ME results we obtain when we tune the energy scale factor α of the quantum signature Hamiltonian. 63 As shown in section 2.18.2, the SD model with thermal fluctuations is described by a (Markovian) spin-Langevin equation [115, 116] with a Landau-Lifshitz friction term [116, 117], d dt ~ M i =− ~ H i + ~ ξ (t) +χ ~ H i × ~ M i × ~ M i , (2.26) with the Gaussian noise ~ ξ ={ξ i } satisfying hξ i (t)i = 0 , hξ i (t)ξ j (t 0 )i = 2k B Tχδ ij δ (t− t 0 ) , (2.27) and ~ H i = 2A(t)ˆ x + 2B(t)   h i + X j6=i J ij ~ M j · ˆ z   ˆ z , (2.28) where ˆ x and ˆ z are unit vectors. For thenth run out of a total ofN r runs we obtain a set of angles{θ (n) j }, which are interpreted in terms of a state in the computational basis by defining the probability of the|0i state for the jth spin (out of N) as cos 2 (θ (n) j /2). Therefore, we define: P C = 1 16N r Nr X n=1 N Y j=N/2+1 cos 2 θ (n) j /2 (2.29a) P I = 1 N r Nr X n=1 N Y j=1 sin 2 θ (n) j /2 , (2.29b) where the product over the lastN/2 spins inP C is 1 if and only if all the core spins are in the|0i state, i.e., a cluster state, and likewise the product over all N spins in P I is 1 if and only if all the spins are in the|1i state, i.e., the isolated state. To incorporate the α -dependence we simply rescale B(t) to αB (t). 64 2.13.3 Master Equation We used an adiabatic Markovian ME in order to simulate the DW2 as an open quantum system. Details of the derivation of the ME can be found in Ref. [45]. The derivation assumes a system-bath Hamiltonian of the form: H =H S (t) +H B +g X α A α ⊗ B α , (2.30) where A α is a Hermitian system operator acting on the α th qubit and B α is a Hermitian bath operator. We restrict ourselves to a model of independent baths of harmonic oscillators, i.e., each qubit experiences its own thermal bath, with a dephasing system-bath interaction, A α =σ z α ; B α = X k b k,α +b † k,α , (2.31) where b k,α and b † k,α are lowering and raising operators and k is a mode index. We use the double-sided adiabatic ME without the rotating wave approximation [45]: d dt ρ S (t) = − i [H S (t),ρ S (t)] + g 2 X αβ X ab Γ αβ (ω ba (t)) [L ab,β (t)ρ S (t),A α ] + h.c. , (2.32) 65 where ω ba =ε b (t)− ε a (t) are differences of instantaneous energy eigenvalues given by H S (t)|ε a (t)i =ε a (t)|ε a (t)i and L ab,α (t) =hε a (t)|A α |ε b (t)i|ε a (t)ihε b (t)| =L † ba,α (t) , (2.33a) Γ αβ (ω ) = Z ∞ 0 e iωt he − iH B t B α e iH B t B βidt = 1 2 γ (ω ) +iS(ω ) , (2.33b) γ αβ (ω ) = Z ∞ −∞ he − iH B t B α e iH B t B βidt , (2.33c) S αβ (ω ) = Z ∞ ∞ dω 0 γ αβ (ω 0 )P 1 ω − ω 0 dω 0 , (2.33d) whereP denotes the Cauchy principal value. Under the assumption of Ohmic independent baths, we have: γ αβ (ω ) =δ αβ 2πg 2 ηω 1− e − βω e −| ω |/ω c , (2.34) where β is the inverse temperature, η is a parameter (with units of time squared) characterizing the bath, and ω c is an ultraviolet cut-off, which we set to 8π GHz to satisfy the assumptions made in deriving the ME [45]. Note that the only remaining free dimensionless parameter is κ ≡ g 2 η/ ~ 2 (2.35) (we have reintroduced the factor of~ here), which controls the effective system-bath coupling. We choose to work with the ME in Eq. (2.32) instead of its counterpart in (completely positive) Lindblad form because it is numerically more efficient to calculate the evolution. Although it does not guarantee positivity of the density matrix, we always make sure to work in a parameter regime where we do not 66 , 0 0.2 0.4 0.6 0.8 1 P I = P C 0 0.5 1 1.5 S A S D S S S V M E (a) , 0 0.2 0.4 0.6 0.8 1 P G S 0 0.2 0.4 0.6 0.8 1 S A S D S S S V M E (b) Figure 2.19: Numerical results distinguishing the quantum ME and classical SA, SD, and SSSV models. (a) Results for the ratio of the isolated state population to the average population in the cluster-states (P I /P C ), and (b) the ground state probability(P GS ), asafunctionoftheenergyscalefactorα , atafixedannealingtime of t f = 20μs . The error bars represent the 95% confidence interval. Two striking features are the “ground state population inversion” between the isolated state and the cluster (the ratio of their populations crosses unity), and the manifestly non-monotonic behavior of the population ratio, which displays a maximum. At the specific value of the system-bath coupling used in our simulations (κ = 1.27× 10 − 4 ), it is interesting that the ME underestimates the magnitude and position of the peak in P I /P C but qualitatively matches the experimental results shown in Fig. 2.9(a), capturing both the population inversion and the presence of a maximum even in the absence of noise. In contrast to the ME results shown, the SA, SSSV, and SD results for the population ratio are not in qualitative agreement with the ME. Specifically, all three classical models miss the population inversion and maximum seen for the ME. Simulation parameters can be found in Section 2.13. observe any violations of positivity. In the ME simulations presented in the text, we truncate the spectrum to the lowest 56 instantaneous energy eigenstates to keep computational costs within reason. We have checked that increasing this number for the smallest α regime does not substantially change our conclusions. 67 2.14 Comparing the models in the noiseless case In this section we present additional numerical findings for SA and SD in the absence of noise on the local fields and couplings. Since we presented evidence in the main text that noise and cross-talk play an important role in the experimental DW2 results, the results presented in this section are limited to a comparison between the models, which behave quite differently in the ideal case. We may expect the ME results to match a future quantum annealer with better noise characteristics and no cross-talk. Therefore we present our findings by contrasting each of the classical models in turn with the ME simulations. 2.14.1 SA The main result showing the dependence of P I /P C as a function of the energy scale α for N = 8 qubits is summarized in Fig. 2.19. We note first that the total ground state probability P GS =P I + 16P C decreases monotonically as α is decreased. This reflects an increase in thermal excitations, whereby the ground state population is lost to excited states, and confirms thatα acts as an effective inverse temperature knob. However, in contrast to SA, the ME result for P I /P C is non-monotonic inα ; see Fig. 2.19(a). Initially, as α is decreased from its largest value of 1, the ratio P I /P C increases and eventually becomes larger than 1; i.e., the population of the isolated state becomes enhanced rather than suppressed. For sufficiently small α , the ME P I /P C ratio turns around and decreases towards 1. The SA results also converge to 1 as α → 0 but do not display a maximum. 68 A close examination of Fig. 2.19(a) shows that even in the “relatively classical” small α region (α . 0.1) the curvature of P I /P C for the ME re- sults [d 2 (P I /P C )/dα 2 < 0] is inconsistent with the curvature of the SA result [d 2 (P I /P C )/dα 2 > 0], as seen in Fig. 2.19(b). We can show that the positive curva- ture of SA is a general result as long as the initial population is uniform. To see this, we expand the SA Markov-chain transition matrix [Eq. (2.25)] in powers of α : T(α ) = T 0 +α T 1 +... , (2.36) where the SA state vector ~ p(K) at the Kth time-step is given by ~ p(K) T = ~ p(0) T T(α ) K , where the T superscript denotes the transpose. At α = 0, all transitions are equally likely, so (T 0 ) ij = 1/N. The first order term satisfies: (T 1 ) i→ j = min (0,− β (E j − E i )) , i6=j , (2.37a) (T 1 ) i→ i =− X j min (0,− β (E j − E i )) . (2.37b) The first order term has the property that P j (T 1 ) ij = 0. Therefore, applying the transition matrix K times, we have to first order inα : ~ p(K) T =~ p(0) T (T 0 ) K +α~ p (0) T K− 1 X i=0 (T 0 ) K− 1− i T 1 (T 0 ) i +... (2.38) Using the fact that we start from the uniform statep i (0) = 1/N, we have~ p(0) T T 1 = 0, but also that (T 1 T 0 ) ij = 1 N X k (T 1 ) ik = 0 . (2.39) Therefore, ~ p(K) =~ p(0) +O(α 2 ) . (2.40) 69 This in turn implies that for SA, P I P C = 1 +α 2 f +O(α 3 ) . (2.41) Since we showed in Section 2.10.1 that this quantity is greater than or equal to 1 for SA, this implies that f≥ 0, and hence the curvature d 2 (P I /P C )/dα 2 at α = 0 is positive. However, we emphasize that this argument for the positivity of the initial curvature of P I /P C requires that the initial state be uniform. If a non-uniform initial state is chosen, there will be a non-zero linear term in α for P I /P C , which prevents us from concluding anything about the curvature. 2.14.2 Results for the SD model Using the DW2 operating temperature and annealing schedules for A(t) and B(t), we find that the SD model does not match the ME data. This can be seen in Fig. 2.19(b), where the SD population ratio (the dashed blue line) fails to reproduce the qualitative features of the ME result, in particular the ground state population inversion peak. Another illustration of the same failure of the SD model is given in Fig. 2.20(a), which shows the distribution of M z for the core and outer qubits for different values of α . We expect the core spins to align in the|1111i state for sufficiently small α (i.e., to each have M z =− 1), when the isolated state becomes enhanced. However, as can be seen from Fig. 2.20(a) the median of the core spins is in fact never close to M z =− 1 for small α values, so that the enhancement of the isolated state is missed by the SD model. Furthermore, the model shows a preference for a particular cluster-state, the one with all of the outer spins in the |1111i state (M z =− 1). In the inset of Fig. 2.20(a), this can be seen in that the 70 median of the data occurs always below M z = 0. The explanation is provided in Section 2.18.2. 2.14.3 Results for the SSSV model To test whether this model matches the results of our quantum signature Hamil- tonian we use similar parameters as given in Ref. [48], apart from the annealing schedule, for which we used that of the DW2. Reference [48] found the best agreement with the DW1 data from Ref. [43] for a temperature of 10.6mK, lower than the 17mK operating temperature of the DW1, and for a total of 1.5× 10 5 Monte Carlo update steps per spin (sweeps). We found negligible differences when we used the operating temperature of the DW2 (17mK) for the SSSV model, or when we varied the number of sweeps. As can be seen in Fig. 2.19, the SSSV model does not reproduce the ground state population inversion and maximum seen in the experimental data. In fact the SSSV results are quantitatively similar to the SD model, even in showing a preference for a specific cluster-state, as shown in Fig. 2.20(b). Furthermore, the SSSV model does not reproduce the ground state population inversion even after the number of qubits is increased to 40 [see Fig. 2.31(d)], which is particularly significant as it shows that the essential quantum features that result in the disagreement are retained beyond the initial “small” N = 8 problem size. 2.15 ME vs Modified SSSV models with “deco- herence” from O(2) rotors to Ising spins In this section we consider variants of the SSSV model where the O(2) rotors are first mapped to qubits and then allowed to decohere. The rationale is that the SSSV 71 0 0.2 0.4 0.6 0.8 1 !1 !0.5 0 0.5 1 α M z 0 0.2 0.4 0.6 0.8 1 !1 !0.5 0 0.5 1 α M z 0 0.2 0.4 0.6 0.8 1 !1 !0.5 0 0.5 1 α M z 0 0.2 0.4 0.6 0.8 1 !1 !0.5 0 0.5 1 α M z 0.1 0.25 0.4 0.55 0.7 0.85 1 !1 !0.5 0 0.5 1 M z 0.1 0.25 0.4 0.55 0.7 0.85 1 !1 !0.5 0 0.5 1 M z (a) SD model 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 α M z 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 α M z 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 α M z 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 α M z 0.1 0.25 0.4 0.55 0.7 0.85 1 −1 −0.5 0 0.5 1 M z 0.1 0.25 0.4 0.55 0.7 0.85 1 −1 −0.5 0 0.5 1 M z (b) SSSV model Figure 2.20: Statistical box plot of the averagez component for all core qubits (main plot) and all outer qubits (inset) at t =t f = 20μs . (a) The SD model. The data are taken for 1000 runs with Langevin parameters k B T/~ = 2.226 GHz (i.e., 17mK, to match the operating temperature of the DW2) and ζ = 10 − 3 . In Section 2.18.2 we show that the results do not depend strongly on the choice of ζ . (b) The SSSV model. The data are taken for 1000 runs with parameters k B T/~ = 1.382 GHz (i.e., T = 10.56mK, as in Ref. [48]) and 5× 10 5 sweeps. “qubits” may be too coherent, and we wish to account for single-qubit decoherence effects. 2.15.1 Strongly decohering SSSV model Because of the large deviation of SSSV from the ME at small α observed in Fig. 2.19(a), we propose to modify the model to fix this. In order to raise theP I /P C value, we note that SA, which uses effectively incoherent qubits, has P I /P C ≥ 1. Therefore, we might consider the scenario where the qubits become more incoherent as α becomes smaller, until in the limit of vanishing α they fully decohere and become Ising spins in the computational basis, we might be able to reproduce similar behavior. To model this we replace the x-component of the magnetization vector of each spin by M x i = e − t/τ α sinθ i and leave the z-component unchanged, i.e., M z i = cosθ i . This is equivalent to a model of single-qubit dephasing in the 72 , 0 0.2 0.4 0.6 0.8 1 P I =P C 0 0.5 1 1.5 F or c e d S S S V W e ak l y D e c oh e r i n g S S S V S t r on gl y D e c o h e r i n g S S S V M E Figure 2.21: “Forced” and strongly or weakly “decohering” SSSV models. Shown are the results for the ratio of the isolated state population to the average population in the cluster-states (P I /P C ) as a function of the energy scale factor α , for N = 8 and at a fixed annealing time oft f = 20μs . The error bars represent the 95% confidence interval. For reference the plot also includes the curves for the ME from Fig. 2.19(a). Additional parameters for the modified SSSV models:g 2 η = 10 − 6 for the strongly decohered model and g 2 η = 2.5× 10 − 7 for the weakly decohered and forced models. computational basis, via the mapping to the density matrix ρ i = 1 2 I + ~ M i ·~ σ , where ~ M i = (M x i , 0,M z i ) and~ σ = (σ x i ,σ y i ,σ x i ). This can be visualized as a gradual squashing of the Bloch sphere (restricted to the x− z plane) into an ellipsoid (ellipse) with major axis in the z-direction and a shrinking minor (x-)axis. It is also equivalent to replacing the transverse field amplitudeA(t) in Eq. (A.22) by A(t)e − t/τ α while leaving the magnetization unchanged, i.e., decreasing the time-scale over which the transverse field plays a role. Next we ensure that τ α is monotonically increasing with α . In this manner, for tt f the range is almost that of the fully “coherent” SSSV, while for t.t f the range is restricted to that of the “incoherent” SA. The “decoherence” time τ α dictates how quickly this transition from one extreme to the other occurs, and to 73 incorporate its α -dependence we set τ α = 1/2γ α (0), where γ α (0) = 2πg 2 η (αβ ) − 1 is the dephasing rate used in our ME calculations [the general expression for γ (ω ) is given in Eq. (2.34)], with a rescaled inverse temperature, i.e., αβ instead of β , to capture the idea that α acts to rescale the energy, or equivalently the inverse temperature. Thus in this model τ α =α β 4πg 2 η . Note that we only replaced β with αβ here and not anywhere else in the simulations, so the physical temperature is still given by β − 1 . Figure 2.21 presents the results of this “strongly decohering SSSV” model. The results are similar to the original SSSV model. Thus, “decoherence” of the coherent O(2) spins fails to improve the agreement with the ME results. 2.15.2 Weakly decohering SSSV model We can consider a weaker version of this dephasing model, which attempts to mimic dephasing in the energy eigenbasis of the ME model. When the transverse field Hamiltonian dominates over the Ising Hamiltonian, the dephasing occurs in the z-component of the magnetization, and when the Ising Hamiltonian dominates over the transverse field Hamiltonians, the dephasing occurs in thex-component of the magnetization. Explicitly, this translates to replacing the magnetization components of the spin by M x i = sinθ i , M z i =e − t/τ α cosθ i , (2.42) if A(t)≥ αB (t) and by M x i =e − (t− tc)/τ α sinθ i , M z i =e − tc/τ α cosθ i , (2.43) 74 if A(t)<αB (t), where t c is the transition time satisfying A(t c ) =αB (t c ). As can be seen in Fig. 2.21, this model also fails to capture the ME results. 2.15.3 A modified SSSV model with a forced transition from O(2) rotors to Ising spins To try to get better agreement of a classical model with the ME we finally consider a somewhat contrived model which simply forces a transition to SA with Ising spins. To implement this, instead of uniformly drawing θ i ∈ [0,π ] as in the SSSV model, we draw θ i ∈ [0, π 2 e − t/τ α ]∪ [π − π 2 e − t/τ α ,π ], where τ α is selected just as in the decohered SSSV model ( Section 2.15). This can be visualized as a restriction of the range of angles to gradually shrinking top and bottom parts of the Bloch sphere (again restricted to the x− z plane). We call this a “forced SSSV” model since it does not originate from a natural model of decoherence. Figure 2.21 also presents the results of this forced SSSV model. In contrast to the original SSSV model result [Fig. 2.19(b)], the population ratio now rises to 1 for α > 0. In this regard the forced SSSV model qualitatively captures the tendency toward ground state population inversion. However, it does not exhibit a pronounced ground state population inversion, and this appears to be a robust feature that is shared by other forced SSSV models we have tried (with different “forcing” rules). Furthermore, it exhibits a noticeable drop inP I /P C atα ≈ 0.1, and the fraction of ground state population is almost one in the ground state population inversion regime, in contrast to the ME. In this sense even the forced SSSV model does not agree with the ME data, and further evidence to this effect is presented in the next subsection. 75 , 0 0.2 0.4 0.6 0.8 1 T r ac e - n o r m d i s t an c e f r om ; G i b b s 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 S A S D S S S V S t r o n gl y D e c oh e r i n g S S S V W e ak l y D e c oh e r i n g S S S V F or c e d S S S V M E Figure 2.22: Trace-norm distance of the ME, SA, SD, SSSV, weakly and strongly decohered SSSV, and forced SSSV states from the T = 17mK Gibbs state at t f = 20μs and N = 8. The error bars represent the 95% confidence interval. Three regions are clearly distinguishable for the ME: (1) 1≥ α & 0.3, whereD is decreasing as α decreases; (2) 0.3&α & 0.1, whereD is increasing as α decreases; (3) 0.1&α ≥ 0, whereD is again decreasing as α decreases. Both SA and SD lack the minimum at α ≈ 0.3. 2.15.4 Distance from the Gibbs state How well does the system thermally equilibrate? In this section we consider how distinguishable the final density matrixρ (t f ) is from the thermal Gibbs state at t f , using the standard trace-norm distance measure [118] D (ρ (t f ),ρ Gibbs ) = 1 2 kρ (t f )− ρ Gibbs k 1 , (2.44) where ρ Gibbs =e − βH (t f ) /Z withZ = Tre − βH (t f ) the partition function, andkAk 1 ≡ Tr √ A † A (the sum of the singular values of the operator A). Note the fact that in 76 the Gibbs state all ground states are equiprobable, so that P I /P C = 1. This simple observation helps to explain many of the experimental results. The trace-norm distance result is shown in Fig. 2.22 for the ME, and the six classical models. Although most of the models exhibit a peak in the trace-norm distance like the ME, none of the six classical models exhibits a minimum like the ME does at the corresponding value of α , thus confirming once more that there is a strong mismatch between these classical models and the ME. This is particularly noticeable for the “forced” SSSV model, which as discussed above exhibited the best agreement with the ground state features among the classical models (Fig. 2.21), but poorly matches the excited state spectrum at low α , as can be inferred from Fig. 2.22. Indeed, this model is designed to transition to SA at low α , and it does so at α ≈ 0.1. It then deviates from SA at even lower α values, presumably since there is no transverse field at all in SA, but the transverse field remains active in the “forced” SSSV model at any α> 0. Furthermore, we observe that the “weakly decohering SSSV” model has a higher trace-norm distance than all other models even at highα , and this is due to its strong preference for a particular cluster state, which is a failure mode of the SD and SSSV models that was discussed earlier (see Fig. 2.20). Let us now focus on the ME results and explain the three regions seen in Fig. 2.22. Large α , region (1). As α decreases from 1 to≈ 0.3, since α is relatively large, thermal excitations are not strong enough to populate energy eigenstates beyond the lowest 17 that eventually become the degenerate ground state. Therefore, the system is effectively always confined to the subspace that becomes the final ground state, as can also be seen from the P GS data in Fig. 2.19(a). However, recalling that the isolated state has overlap with excited states higher in energy than the 77 cluster-states for t < t f (see Section 2.10.2), thermal excitations populate the isolated state. Thus as α decreases, P I /P C approaches 1, which is also the ratio satisfied by the Gibbs state, and henceD decreases as observed. At the same time, Fig. 2.19(b) shows that at α = 0.3 both the SD and SSSV models have P I /P C ≈ 0, i.e., these models fail to populate the isolated state. This therefore suggests that the quantum spectrum makes it easier for the system to thermally hop from one eigenstate to another. Intermediate α , region (2). Fig. 2.19(a) shows thatP GS begins to decrease from 1 at α ≈ 0.3, meaning that thermal excitations are now strong enough to populate energy eigenstates beyond the lowest 17. A loss in ground state population to excited states results, and the growth ofP I /P C beyond 1 seen in Fig. 2.19(a) results in the increase ofD observed in Fig. 2.22. At α ≈ 0.1, the maximum distance from the Gibbs state is reached. Beyond this value of α , the energy scale of the Ising Hamiltonian is always below the temperature energy scale, as shown in Fig. 2.1. Small α , region (3). As α → 0 there is only a transverse field left, which is gradually turned off. Thus the system approaches the maximally mixed state (which is the associated Gibbs state). In light of this, for 0.1&α ≥ 0 the energy gaps are sufficiently small that there is a large loss of population from the ground state; thermal excitations become increasingly more effective at equilibrating the system, thus pushing it towards the Gibbs state. 78 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 t/tf Populations in 17 Ising ground states HD= 8 HD= 7 HD= 6 HD= 5 HD= 4 HD= 0 Figure2.23: Masterequationresultsforthepopulationsofthe 17Isinggroundstates, withα = 1,|h| = 0.981|J|,t f = 20μs , andκ = 1.27× 10 − 4 . The cluster-states split by Hamming distance from the isolated state (bottom curve), in agreement with the experimental results shown in Fig. 2.24(a). 2.16 An alternative model for breaking the sym- metry of the cluster states 2.16.1 The effect of an h vsJ offset Under a closed system evolution all the cluster states end up with an identical population. The same is true if we compute the populations using the ME for an independent bath model. In the main text we discussed how cross-talk breaks the symmetry between the cluster states. In this section we discuss another mechanism, of making|J| and|h| unequal, that also breaks the symmetry between the cluster states. This is shown in Fig. 2.23 using the ME, where the population of the cluster-state splits by Hamming distance from the isolated state. Thus this can be viewed as an alternative explanation for the cluster state distribution observed on 79 the DW2, although as we will show, the fit to the experiment is not as good as the cross-talk model discussed in the main text. Tounderstandtheoriginofthisphenomenon, considerthefollowingperturbation theory argument for the N = 8 case. Assume that all the local fields are perturbed by δh> 0 so that for the outer spins h i =− 1 +δh (1≤ i≤ 4) and for the core spins h i = 1− δh (5≤ i≤ 8). Therefore|h i |<|J ij | = 1 and the perturbation to H I [Eq. (2.2)] can be written as: V =− δh 4 X i=1 σ z i − 8 X i=5 σ z i ! (2.45) All the cluster states have their core spins in the|0i state, so V increases all their energies by 4δh . The perturbation acting on the outer spins, however, breaks the degeneracy by Hamming weight. The contribution from this term is given by − (n 0 − n 1 )δh where n 0 or n 1 is the number of outer spins in the|0i or|1i state, respectively. Therefore the energy of the|0000 0000i state is unchanged (it becomes the unique ground state), while the energy of the|1111 0000i state increases by 8δh , so it becomes the least populated among the cluster states. Consequently the final population of the cluster states becomes ordered by Hamming distance from the isolated state|1111 1111i. Interestingly, the experimental data for the final populations of the cluster-states displays a pronounced “step” structure, clearly visible in Fig. 2.24(a). The observed steps correspond to an organization of the cluster-states in terms of their Hamming distance from the isolated state, and agrees with the ordering observed in Fig. 2.23 and the perturbation theory argument. Thus, the step structure can be explained if, in spite of the fact that for all gauges we set|h i |/|J ij | = 1, in reality there is a systematic error causing|h i | <|J ij |. Such an error would arise if the ratio of 80 |h i | |J ij | % Change Absolute Change 1 0.9810 -1.90 -0.0190 6/7 0.8440 -1.53 -0.0131 5/7 0.7040 -1.44 -0.0103 4/7 0.5655 -1.04 -0.0059 3/7 0.4265 -0.48 -0.0021 2/7 0.2850 -0.25 -0.0007 1/7 0.1420 -0.60 -0.0009 Table 2.1: Optimized|J ij | values for a given|h i | value, yielding the flat population structure shown in Fig. 2.24(b). The systematic corrections are of the order of 1%, smaller than the random control errors of 5% at α = 1. B(t)|h i | and B(t)|J ij | is not kept fixed throughout the annealing, whereB(t) is the annealing schedule shown in Fig. 2.1. Moreover, such an error would not be unexpected, as the local fields (an inductance) and couplers (a mutual inductance) are controlled by physically distinct devices [30]. A natural question is whether we can mitigate this type of error. To do so we introduce a simple optimization technique. Specifically, we can compensate for|h i | <|J ij | and fine-tune the values ofh and J to nearly eliminate the step structure for each value of the energy scale factor α . We show this in Fig. 2.24(b). The corresponding optimized values are given in Table 2.1, where the compensation reverses the inequality to|h i | >|J ij |. We have further checked that the same fine-tuning technique suppresses the step structure seen for largerN, as shown in Fig. 2.24(c). The step structure in the N = 16 case is even more pronounced than in the N = 8 case. By adjusting the value of J while keeping|h| = 1 we can reduce the step structure, as shown in Fig. 2.24(d). (We note that this gives rise to the interesting possibility of using this “step-flattening” technique to more precisely calibrate the device.) This control error has little effect on the suppression or enhancement of the isolated state, as can be seen in Fig. 2.25. 81 (H, M) (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) 2000 4000 6000 8000 10000 12000 Gauge-averaged instances (a) (H, M) (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Gauge-averaged instances (b) (0,1) (8,1) (9,8) (10,28)(11,56) (12,70)(13,56)(14,28) (15,8) (16,1) 0 200 400 600 800 1000 1200 1400 1600 1800 Gauge-averaged instances (H, M) (c) (0,1) (8,1) (9,8) (10,28)(11,56) (12,70)(13,56)(14,28) (15,8) (16,1) 0 100 200 300 400 500 600 700 Gauge-averaged instances (H, M) (d) Figure 2.24: Statistical box plot of the gauge-averaged ground states population (a,c) before and (b), (d) after optimization ofJ ij as per Table 2.1, for (a),(b)N = 8 and (c),(d) N = 16, α = 1 and t f = 20μs . Only the N = 8,H = 6 case splits into two rotationally inequivalent sets. Note the clear step structure in the cluster-states (H > 0) in (a),(c), while in (b),(d) the population of the cluster-states is fairly equalized (less so in the N = 16 case since Table 2.1 is optimized for N = 8). (a) Data taken with the random parallel embeddings strategy. (b) Data taken using the in-cell embeddings strategy, with the optimized values of the couplings given in Table 2.1. The same optimization removes the step structure from data taken with the random parallel embeddings strategy (not shown). 82 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 α PI/PC Corrected Uncorrected Figure 2.25: Ratio of isolated state population to average cluster-state population as a function of the energy scale factor α , for t f = 20μs and N = 8. Shown are the ratios calculated with both uncorrected and corrected values ofJ (as per Table 2.1), the latter tuned to flatten the steps seen in the population of the cluster states. Error bars represent the standard error of the mean value of the ratio estimated using bootstrapping. 83 , 0 0.2 0.4 0.6 0.8 1 p = ( M p G S ) 0 0.02 0.04 0.06 0.08 0.1 0.12 (a) SSSV, offset h=0.97J 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 α p/(MpGS) (b) ME, offset h=0.985J , 0 0.2 0.4 0.6 0.8 1 p = ( M p G S ) 0 0.02 0.04 0.06 0.08 0.1 0.12 (c) Perturbation theory, off- set h=0.975J Figure 2.26: (a) Noisy SSSV; (b) ME, (c) perturbation theory [Eq. (2.46)] with a population ordering correction (offset of h vsJ) atα = 1. The error bars represent the 95% confidence interval. The SSSV model now has the right ordering of the cluster states but clearly disagrees with the DW2 result [Fig. 2.9(a)] for α . 0.3, near where the isolated state has its maximum. The ME result is in qualitative agreement with the DW2 result except that the cluster state populations do not equalize for small α , which is a consequence of not including the α -dependence of the offset. 2.16.2 Using the distribution of cluster states to rule out the noisy SSSV model As we saw in Figs. 2.24, without the Table 2.1 correction, the DW2 results at α = 1 exhibit a non-uniform distribution over the cluster states. We now demonstrate that the noisy SSSV model is incapable of correctly capturing this aspect of the experimental results. We do so for various scenarios differing in how we treat the control error. 2.16.2.1 Population ordering correction at α = 1 The noisy SSSV and ME results after calibrating h and J so as to match the DW2 ordering of the distribution of cluster states at α = 1 are shown in Fig. 2.26(a) and 2.26(b), respectively. Like the DW2, the ME cluster state populations converge as α goes to zero, whereas the SSSV populations converge up to α ≈ 0.3 and 84 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 α p/(MpGS) (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 α p/(MpGS) (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 α p/(MpGS) (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 0.12 α p/(MpGS) (0,1) (4,1) (5,4) (6,4) (6,2) (7,4) (8,1) (a) DW2, offset J =0.981h , 0 0.2 0.4 0.6 0.8 1 p = ( M p G S ) 0 0.02 0.04 0.06 0.08 0.1 0.12 (b) SSSV, offset h=0.988J , 0 0.2 0.4 0.6 0.8 1 p = ( M p G S ) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 (c) SSSV, offset h=0.94J Figure 2.27: (a) DW2 and (b) noisy SSSV model with a population equalizing correction atα = 1. (c) Noisy SSSV model with offset chosen to equalize the cluster state populations at α = 0.2. The error bars represent the 95% confidence interval. then diverge again. Therefore, with this calibration, the noisy SSSV model fails to capture the DW2 cluster state populations at low α . We can understand the equalization of the cluster state populations in the ME as follows: as α is made smaller, the spacing of the quantized, discrete energy levels (when the Ising Hamiltonian dominates) shrinks withα . Thermal excitations between the levels will be less suppressed, allowing for a redistribution of the population. We check this intuition with a generalization of our perturbation theory argument that was used explain the suppression of the isolated state in the closed system setting (see Section 2.10.2). We diagonalize the Hamiltonian: H =α (H I + 0.01H X ) +η (2.46) where H I is the detuned Ising Hamiltonian, η is Gaussian noise (independent of α ) introduced on the couplings and local fields, andH X is the transverse field, whose small magnitude models the end of the annealing evolution. We then populate the lowest 17 energy eigenstates of this Hamiltonian by a Boltzmann distribution, i.e., p n = e − βE n /Z, where Z = P 256 n=1 e − βE n . We pick β/ ~ = 10.7ns and choose a calibration ofh andJ in order to best match the DW2 results atα = 1. The cluster 85 state populations are then extracted from their overlap with these Boltzmann populated 17 energy levels. As shown in Fig. 2.26(c), this perturbation theory argument reproduces the behavior of the ME for the cluster states very well: it shows the cluster state populations converging to an equal population as α goes to zero. This at least suggests that the intuition presented above is consistent. However, this method does not reproduce all the data. The isolated state shows a very large population (it is off the scale of the graph), which does not match the ME or the DW2 results. This is not entirely surprising since the Boltzmann distribution of course does not take into account the annealing evolution. The reason that the ME does not exhibit a uniform population on the cluster states for small α [as seen in the DW2 results of Fig. 2.9(a)] was addressed in Sec. 2.3, where we discussed a cross-talk mechanism that generates anα -dependence ofh andJ. With this dependence the ME reproduces this feature of the DW2 data as well. 2.16.2.2 Population equalizing correction at α = 1 orα = 0.2 As an example of a different calibration procedure, we can calibrate the DW2 and the noisy SSSV model to have equal populations atα = 1. As shown in Fig. 2.27(a) and 2.27(b), we observe that initially asα is decreased, both SSSV and DW2 behave in a similar manner whereby the cluster state populations diverge, but whereas DW2 converges again for small α and is almost uniform at α ≈ 0.15, the noisy SSSV model populations do not start to re-converge until much closer to α = 0. Therefore, once again, we find a qualitative difference between the noisy SSSV model and the DW2. We note that for this calibration (i.e., having the cluster states equal at α = 1), the ME would not need to be offset and would be as shown in Fig. 2.6(a). 86 Since the no-offset DW2 results [Fig. 2.9(a)] show the cluster state populations equalizing at α ≈ 0.2 we can alternatively attempt to calibrate the noisy SSSV model to match the no-offset DW2 results at this value of α , i.e., we can choose an offset for SSSV such that it has an almost equal population at α = 0.2. This is shown in Fig. 2.27(c). The cluster states continue to diverge as α decreases, while they diverge in the opposite order as α grows. If we continue this procedure, i.e., make the populations equal for smaller and smaller α , this requires a larger offset which will make the staircase structure at α = 1 even further pronounced, further increasing the mismatch with the DW2 in this regime. 2.16.2.3 Excited states ordering As in the main text, we now go beyond the ground subspace and consider an 8-dimensional subspace of the subspace of first excited states. We arrange these according to permutations of the core or outer qubits, i.e., we group the states as |1111 Π(0001)i and|Π(1110) 1111i, where Π denotes a permutation. As shown in Fig. 2.28, the DW2 prefers the set|Π(1110) 1111i, and the perturbation theory analysis based on the noisy quantum signature Hamiltonian [Eq. (2.46)] agrees. However, for all values of the offset considered in the previous two subsections, the noisy SSSV model prefers the set|1111 Π(0001)i, as seen in Fig. 2.28(b)-2.28(d). 2.17 Effect of varying the annealing time or the total number of spins In the main text we discussed the effect of varying the energy scale α of the final Hamiltonian as a means to control thermal excitations. In this section we consider 87 , 0 0.2 0.4 0.6 0.8 1 p 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 1 11 1 &( 0 0 01 ) &( 1 1 10 ) 1 1 1 1 (a) Perturbation theory, offset h = 0.975J , 0 0.2 0.4 0.6 0.8 1 p 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 1 11 1 &( 0 0 01 ) &( 1 1 10 ) 1 1 1 1 (b) SSSV, offset h=0.97J , 0 0.2 0.4 0.6 0.8 1 p 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 1 11 1 &( 0 0 01 ) &( 1 1 10 ) 1 1 1 1 (c) SSSV, offset h=0.988J , 0 0.2 0.4 0.6 0.8 1 p 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 1 11 1 &( 0 0 01 ) &( 1 1 10 ) 1 1 1 1 (d) SSSV, offset h=0.94J Figure 2.28: Subset of the first excited state populations for (a) perturbation theory [as in Eq. (2.46)], (b), (c) and (d) SSSV with offsets matching Figs. 2.26(a), 2.27(b) and 2.27(c) respectively. Panel (f) shows the case with qubit cross-talk discussed in Sec. 2.3. The Π symbol denotes all permutations. Whereas the perturbation theory result for a QA Hamiltonian [Eq. (2.46)] reproduces the correct ordering, none of the three SSSV cases shown does. These three SSSV cases were chosen to optimize the fit for the cluster state populations. The error bars represent the 95% confidence interval. 88 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 α P I /P C (DW2) t f =20μs t f =200μs 0 0.5 1 0 0.5 1 1.5 α PI/PC (ME) tf = 20μs tf = 200μs Figure 2.29: Ratio of the isolated state population to the average population in the cluster (P I /P C ) as a function of the energy scale factor α , for two different values of t f , and N = 8. The inset shows the ME results. Data were collected using the “in-cell embeddings” strategy (see Section 2.11.1 for details). Error bars are one standard deviation above and below the mean. 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 α Energy (GHz) 8-spin 12-spin 16-spin Gap between ground and first excited state (GHz) t/t f Figure 2.30: Numerically calculated instantaneous energy gap between the ground and first excited state for the 8, 12 and 16 spin Hamiltonians. The gap vanishes since the first excited state becomes part of the 2 N/2 + 1-fold degenerate ground state manifold at t =t f . 89 two alternative approaches, namely varying the annealing time or the total number of spins and provide our experimental results. 2.17.1 Increasing the total annealing time t f As reported in Ref. [44] (which only studied the α = 1 case), increasing the annealing time reduced the suppression of the isolated state, which is consistent with the effect of increased thermal excitations. To understand this, note that in general the requirement of high ground state fidelity generates a competition between adiabaticity (favoring long evolution times) and suppression of thermal effects (favoring short evolution times) [65]. Since the shortest annealing time of the DW2 (20μ s) is already much longer than the inverse of the minimal gap (∼ (25GHz) − 1 at α = 1; see Fig. 2.4), increasing the annealing time does not suppress non-adiabatic transitions, but does increase the probability of thermal fluctuations. Figure 2.29 shows that, as expected, an increase in the annealing time is consistent with stronger thermalization, and indeed, over the range of α where we observe suppression of the isolated state (P I /P C < 1), this suppression is weaker for the larger total annealing time. The ME result is in qualitative agreement with the experimental data: The larger annealing time curve is the higher of the two, and the peak values of P I /P C at the two different annealing times coincide, which also agrees with the experimental result, within the error bars. 2.17.2 Increasing the number of spins N There are two important effects to keep in mind when considering larger numbers of spins N (even). First, increasing the number of spins does not change our previous argument that the instantaneous ground state has vanishing support on the isolated state towards the end of the evolution. We showed this explicitly using 90 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 α PI/PC (a) N =12 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 α PI/PC (b) N =16 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 α PI/PC (c) N =20 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 35 40 α PI/PC (d) N =40 Figure 2.31: Ratio of the isolated state population to the average population in the cluster-states (P I /P C ) as a function of the energy scale factor α , for different values ofN, at a fixed annealing time oft f = 20μs . The non-monotonic dependence of the population ratio onα is observed for all values ofN. The growth of theP I /P C peak with increasing N is consistent with the discussion presented in Section 2.17.2. The increasingly large error bars are due to the smaller amount of data collected as N grows. For N = 12, 16, 20 data were collected using the “random parallel embeddings” strategy and for N = 40 using the “designed parallel embedding” strategy (see Section 2.11.1 for details). Error bars are one standard deviation above and below the mean. 91 first order perturbation theory in Section 2.10.2. This means that we should still expect that P I /P C < 1 for QA, unless thermal excitations dominate. Second, the degeneracy of the instantaneous first excited state grows withN, while the energy gap to the ground state remains fixed withN. The latter is illustrated in Fig. 2.30. Consequently there is an enhancement of the thermal excitation rate out of the instantaneous ground state into the first excited state, eventually feeding more population into the instantaneous excited states that have overlap with the isolated state. Thus we expect P I /P C to grow with N (as we indeed find experimentally; see Fig. 2.31). We have studied the simplest extensions beyond N = 8 (examples are shown in Fig. 2.3) with 12, 16 and 40 spins. We expect the same qualitative features observed for N = 8 to persist, and this is confirmed in Fig. 2.31, which displays the same qualitative non-monotonic behavior as a function of α . The main difference is that the enhancement of the isolated state (when P I /P C > 1) becomes stronger as N is increased. This is a manifestation of the growth, with N, in the number of excited states connected to the isolated state as compared to the number connected to the cluster-states. This implies that the excitation rate due to thermal fluctuations is proportionally larger for the isolated state than for the cluster states. Going to even larger N on the DW2 is prohibitive, since it requires the number of readouts to be O(2 N/2 ) in order to collect a statistically significant amount of data. This is due to the growth of the number of cluster states as described in Eq. (2.4). 92 2.18 DerivationoftheO(3)spin-dynamicsmodel 2.18.1 Closed system case Here we present the standard path integral derivation of the O(3) model, which is closely related to the O(2) SD model of Ref. [46]. Let us introduce the tensor product state of coherent spin-1/2 states |Ω(t)i =⊗ i cos(θ i (t)/2)|0i i + sin(θ i (t)/2)e iφ i (t) |1i i . (2.47) We consider the amplitude associated with beginning in|Ω(0)i =⊗ i |Ω i (0)i and ending in|Ω(t f )i, A =hΩ(t f )|T + e − i ~ R t f 0 H(t)dt |Ω(0)i, (2.48) where T + represents time-ordering. We write the integral in terms of a Riemann sum: Z t f 0 H(t)dt = lim ν →∞ ν − 1 X n=0 H(t n )Δt, (2.49) where Δt =t f /ν and t n =nΔt, and then perform a Trotter slicing: T + e − i ~ R t f 0 H(t)dt = ν − 1 Y n=0 e − i ~ H(tn)Δt +O(Δt 2 ) (2.50) We now introduce an over-complete set of spin-coherent states (2.47) between the Trotter slices 1 = Z dΩ|ΩihΩ|, (2.51) where for general spin S dΩ = Y i 2S + 1 4π sinθ i dφ i dθ i , (2.52) 93 so that we have: A = Z dΩ 1 ··· Z dΩ ν − 1 ν Y n=1 hΩ n |e − i ~ H n− 1 Δt |Ω n− 1 i +O(Δt 2 ), (2.53) where we have denoted Ω ν ≡ Ω(t f ), Ω 0 ≡ Ω(0), and H n ≡ H(t n ). To the same order of approximation we can write hΩ n |e − i ~ H n− 1 Δt |Ω n− 1 i (2.54a) =hΩ n | 1− i ~ H n− 1 Δt |Ω n− 1 i +O(Δt 2 ) (2.54b) =hΩ n |Ω n− 1 i 1− iΔt ~ hΩ n |H n− 1 |Ω n− 1 i hΩ n |Ω n− 1 i ! +O(Δt 2 ). (2.54c) Let us assume differentiability of the states Ω n and the Hamiltonian H n so that we can write: |Ω n− 1 i =|Ω n i− Δt∂ t |Ω n i +O(Δt 2 ) (2.55a) H n− 1 =H n − Δt∂ t H n +O(Δt 2 ). (2.55b) Using this differentiability on the overlap, we have: hΩ n |Ω n− 1 i =hΩ n | (|Ω n i− Δt∂ t |Ω n i) +O(Δt 2 ) (2.56a) = 1− ΔthΩ n |∂ t |Ω n i +O(Δt 2 ) (2.56b) = exp (− ΔthΩ n |∂ t |Ω n i) (2.56c) 94 Likewise, using this differentiability on the matrix element of the Hamiltonian, we have: ΔthΩ n |H n− 1 |Ω n− 1 i (2.57a) = ΔthΩ n | (H n − Δt∂ t H n )(|Ω n i− Δt∂ t |Ω n i) +O(Δt 2 ) (2.57b) = ΔthΩ n |H n |Ω n i +O(Δt 2 ) (2.57c) Putting these results together, we have for the amplitude: A = Z dΩ 1 ··· Z dΩ ν − 1 × (2.58a) e i ~ Δt P ν n=1 (i~hΩn|∂ t|Ωni−h Ωn|Hn|Ωni) +O(Δt 2 ) (2.58b) = Z DΩ exp i ~ Z dt (i~hΩ|∂ t |Ωi−h Ω|H(t)|Ωi) (2.58c) = Z DΩ e i ~ S[Ω] (2.58d) where we have taken the continuum limit such that Ω n → Ω(t) and introduced the action S[Ω] = Z dtL = Z dt (i~hΩ|∂ t |Ωi−h Ω|H(t)|Ωi) . (2.59) For simplicity, let us now work in units of~ = 1. Using Eq. (2.47) we can write the first term in the action as: ihΩ|∂ t |Ωi =− 1 2 X i (1− cosθ i ) dφ i dt (2.60) 95 0 0.2 0.4 0.6 0.8 1 −0.2 0 0.2 0.4 0.6 0.8 1 t/tf M z Core Outer (a) Closed system 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 t/tf M z Core Outer (b) Open system Figure2.32: Evolutionofacore(blue)andouter(green)spinwitht f = 20μs , subject to the O(3) SD model with α = 1. All spins start with M x = 1, M y =M z = 0, i.e., point in the x direction. (a) Closed system case given by Eq. (2.63). (b) Open system case given by Eq. (2.65). Rapid oscillations at the beginning of the evolution in (a) are because the initial conditions used are not exactly the ground state of the system [because of the finiteB(0)]. In (b), Langevin parameters are k B T/~ = 2.226 GHz and ζ = 10 − 6 . The Euler-Lagrange equations of motion d dt ∂ L ∂ ˙ φ i ! − ∂ L ∂φ i = 0, (2.61a) d dt ∂ L ∂ ˙ θ i ! − ∂ L ∂θ i = 0 (2.61b) extremize the action and yield the semi-classical saddle point approximation: 1 2 sinθ i d dt θ i − ∂ ∂φ i hΩ|H(t)|Ωi = 0, (2.62a) − 1 2 sinθ i d dt φ i − ∂ ∂θ i hΩ|H(t)|Ωi = 0. (2.62b) These are the equations of motion for theO(3) model, wherehΩ|H(t)|Ωi plays the role of a time-dependent potential. 96 For a Hamiltonian of the form of Eq. (2.2), the equations of motion (2.62) become, in terms of the magnetization ~ M = Tr (~ σρ ), d dt ˙ ~ M i =− ~ H i × ~ M i (2.63a) ~ H i ≡ 2A(t)ˆ x + 2αB (t)   h i + X j6=i J ij ~ M j · ˆ z   ˆ z, (2.63b) where we have already included the α dependence. Using the DW2 annealing schedule in Fig. 2.1 we plot the evolution of the spin system in Fig. 2.32(a). This figure shows that the system evolves to a cluster state. Namely, the core spins have M z = 1, i.e., are in the|0i state, and the outer spins have M z = 0. Since the outer spins have eigenvalues±1 underσ z with equal probability, having the average equal zero is consistent with having an equal distribution among the cluster states. This suppression of the isolated state result is consistent with the QA evolution, and was used in Ref. [46] to critique the conclusion of Ref. [44] that the experimental evidence is consistent with quantum evolution. In the next subsection we discuss the effect of adding thermal noise and a dependence on the energy scale factor α . 2.18.2 Open system case: Langevin equation Now that we have our “classical” model, we introduce a thermal bath by extending the equations of motion to an appropriately generalized (Markovian) spin-Langevin equation [115, 116], d dt ~ M i =− ~ H i + ~ ξ (t)− ζ d dt ~ M i ! × ~ M i , (2.64) 97 −1 −0.5 0 0.5 1 0 500 1000 1500 2000 2500 3000 M z # of instances (a) Core spins −1 −0.5 0 0.5 1 0 100 200 300 400 500 600 M z # of instances (b) Outer spins Figure 2.33: Distribution of M z at the end of the evolution for all (a) core spins and (b) all outer spins for α = 1. Langevin parameters are k B T/~ = 2.226 GHz and ζ = 10 − 3 . Data collected using 1000 runs of Eq. (2.65). with the Gaussian noise ~ ξ satisfyinghξ i (t)i = 0 andhξ i (t)ξ i (t 0 )i = 2k B Tζδ (t− t 0 ). One can simplify Eq. (2.64) by perturbatively inserting d dt ~ M i =− ~ H i × ~ M i into the “friction” term to get: d dt ~ M i =− ~ H i + ~ ξ (t) +ζ ~ H i × ~ M i × ~ M i , (2.65) which gives rise to a “Landau-Lifshitz” friction term [116, 117] and is the evolution equation (2.26). An example of the resulting evolution for α = 1 is shown in Fig. 2.32(b). Note that the M z value of the outer spins does not converge to 0, unlike the closed system case shown in Fig. 2.32(a). This is not accidental: While the core spins prefer the|0i state [M z = 1, Fig. 2.33(a)], the outer spins prefer the|1i state, i.e., the median occurs at M z < 0, as is clearly visible in Fig. 2.33(b). An explanation in terms of the effective Ising potential between a core-outer spin pair is given in the main text (Sec. V) for the noisy SSSV model, but the same applies to the SD model. 98 0.1 0.25 0.4 0.55 0.7 0.85 1 −1 −0.5 0 0.5 1 M z 0.1 0.25 0.4 0.55 0.7 0.85 1 −1 −0.5 0 0.5 1 M z 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 α M z 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 α M z 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 α M z 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 α M z (a) ζ =10 − 1 0.1 0.25 0.4 0.55 0.7 0.85 1 −1 −0.5 0 0.5 1 M z 0.1 0.25 0.4 0.55 0.7 0.85 1 −1 −0.5 0 0.5 1 M z 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 α M z 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 α M z 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 α M z 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 α M z (b) ζ =10 − 5 Figure 2.34: Statistical box plot of the z component for all core qubits (main plot) and all outer qubits (inset) at t =t f = 20μs . The data is taken for 1000 runs of Eq. (2.65) with Langevin parameters k B T/~ = 2.226 GHz (to match the operating temperature of the DW2) and (a) ζ = 10 − 1 and (b) ζ = 10 − 5 . The ζ = 10 − 3 is shown in Fig. 2.20(a). This illustrates that the results do not depend strongly on the choice of ζ . The dependence on α is given in Fig. 2.20. We observe that as α is decreased, the median value of the core spins and outer spins does not significantly change. However, we do observe a very slow decrease away from M z = 1 for the core spins. A larger effect is the appearance of more outliers asα decreases, which is consistent with the system being able to explore states away from the cluster states. However, we emphasize that the majority of the states observed are cluster states and not the isolated state (in contradiction with the ME results). We have checked the dependence of our results on the friction parameter ζ . In Fig. 2.34(a) we see that for sufficiently large ζ (> 10 − 3 ), the median values are not affected significantly by changingζ . For sufficiently small ζ [Fig. 2.34(b)] we observe that the median value of the core spins does not deviate very far from 1, and the median of the outer spins appears to shift even further towards M z o =− 1. We have also checked the dependence on the annealing time. As shown in Fig. 2.35, there is no significant change in the median for either the core or the 99 20 50 100 200 500 −1 −0.5 0 0.5 1 tf (μs) M z (a) Core spins 20 50 100 200 500 −1 −0.5 0 0.5 1 tf (μs) M z (b) Outer spins Figure 2.35: Distribution ofM z at the end of the evolution for (a) all core spins and (b) all outer spins (i.e., the values of all core spins and all outer spins are included in each respective box plot) forα = 2/7. Langevin parameters arek B T/~ = 2.226 GHz and ζ = 10 − 3 . Data collected using 1000 runs of Eq. (2.65). Note that the t f -axis scale is not linear. outer spins, suggesting that (over the range of annealing times studied) the system does not fully thermalizes. 100 Chapter 3 Performance of two different quantum annealing correction codes 3.1 Introduction Steady progress is being made towards the realization of a universal fault-tolerant quantum computer (e.g., [119, 120]), yet the development of a scalable architecture remains a tenacious obstacle. This has spurred the arrival of alternative quantum computing devices that sacrifice universality in order to allow for more rapid progress and thus hopefully usher in the era of quantum computing, albeit for a limited set of computational tasks, such as quantum simulation [121–123]. Another example is quantum annealing [13, 24, 25, 63, 97, 98], an analog quantum approach for solving optimization problems [17, 18], that may offer a quicker route than the standard circuit model towards the demonstration of the utility of quantum computation. The task addressed by quantum annealing is the well-known NP-hard problem of solving for the ground state of a classical Ising Hamiltonian [124], H I = X i∈ V (G) h i Z i + X {(i,j)}∈ E(G) J ij Z i Z j , (3.1) 101 where{h i } are the local fields on the verticesV of the connectivity graph G,{J ij } are the couplings along the edges E of G, and Z i is the Pauli-Z operator on the ith qubit. This is done, ideally, by evolving the system for a total annealing time t f according to the time-dependent Hamiltonian H(t) =− A(t) X i X i +B(t)H I , (3.2) where X i is the Pauli-X operator on the ith qubit, and where A(t) and B(t) are the annealing schedules, satisfying A(0) B(0) and A(t f ) B(t f ). In the closed system setting, starting in the ground state of H(0) and evolving adiabatically, the system is guaranteed to reach the ground state of H I with high probability [15, 125, 126]. Although adiabatic dynamics is robust against certain forms of decoherence appearing in the more realistic open system setting [63–69], it remains susceptible to thermal noise and specification errors [70] Quantum annealers, such as the D-Wave device (Section 1.2) are now commercially available. Can such devices benefit from some form of error correction? Here we address this question by comparing two codes and two quantum annealing devices in the setting of the toy optimization problem of finding the ground states of antiferromagnetic chains. Work on the first generation D-Wave 1 (DW1) “Rainier” and second generation D-Wave 2 (DW2) “Vesuvius” processors has already demonstrated that error correc- tion can substantially benefit quantum annealing [50, 86, 87]. Namely, it was shown that even a relatively simple quantum repetition code incorporating energy penalty terms and a decoding procedure, can significantly improve the success probability of finding ground states, as well as overcome precision issues in the specification of the Ising Hamiltonian. This technique is known as quantum annealing correction 102 (QAC). QAC is designed to work within the present technological restrictions of available quantum annealers whereby only the problem Hamiltonian (H I ) and not the transverse field Hamiltonian ( P i X i ) can be encoded. This differs from the standard scheme of error-suppression via energy penalties for AQC wherein the entire Hamiltonian is encoded [72, 76]. This type of encoding in principle allows for the suppression of arbitrary errors, but requires k-local interactions withk≥ 3 [79]. QAC provides a pragmatic compromise, in that it only suppresses errors that do not commute with the repetition code stabilizers (i.e., it does not suppress pure dephasing errors), but is directly implementable since it only requires 2-local Z i Z j interactions. We use the notation [n,k,d] n P to denote a distance d code that uses n physical data qubits and n P dedicated penalty qubits to encode k qubits [87]. The original QAC code introduced in Ref. [86] is a [3, 1, 3] 1 code, designed to be compatible with the D-Wave Chimera hardware graph (shown in Fig. 3.1). Recently, Ref. [87] introduced a new [4, 1, 4] 0 code that uses the same physical resources as the [3, 1, 3] 1 code, without a dedicated penalty qubit. The two key benefits of the new code are that (i) the encoded hardware graph corresponding to the [4, 1, 4] 0 code has a higher degree (of connectivity), and (ii) it can be concatenated to give higher distance codes. Given that the [3, 1, 3] 1 and [4, 1, 4] 0 codes consume the same physical resources, it is natural to ask for a comparison between the two. Here we address this by testing the two codes on uniform antiferromagnetic chains, the same problem first studied in Ref. [86]. These problems are simple—their ground state can be trivially written down—but they are instructive since they are particularly error-prone in quantum annealers because of the existence of numerous low energy excitations (domain walls). We demonstrate that for sufficiently long chains and sufficiently high noise rates, the [4, 1, 4] 0 code outperforms the classical strategy of 103 0 1 2 3 4 5 6 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 361 362 363 364 365 367 368 369 370 371 372 373 374 375 376 377 378 379 380 382 383 384 385 386 387 388 389 390 391 392 393 394 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 7 122 140 360 366 381 395 423 (a) DW2-ISI hardware graph 0 1 2 3 4 5 6 7 8 9 10 12 13 14 16 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 85 86 87 88 89 90 91 92 93 94 95 96 97 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 116 117 118 119 120 121 122 123 124 125 126 127 128 130 131 132 133 135 136 137 138 139 140 141 144 145 146 147 148 149 150 151 152 153 154 155 156 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 304 305 306 307 308 309 310 312 313 314 315 316 317 318 319 320 321 323 324 325 326 327 328 329 330 331 332 333 334 336 337 338 339 340 341 342 343 344 345 347 348 349 350 351 352 353 354 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 388 389 390 391 392 393 394 395 396 397 398 400 401 402 403 404 405 406 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 428 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 460 461 462 463 464 465 466 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 493 494 495 496 497 498 499 500 501 502 504 505 506 507 508 509 510 511 11 15 17 66 84 98 115 129 134 142 143 157 227 249 274 303 311 322 335 346 355 387 399 407 424 425 426 427 429 430 431 459 467 491 492 503 (b) S6 hardware graph Figure 3.1: The DW2 processors have unit cells arranged in an 8× 8 grid, each containing 8 qubits forming aK 4,4 bipartite graph. The active (inactive) qubits are shown in green (red), and active couplers are shown as solid black lines. Out of the 512 qubits on the full DW2 Chimera graph, 504 and 476 were functional on the DW2-ISI and S6 devices, respectively. A comparison of the physical parameters of the two devices is given in Table 3.1. running four chains in parallel and selecting the best, which also consumes the same physical resources. The [4, 1, 4] 0 code was shown in Ref. [87] to significantly improve the performance of quantum annealing in a minor embedding setting, a technique enabling the embedding of a given graph of interactions G into one of a smaller degree, by using several physical qubits to represent a single logical qubit. This is crucial for applications, where one often starts from a logical problem defined on 104 a high-degree (even complete) graph [35]. In particular, the [4, 1, 4] 0 code can be viewed as a minor embedding on the Chimera graph of two interconnected square graphs, as shown in Figure 3.2(b). Here we find that the [4, 1, 4] 0 code is bested by the [3, 1, 3] 1 code in the setting of chains. We provide and verify an explanation for this performance difference in terms of the different effective energy scales generated by the two codes. A novel aspect of the work in this chapter is that we compare two different quantum annealing devices, namely two DW2 devices with somewhat different operating characteristics. This allows us to observe the role of temperature effects, among others. This chapter is organized as follows. In Sec. 3.2, we briefly review QAC in the context of the [4, 1, 4] 0 and [3, 1, 3] 1 codes, including decoding strategies. In Sec. 3.3 we describe our benchmarking procedure and define the strategies that QAC is compared against. Section 3.4 presents our experimental results. Theoretical analysis is provided in Sec. 3.5. We conclude in Sec. 3.6. Additional details are provided in Sections 3.7 and 3.8. Annealer N f Temperature(mK) M AFM (pH) 1/f Amplitude DW2-ISI 504 16± 1 1.33 7.5± 1 S6 476 11± 1 1.92 4.1± 0.3 Table 3.1: Physical parameter of the two quantum annealing devices used in our study. Both devices belong to the same Vesuvius generation, with the major difference being a lower temperature and lower noise on the S6 device, and a higher qubit yield on the DW2-ISI device. M AFM is the inter-qubit coupling energy when a coupler is set to provide the maximum antiferromagnetic (AFM) coupling. N f is the number of functional qubits on the annealer. 1/f is the low frequency flux noise in units of flux quanta, Φ 0 = h/2e, where h is the Planck constant and e is the electron charge. Details about these physical parameters can be found in Ref. [127]. 105 1 2 3 4 a b c d e f (a) Arrangement of physical qubits a b c d e f (b) Logical qubit lattice Figure 3.2: Construction of the [4, 1, 4] 0 code. Two encoded qubits are con- structed using the four physical qubits in the upper and the lower halves of the unit cell. In (a), the dotted lines represent the penalty terms. The solid lines form the encoded Hamiltonian couplings. Since each encoded coupling is formed by 2 physical couplings, the energy scale of the Ising problem is boosted by factor of 2. In (b) we show the section of the encoded graph formed by (a), with the same color scheme for the couplings. Roman letters labels the same encoded qubits in (a) and (b). 3.2 QAC using the [4, 1, 4] 0 and [3, 1, 3] 1 codes 3.2.1 Layout We first briefly review the layout of the[4, 1, 4] 0 and [3, 1, 3] 1 codes, which are both quantum repetition codes against bit-flip errors. The qubits on the Chimera graph of the D-Wave device (depicted in Fig. 3.1) are arranged in a square grid of unit cells, where each unit cell forms a complete K 4,4 bipartite graph. This graph supports a number of QAC codes wherein each encoded qubit is represented by several physical qubits, which we call an “encoded group.” Figure 3.2 describes the 106 1 2 3 p 1 2 3 p 1 2 3 p 1 2 3 p 1 2 3 p 1 2 3 p a b c d e f (a) Arrangement of physical qubits a c e b d f (b) Logical qubit lattice Figure 3.3: Construction of the [3, 1, 3] 1 code. An encoded qubit is constructed using three data qubits from each vertical half of the unit cell and a penalty qubit from the opposite half. In (a), the four physical qubits forming the encoded group are shown in the same color, and the dashed lines represent the stabilizer couplings. The solid lines form the encoded Hamiltonian coupling. Since each encoded coupling comprises 3 physical couplings, the energy scale of the encoded problem is boosted by a factor of 3. In (b) we show the section of the encoded graph formed by (a), with the same color scheme for the couplings. In both (a) and (b), the Roman alphabet labels the respective encoded qubits. [4, 1, 4] 0 code. We split the unit cell horizontally into two halves. The top and the bottom halves separately form two encoded qubits, where each of the four physical qubits are maximally connected via intra-cell ferromagnetic penalty couplings. The encoded qubit connects to the encoded qubits on each side via inter-cell problem couplings and also connects to the other encoded qubit in its unit cell via intra-cell problem couplings. On the Chimera graph, this generates an 8× 8× 2 lattice, as shown in Fig. 3.4. Incidentally, this is the two-level-grid (2LG) used in the original proof of the NP-hardness of the Ising model [124]. 107 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 (a) DW2-ISI device 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 (b) S6 device Figure 3.4: The [4, 1, 4] 0 encoded graph. Each encoded qubit is composed of four physical qubits, and the encoded couplings are formed from two physical couplers. The green (red) circles denote functional (inactive) qubits. Out of 128 possible encoded qubits on the complete graph, 120 were functional on the DW2-ISI device (a) and 99 on the S6 device (b). The encoded graph is a 2-level grid [124] and has degree 5. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 (a) DW2-ISI device 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 (b) S6 device Figure 3.5: The [3, 1, 3] 1 code encoded graph. Each encoded qubit is composed of four physical qubits, and the encoded couplings are formed from three physical couplers. The green (red) circles denote functional (inactive) qubits. Orange circles indicate encoded qubits that have all three data qubits but are missing their penalty qubits. Out of 128 possible encoded qubits on the complete graph, 120 were fully functional while 3 were missing penalty qubits on the DW2-ISI device (a); 95 were fully functional and 18 were missing penalty qubit on the S6 device (b). We only used fully functional encoded qubits in our experiments. The encoded graph has degree 3. 108 The [3, 1, 3] 1 code splits a unit cell vertically into two halves. As shown in Fig. 3.3, the three qubits on either half of the unit cell are combined with one qubit on the opposite half of the cell, which plays the role of a dedicated penalty qubit, to form an encoded qubit. Each unit cell thus contains two encoded qubits. This construction gives rise to another encoded graph, which is shown in Fig. 3.5. While both the [4, 1, 4] 0 code and the [3, 1, 3] 1 code use the same number of physical qubits and generate a non-planar encoded graph, their encoded graphs differ in connectivity, with the [4, 1, 4] 0 code having the advantage of degree 5 over the degree 3 of the [3, 1, 3] 1 code. 3.2.2 Encoding Encoding is achieved by replacing the Pauli-Z operators in the Ising Hamiltonian in Eq. (3.1) by their encoded counterparts. Thus, the encoded Ising Hamiltonian can be written as H I = X i∈ V (G) h i Z i + X {(i,j)}∈ E(G) J ij Z i Z j , (3.3) where the h i and J ij values are inherited from the original problem Hamiltonian, Eq. (3.1), and where G⊂ G is the encoded graph. G is also a minor of G, i.e., it is formed by collapsing vertices and removing certain edges. To tie the encoded group together, we introduce an energy penalty Hamiltonian H P , which is the sum of the stabilizer generators of the code. The energy penalty Hamiltonian serves to energetically penalize differences among the physical qubits in the encoded group, which helps to suppress bit flip errors. With the energy penalty term included, the overall time-dependent Hamiltonian during the evolution is H(t) =A(t)H X +B(t)(α H I +γH P ) , (3.4) 109 whereH X =− P i X i is the original (unencoded) transverse field, and(α,γ ) are two controllable experimental parameters that can be varied in the range [0, 1] to control the relative strength of the problem and the penalty Hamiltonians. Because H P is a part of physical problem Hamiltonian H I it inherits the latter’s time-dependence, i.e., is turned on via the annealing schedule B(t). This aspect of QAC differs from standard error suppression [72]. Note that H X is itself a sum of bit-flip operators, so it plays a dual role: it is used to prepare the initial superposition state (its ground state), and is an “error” from the perspective of the penalty Hamiltonian. This is unavoidable in the setting of the D-Wave device, which (also unlike Ref. [72]) preventsH X from being encoded, as this would require many-body X ⊗ n terms, which are experimentally unavailable. Because of this tension there is an optimal penalty value γ that depends on α , the problem instance, and other variables. In particular, the optimal γ keeps decodable states lower in the energy spectrum. We shall return to this point later. 3.2.2.1 [4, 1, 4] 0 code For the [4, 1, 4] 0 code, the encoded Pauli-Z operators can be constructed from physical operators as follows: Z i 7→ Z i = 1 2 (Z 1 i +Z 2 i +Z 3 i +Z 4 i ) , (3.5) Z i Z j 7→ Z i Z j = X k Z k i Z k j , (3.6) where k runs over two of the four physical qubits depicted in Fig. 3.2(a), i.e., solid lines of the same color in that figure. By encoding in this fashion, we boost the 110 Ising problem energy scale uniformly by a factor of two. The penalty Hamiltonian is chosen as indicated by the dotted couplings in Fig. 3.2(a), i.e.: H P =− N X i=1 (Z 1 i Z 2 i +Z 1 i Z 3 i +Z 2 i Z 4 i +Z 3 i Z 4 i ) , (3.7) where henceforth N =|V (G)| denotes the number of encoded qubits. 3.2.2.2 [3, 1, 3] 1 code The [3, 1, 3] 1 code uses a similar construction, except a distinction is made between the four physical qubits in the encoded group. They are categorized into a single “penalty qubit” and three “data qubits,” depicted in Fig 3.3(a). Now the encoded Pauli-Z operators are constructed from physical operators as follows: Z i 7→ Z i =Z 1 i +Z 2 i +Z 3 i (3.8) Z i Z j 7→ Z i Z j = X k Z k i Z k j (3.9) where k runs over the three data qubits. This encoding boosts the energy scale by a factor of three, which is more than the boost provided by the [4, 1, 4] 0 code. The importance of this difference is discussed in detail below. The penalty Hamiltonian, which is again the sum of the stabilizer generators of the code, is formed by coupling the data qubits to the penalty qubit, i.e.: H P =− N X i=1 (Z 1 i Z P i +Z 2 i Z P i +Z 3 i Z P i ) . (3.10) 111 3.2.3 Decoding strategies The encoded state is decoded via a majority vote on the physical qubits in an encoded group. Since the number of qubits in the [4, 1, 4] 0 code is even, a majority vote alone does not suffice since ties are possible. To decode in such cases we follow two different decoding schemes (see also Ref. [87]): • Coin tossing (CT). We flip an unbiased coin to break each tie, i.e., we assign a random±1 value to each encoded qubit. This random decoding strategy serves as a baseline against which we can compare other strategies. • Energy Minimization (EM). The tied qubits can be treated as an Ising system with effective local fields (due to the now fixed decoded qubits) and couplings to other tied qubits. This (hopefully small) system is then solved exactly by explicitly checking the energy of all possible configurations. EM is guaranteed to give the lowest possible energy from decoding, and it remains a feasible decoding scheme as long as the size of the tied clusters does not scale with the size of the problem 1 . We call an excited physical state “decodable” if, upon applying either of the decoding procedures described above, the decoded state is an encoded ground state. When this happens, we declare a success. For a given decoding scheme at a given problem energy scale α , we always locate the optimal energy penalty strength γ opt that maximizes the success probability. 1 It was shown in Ref. [87] that in general this is related to the per-site percolation threshold of the encoded graph, though this is not relevant in the case of chains. 112 3.3 Benchmarking using antiferromagnetic chains We implement an (unencoded) N-qubit antiferromagnetic chain with the following Ising Hamiltonian: H I = N− 1 X i=1 Z i Z i+1 . (3.11) In order to quantify the performance of the QAC scheme, we also test two classical strategies and one additional quantum strategy [50, 86, 87]. • Unprotected (U): In this case we directly embed the Hamiltonian in Eq. (3.11) on the device hardware graph. Each run in which either one of the two degenerate ground states of the chain is found is then declared a success. • Classical (C): Here we use simple classical repetition, i.e., we run four copies of the chain in parallel. This uses an equal number of physical qubits as our QAC scheme. A run is then considered a success if at least one of the four copies is in one of the two ground states. If p is the success probability for the U case, then, assuming the chains running in parallel are independent, the success probability for the C strategy is 1− (1− p) 4 (i.e., at least one chain is correct). • Energy Penalty (EP): Here we encode into either the [4, 1, 4] 0 or the [3, 1, 3] 1 code, including the energy penalty, but we do not decode. That is, we declare a success only when the physical ground state of the physical graph obtained after encoding is observed (note that this graph is not a chain). We refer to this as the EP strategy since it relies only on the energy penalty, but not on decoding, to increase the success probability. 113 The comparison of the U, C, EP and QAC strategies allows us to isolate different aspects of our overall error correction strategy. The U case is the baseline against which all other strategies are measured. EP informs us about whether the energy penalty is helping. In order for QAC to be considered successful, it should exhibit better performance than the C strategy. The experiments detailed next were performed on two different programmable quantum annealing devices. The DW2 processor at the USC Information Sciences Institute (DW2-ISI) has 504 functional qubits with an operating temperature of 16± 1 mK. Another DW2 processor, at D-Wave Inc. in Burnaby (S6) had 476 operational qubitsandoperatedat 11±1mK 2 .Thesedevicesandtheunderlyingtechnologyhave been described before in detail in various publications (e.g., Refs. [29, 30, 52, 127]). All experiments were performed with 30 instances of randomly placed chains on the hardware graph. The error bars in all figures below are the standard error of the means calculated over these 30 instances. Each chain instance was run 1000 times (in a single programming cycle), and the fraction of successful runs was taken to be the success probability of an instance. 3.4 Experimental Results In this section we present our success probability results for the various strategies and the two devices tested, and we analyze these results from a number of different angles. All our results use optimized penalty values. 2 Both processors have meanwhile been dismantled. 114 3.4.1 Success Probability Comparison Figure 3.6 displays the DW2-ISI results. It shows that for the highest chain length studied ( ¯ N = 98) the [4, 1, 4] 0 code (with EM) is bested by the C strategy at higher α , but outperforms the C strategy at lower α , corresponding to a transition from a regimes of low to high error rates. The cross-over point occurs around α = 0.6. The [3, 1, 3] 1 code provides superior error correction at all scales for this largest size, confirming and extending the results of Ref. [86]. Figure 3.7 displays the same for the S6 device. Since this device operated at a lower temperature than the DW2-ISI device, we expect it to have a lower thermal excitation error rate. This means that the probability of multiple bit-flips per encoded qubit will decrease, i.e., more errors will be decodable, and so we can expect that—all else being equal—QAC will be more effective. This explains why the α cross-over point for the [4, 1, 4] 0 code shifts to higher values; it is now closer to α = 0.9. In order to demonstrate the independence of the four copies in the C strategy, we compare the performance of the U and C strategies in Fig 3.8. The C strategy’s performance is close to the one predicted for independent runs, indicating that chains indeed behave independently. The success probabilities are high for small chains, but rapidly drop as we increase the chain length. The same conclusion holds across the range of scaling parameter α . In Fig 3.9, we compare the results for the EP, CT, and EM decoding strategies. As expected, the EM strategy outperforms the EP and CT strategies when decoding chains. The small enhancement in the success probability of the EM strategy over CT indicates that the number of ties is correspondingly small. This is confirmed, along with additional results compare the various decoding strategies at other α values, in Section 3.8. 115 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer Unprotected(U) Classical(C) [3,1,3] 1 [4,1,4] 0 (a) α =1 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer Unprotected(U) Classical(C) [3,1,3] 1 [4,1,4] 0 (b) α =0.5 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer Unprotected(U) Classical(C) [3,1,3] 1 [4,1,4] 0 (c) α =0.4 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 α Probabilityofcorrectanswer Unprotected(U) Classical(C) [3,1,3] 1 [4,1,4] 0 (d) N =98 Figure 3.6: Results for the DW2-ISI device. Panels (a), (b) and (c) compare the results for chains using the U, C, [3, 1, 3] 1 code and [4, 1, 4] code at scaling parameters α = 1, 0.5 and 0.4 respectively. Panel (d) shows a comparison at a fixed chain length ofN = 98 as α is varied. For α . 0.5 and sufficiently long chain lengths, the [4, 1, 4] 0 code starts to outperform the C strategy. The [3, 1, 3] 1 code outperforms all other strategies at all values ofα , for sufficiently long chain lengths. 3.4.2 The role of energy scaling We now address the performance mismatch between the two QAC codes. Recall that QAC boosts the problem scale via a redundant representation of the Z i and Z i Z j operators. One may expect this energy boost to reduce errors due to the combination of two effects: thermal excitations are suppressed via the Boltzmann factor, and raising the overall problem energy scale reduces diabatic transitions 116 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer Unprotected(U) Classical(C) [3,1,3] 1 [4,1,4] 0 (a) α =1 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer Unprotected(U) Classical(C) [3,1,3] 1 [4,1,4] 0 (b) α =0.5 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer Unprotected(U) Classical(C) [3,1,3] 1 [4,1,4] 0 (c) α =0.4 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 α Probabilityofcorrectanswer Unprotected(U) Classical(C) [3,1,3] 1 [4,1,4] 0 (d) N =80 Figure 3.7: ResultsfortheS6device. Panels (a), (b) and (c) compare the results for chains using the U, C, [3, 1, 3] 1 code and [4, 1, 4] code at scaling parameters α = 1, 0.5 and 0.4 respectively. Panel (d) shows a comparison at a fixed chain length of N = 80 as α is varied. For α . 0.90 and for sufficiently long chains, the [4, 1, 4] 0 code starts to outperform the C strategy. The [3, 1, 3] 1 code outperforms all other strategies at all values of α , for sufficiently long chain lengths. by increasing the minimum gap during the evolution, though the latter effect is difficult to quantify without diagonalizing the full Hamiltonian H(t) 3 . As noted above, the [3, 1, 3] 1 code boosts the problem energy scale by a factor of three, while 3 See Ref. [50] for an analytically solvable model that exhibits an increased gap via this mechanism. 117 20 40 60 80 100 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer Unprotected(U) BestClassical(C) Binomialtheory Figure 3.8: Independence test. Using the data from the DW2-ISI device, we compare the performance of the U strategy to the C strategy at α = 0.45. The C strategy agrees with the prediction from binomial theory that assumes independent chains. the [4, 1, 4] 0 code boosts the problem energy scale by a factor of two, so we might expect the [3, 1, 3] 1 code to outperform the [4, 1, 4] 0 code on this basis alone. To compare the performance of the two codes we can equalize their effective problem energy scales, defined asα times the boost factor due to the redundancy in representing the Z i and Z i Z j operators. We set α = 0.3 for the [3, 1, 3] 1 code and α = 0.45 for the [4, 1, 4] 0 , so that, after the energy boost is accounted for, the effective scale of both is 0.9. We first compare the two using the EP strategy in order to eliminate the role of decoding. Figure 3.10(a) reveals that the two codes perform almost identically when tested at the same effective energy scale, indicating that the protection offered by both codes is quantitatively determined by this scale. Figure 3.10(b) shows the results after decoding, with a slight advantage for the majority vote decoding of the [3, 1, 3] 1 code over the energy minimization of the [4, 1, 4] 0 code. 118 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer EP, CTQAC, EnergyMinQAC Figure 3.9: Decoding strategies for the [4, 1, 4] 0 code. DW2-ISI device at α = 0.70. The EP strategy is marginally improved upon by the use of decoding. Ties are broken by either coin tossing or energy minimization; the latter performs slightly better at all chain lengths. 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 ChainlengthN Probabilityofcorrectanswer [3,1,3] 1 α=0.3 [4,1,4] 0 α=0.45 (a) EP 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 ChainlengthN Probabilityofcorrectanswer [3,1,3] 1 α=0.3 [4,1,4] 0 α=0.45 (b) QAC Figure 3.10: Code comparison at equalized effective energy scales for the DW2-ISI device. Panel (a) compares the EP performance of the two codes at equivalent effective energy scales: 3× 0.3 and 2× 0.45 for the [3, 1, 3] 1 and [4, 1, 4] 0 code, respectively. Panel (b) compares the two codes after decoding. Code performance is essentially indistinguishable in the EP case, indicating that the effective energy scale is the dominant performance-determining factor. The [3, 1, 3] 1 code exhibits a slight advantage across all chain lengths after decoding. 119 3.4.3 Decodable states: energy vs Hamming weight We expect both codes to enable correct decoding of physical states that are a small Hamming distance away from the physical ground state, but not when the physical state is far away in Hamming distance from the physical ground state. Figure 11 confirms this intuition. Additionally, we note that for both codes we can correct highly excited states, as long as they are within a small Hamming distance from the physical ground state. Thus, the requirement of remaining in the ground state throughout the evolution, which is impractical for non-zero temperature quantum annealers but is typically the condition imposed by adiabaticity in closed-system AQC, is seen to be overly restrictive in the present setting, since QAC is able to tolerate certain excitations out of the ground state. The observation that error correction for AQC or quantum annealing is designed to tolerate excitations has of course been made before, e.g., in Refs. [70, 79, 86]. Figure 3.11 reveals a striking difference between the [3, 1, 3] 1 and [4, 1, 4] 0 codes. The latter exhibits many undecodable states over the entire range of Hamming distances from the encoded ground state, starting from the second excited state. The [3, 1, 3] 1 code, on the other hand, exhibits a large Hamming distance separation between decodable and undecodable states, with the latter appearing only for relatively high excited states. This reflects the higher effectiveness of the penalty term in the [3, 1, 3] 1 code, and at the same time gives a detailed view of the different failure mechanisms of both codes. 3.4.4 The role of temperature Having access to two quantum annealers operating at two different temperatures and with different characteristics (see Table 3.1), we can compare the performance of the two devices at equivalent programming parameters. We show in Fig. 3.12 a 120 Physical Hamming distance 0 10 20 30 40 50 Energy relative to ground state 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) [3,1,3] 1 Physical Hamming distance 0 10 20 30 40 50 Energy relative to ground state 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) [4,1,4] 0 Figure 3.11: Decodable states. Panels (a) and (b) display the performance of the [3, 1, 3] 1 and [4, 1, 4] 0 codes, respectively, for the longest chain of length 100, α = 1, and optimal γ observed on the DW2-ISI device. We show the energies of all the states observed relative to the encoded ground state along the vertical axis, while the horizontal axis is the Hamming distance from the encoded ground state. We did not find any decodable states of Hamming distance higher than 50. Color indicates the fraction of decodable states at each observed energy and Hamming distance. States with a small Hamming distance are mostly decodable. The [3, 1, 3] 1 code is decoded via majority vote, while the [4, 1, 4] 0 code uses the EM strategy. correlation plot for instances encoded using the [4, 1, 4] 0 code, with equal γ opt for a given α . We observe a clear advantage for the S6 device, which we attribute to its lower operating temperature. 3.4.5 Behavior of the optimal energy penalty for the [4, 1, 4] 0 code It is instructive to study the dependence of the optimal penalty value onα and chain length N. Figure 3.13 shows the results of the optimization of success probabilities for the [4, 1, 4] 0 code on the two quantum annealing devices. The optimal penalty value scales with α , i.e., γ opt ∝ α , which is quite unlike the behavior of the [3, 1, 3] 1 code reported in Ref. [86], for which γ opt was found to be essentially constant (this 121 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Probabilityofcorrectanswer(DW2-ISI) Probabilityofcorrectanswer(S6) (a) EP 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Probabilityofcorrectanswer(DW2-ISI) Probabilityofcorrectanswer(S6) (b) EM Figure 3.12: Comparison of the performance of the ISI and S6 devices. Correlation plot of the success probability on the S6 and ISI devices for instances that have the same γ opt (for the same α and chain length) using the EP strategy [panel (a), with 264 instances] and the QAC with EM strategy [panel (b), with 376 instances]. Instances to the left (right) of the diagonal line have a higher success probability on the S6 (DW2-ISI) device. Virtually all instances were solved with a higher success probability on the S6 device. is reproduced in Section 3.7). We may perhaps attribute this difference to the fact that the [3, 1, 3] 1 code has a dedicated penalty qubit which is therefore not as sensitive to values of the problem couplings as are the data qubits of the [4, 1, 4] 0 code, which participate simultaneously in the penalty and problem Hamiltonians. Figure 3.13 also shows that lower values of γ opt were required on the S6 device, which can again be attributed to its lower operating temperature. Section 3.7 provides additional results, showing the full dependence of the success probabilities on the penalty values, α , and chain length. 3.5 Theoretical Analysis In this section we provide a theoretical analysis of some of our results. In particular, we provide a simple thermodynamic explanation of the performance difference between the [4, 1, 4] 0 and [3, 1, 3] 1 codes, which we can attribute primarily to the 122 1 2 3 p 2v m mult dec 0 0 0 0 0 -4 1 y 0 0 1 0 2 -2 3 y 0 1 1 0 4 0 3 n 1 1 1 0 6 2 1 n 0 0 0 1 6 -2 1 y 0 0 1 1 4 0 3 y 0 1 1 1 2 2 3 n 1 1 1 1 0 4 1 n (a) 1 2 3 4 2v m mult dec 0 0 0 0 0 -4 1 y 0 0 0 1 4 -2 4 y 0 0 1 1 4 0 2 t 0 1 0 1 4 0 2 t 0 1 1 0 8 0 2 t 0 1 1 1 4 2 4 n 1 1 1 1 0 4 1 n (b) Table 3.2: Comparison of the two codes for a 1-qubit encoded problem with a local field. (a) [3, 1, 3] 1 code. The first three columns are the values of the data qubits; the fourth column is the penalty qubit; the fifth is the energy penalty counted as twice the number of violated couplings v; the sixth is the magnetization m =− P i s i ; the seventh is the multiplicity, which is the number of states that are topologically equivalent to the state shown, up to relabeling of qubits; and the last is the decodability of the state. (b) [4, 1, 4] 0 code. The first four columns are the values of the data qubits; the fifth is the energy penalty; the sixth is the magnetization m =− P i s i ; the seventh is the multiplicity, which is number of states that are topologically equivalent to the state shown, up to relabeling of qubits; and the last is the decodability, where “t” denotes a tie. Rows 3–5 denote three distinct ways of placing the two plus and two minus states, where the qubits are numbered 1–4 clockwise, starting from top-left [as in Fig. 3.2(a)]. effective energy scale. In addition, we explain the decodability of the [4, 1, 4] 0 code in terms of an intuitively appealing criterion of the ordering of decodable vs undecodable excited states. 3.5.1 Thermodynamic comparison To gain a better understanding, we compare the theoretical performance of the two codes for decoding a single encoded qubit using a simple thermodynamic argument (see Ref. [89] for a much more detailed analysis of the QAC partition function along the annealing evolution, for a fully connected ferromagnetic transverse field Ising model). We assume the presence of a local field of strengthh acting on the encoded 123 qubits; this would translate to a local field of ˜ h =h/2 on all physical qubits of the [4, 1, 4] 0 code [by Eq. (3.5)], and a local field of ˜ h =h on the three data qubits of the [3, 1, 3] 1 code [by Eq. (3.8)]. If h< 0, then the state|0000i is the ground state of the system. Success would be declared if the evolution takes the system to this state, or if of the final state is correctly decodable to|0i. Tables 3.2(a) and 3.2(b) enumerate all 16 cases along with their energy penalty and decodability for the two codes. Let us now assume that the state at the end of the anneal is thermal. The Boltzmann weight of any of the 16 states is given by e − β (2vγ − ˜ hm) /Z where m = − P i s i is the magnetization, v counts the number of violated couplings, β = 1/kT is the inverse temperature, and Z is the partition function. The probability p err for an error in the [3, 1, 3] 1 code case is the sum of the Boltzmann factors of the undecodable states, while for the [4, 1, 4] 0 code we must also include the tied cases with a factor of 1/2, assuming that these cases are decoded by coin tossing. We write Z =W +W 0 where W is the sum of the unnormalized Boltzmann factors for the encoded error cases (rows with ‘n’ in the decodability column of tables 3.2(a) and 3.2(b), and half of the ‘t’ cases in table 3.2(b)) and W 0 is the sum of the decodable cases (rows with ‘y’ in the decodability column of tables 3.2(a) and 3.2(b), and the other half of the ‘t’ cases in table 3.2(b)). The error probabilities p [3]/[4] err are functions of γ and h, where the labels [3]/[4] label the [3, 1, 3] 1 /[4, 1, 4] 0 code respectively: 124 p [3] err = W [3] W [3] +W 0 [3] , p [4] err = W [4] W [4] +W 0 [4] , (3.12) W [3] = 3e − β (4γ +h) +e − β (6γ +3h) + 3e − β (2γ +2h) +e − 3βh , (3.13) W 0 [3] = 3e − β (4γ − h) +e − β (6γ − 3h) + 3e − β (2γ − 2h) +e 3βh , (3.14) W [4] = 1 2 4e − 4βγ + 2e − 8βγ + 4e − β (4γ +h) +e − 2βh , (3.15) W 0 [4] = 1 2 4e − 4βγ + 2e − 8βγ + 4e − β (4γ − h) +e 2βh . (3.16) We minimize the error probabilities with respect toγ for each value ofh, noting that the optimal γ value is different for the two codes. Figure 3.14(a) shows the error rates of the two codes asβh is varied. We note that the [3, 1, 3] 1 code exhibits a lower error rate than the [4, 1, 4] 0 code. This agrees with our experimental findings and is a simple consequence of the [3, 1, 3] 1 code operating at a higher boosted energy scale than the [4, 1, 4] 0 code. We also compare the error rates at equivalent effective energy scales, i.e., 2h/3 for the [3, 1, 3] 1 code and h for the [4, 1, 4] 0 code. Figure 3.14(b) shows that at equivalent effective energy scales the two codes have similar error rates, with the [3, 1, 3] 1 code performing slightly worse for all βh values. This is the opposite of the experimental findings presented in Fig. 3.10 and suggests that the output of the D-Wave devices for these problems is not fully captured by the thermal model. Nevertheless, this analysis confirms that the error rate due to a thermal bath would be similar for the two codes, when operated at equivalent effective energy scales. 125 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 ChainlengthN γopt α = 0.4, α = 0.5, α = 1.0 (a) DW2-ISI device — EP 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 ChainlengthN γopt α = 0.4, α = 0.5, α = 1.0 (b) DW2-ISI device — QAC with EM decoding 20 40 60 80 0 0.2 0.4 0.6 0.8 1 ChainlengthN γopt α = 0.4, α = 0.5, α = 1.0 (c) S6 device — EP 20 40 60 80 0 0.2 0.4 0.6 0.8 1 ChainlengthN γopt α = 0.4, α = 0.5, α = 1.0 (d) S6 device — QAC with EM decoding Figure 3.13: Optimal γ for the [4, 1, 4] 0 code. Panels (a) and (b) show the optimal γ values for the DW2-ISI device. Panels (c) and (d) show the same for the S6 device. For three representative values of the scaling parameter α , we note that the EP strategy consistently requires a higher value for the optimal γ . There is a slight tendency for longer chains to have a larger optimal penalty. Additionally, since the DW2-ISI device operates at a higher temperature and hence is more prone to errors, it requires a higher value for the optimal γ than the S6 device. The difference is most prominent in the QAC case, i.e., when comparing panels (b) and (d). 126 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 h/kT Errorrate [4,1,4] 0 code, [3,1,3] 1 code (a) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 h/kT Errorrate [4,1,4] 0 code, Scaled[3,1,3] 1 code (b) Figure 3.14: Thermodynamic comparison of codes. Panel (a) shows the thermal error rates of the two codes at the same encoded energy scale h. Panel (b) compares the two codes at equivalent effective energy scales, i.e., h for the [4, 1, 4] 0 code and 2h/3 for the [3, 1, 3] 1 code. 127 ↑ ↑ ↑ ↑ ↓ ↓ ↓ ↓ (a) The ground state. ↑ ↑ ↑ ↑ ↓ ⇑ ↓ ↓ (b) One flip ( ×2). 2α + 4γ ↑ ↑ ↑ ↑ ⇑ ↓ ↓ ↓ (c) One flip ( ×2). 4γ ↑ ↑ ↑ ↑ ↓ ⇑ ⇑ ↓ (d) Two flips ( ×1). 4α +8γ ↑ ↑ ↑ ↑ ⇑ ↓ ↓ ⇑ (e) Two flips ( ×1). 8γ ↑ ↑ ↑ ↑ ↓ ⇑ ↓ ⇑ (f) Two flips ( ×4). 2α +4γ ↑ ↑ ↑ ↑ ⇑ ⇑ ↓ ⇑ (g) Three flips ( ×2). 2α +4γ ↑ ↑ ↑ ↑ ⇑ ⇑ ⇑ ↓ (h) Three flips ( ×2). 4α +4γ ↑ ↑ ↑ ↑ ⇑ ⇑ ⇑ ⇑ (i) Four flips ( ×1). 4α Figure 3.15: Two coupled encoded [4, 1, 4] 0 code qubits. The two encoded [4, 1, 4] 0 code qubits consists of 8 physical qubits. We show one of the two degenerate ground states, and all inequivalent ways in which bit-flip errors might accumulate on one of the encoded qubits. Each flip leads to an excited state. The number in parentheses denotes the multiplicity of such states, followed by the energy separation from the shown ground state. The↑ symbol denotes the correct state of the qubit while the⇑ symbol denotes the occurrence of a flip. 128 3.5.2 Decodability of the [4, 1, 4] 0 code In order to study the decodability of the [4, 1, 4] 0 code, and in particular the effect of varying the penalty strength γ , we consider two antiferromagnetically coupled encoded qubits, decoded via EM. In Fig. 3.15, we show how different bit-flip errors can accumulate on a two qubit chain, pushing the system into one of the excited states, and the energy gap of these excited state from one of the ground state. We show in Fig. 3.16(a) the spectrum of these excited states and whether they can or cannot be decoded at α = 0.3, as a function of γ . For sufficiently high γ a non-decodable state becomes lower in energy than a decodable state. This coincides with the optimalγ value from a quantum adiabatic master equation simulation [45]. When the EP strategy is used instead of EM, the optimalγ occurs at a larger value as shown in Fig. 3.16(b). 3.6 Conclusions Quantum annealing will require error correction in order to become a scalable form of quantum information processing. While our results depend heavily on the Chimera architecture of the D-Wave devices, it is only possible to make progress in the field of experimental quantum error correction by studying specific devices that provide snapshots of evolving technologies (e.g., Refs. [119, 120, 128]). With this caveat in mind, our study contains several valuable long-term lessons. Specifically, in this chapter we studied two quantum annealing correction codes— the [4, 1, 4] 0 and [3, 1, 3] 1 codes—and compared their performance using the simple case of antiferromagnetic chains, on two different experimental platforms belonging to the same generation of D-Wave Two devices. The two codes differ in the energy boost they provide by redundantly encoding their logical operators, and the two 129 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 γ EnergyGap Yes(c) Yes/No(b),(f),(g) Yes(e) Yes/No(d),(h) No(i) (a) 0 0.2 0.4 0.6 0.8 1 0.9 0.92 0.94 0.96 0.98 1 γ Probabilityofcorrectanswer EP QAC 0 1 0 1 (b) Figure 3.16: Decodability analysis forα = 0.3. (a) The first few excited states at the final time for two coupled[4, 1, 4] 0 code qubits. The legend labels refer to the states in Fig. 3.15, indicating their decodability. The optimal γ value occurs right before the first excited state goes from being decodable to undecodable. Logical error states are represented by solid lines, decodable states by dashed lines. Thick lines indicate degenerate excited states, some of which are decodable, and some of which are encoded errors. (b) Success probability calculated using the adiabatic master equation [45]. Inset: a zoomed out version. The success probability of the QAC strategy is maximized near γ ≈ 0.3, which agrees with the value of γ in (a) where an undecodable state becomes the first excited state. Population lost to this state in the simulations cannot be recovered after decoding. This explains the comment made in Sec. 3.2.2 that the optimal γ keeps the decodable states lower in the energy spectrum. The EP strategy is optimized at a larger value of γ . quantum annealers differ in operating temperature. We have shown that these differences translate into performance gains as expected, i.e., both a higher energy boost and a lower operating temperature result in improved success probabilities. This conclusion has immediate implications for the design of future quantum annealing devices: despite results indicating that thermal effects can assist AQC [101], and our (and previous [70, 79, 86]) results supporting the notion that error correction can tolerate thermal excitations, significant performance gains are to be realized via the straightforward mechanisms of cooling and increasing the energy scale. 130 Despite delivering lower success probabilities, the [4, 1, 4] 0 code with the smaller energy boost is interesting, since it gives rise to an encoded graph with higher degree than the [3, 1, 3] 1 code, and physical implementations of quantum annealers are likely to be subject in general to constraints that reduce connectivity. The trade-off between code performance and the degree of the encoded graph may thus be worthwhile, as long as an improvement over purely classical error correction strategies is achieved, as we have demonstrated here for sufficiently high noise levels and problem sizes. An intriguing question is whether this trade-off is necessary. Our work will hopefully inspire the design of quantum annealing architectures with higher connectivity and of codes that better leverage encoded graph degree and energy boosts. 131 3.7 Optimizing γ For each chain instance, we identified the optimal penalty coupling strengthγ by varying it in increments of 0.1 in the range [0, 1]. This is shown in Figs. 3.17-3.20 where we plot the success probability as a function of γ and N. We note that for the [4, 1, 4] 0 code the optimal penalty scales with α , i.e., γ opt ∝ α . Lower values of γ opt are observed on the S6 device. For the [3, 1, 3] 1 code, the optimal γ is around γ ≈ 0.2− 0.3 for all α values studied, and the optimal values are unchanged across the two devices. 3.8 Comparing decoding strategies In the main text we compared four strategies: U, C, the [4, 1, 4] 0 code, and [3, 1, 3] 1 code. We also used different decoding strategies: EM, EP, and CT. Figure 3.21 and Fig. 3.22 show all these strategies for a few chosen values of the scaling parameter α for the DW2-ISI and S6 devices, respectively. The U strategy is always worst. The [3, 1, 3] 1 code can be seen to outperform all other strategies at each α value for sufficiently long chains. The [4, 1, 4] 0 code outperforms the C strategy below a device-dependentα value and for sufficiently long chains. The fact that the success probabilities of the CT and EM strategies are nearly equal suggests that there are very few tied qubits in the [4, 1, 4] 0 -encoded chains, an observation that holds for both devices. In the main text we also presented indirect evidence for the small number of ties in the the [4, 1, 4] 0 code. Figure 3.23 shows this directly. 132 0 0.2 0.4 0.6 0.8 1 20 40 60 80 100 0 0.5 1 AFMchainlengthN Penaltystrengthγ (a) α =0.4 EP 20 40 60 80 100 0 0.5 1 AFMchainlengthN Penaltystrengthγ (b) α =0.5 EP 20 40 60 80 100 0 0.5 1 AFMchainlengthN Penaltystrengthγ (c) α =1.0 EP 20 40 60 80 100 0 0.5 1 AFMchainlengthN Penaltystrengthγ (d) α =0.4 QAC (EM) 20 40 60 80 100 0 0.5 1 AFMchainlengthN Penaltystrengthγ (e) α =0.5 QAC (EM) 20 40 60 80 100 0 0.5 1 AFMchainlengthN Penaltystrengthγ (f) α =1.0 QAC (EM) Figure 3.17: Optimalγ for the [4, 1, 4] 0 code on the DW2-ISI device. The color scale represents the success probability, while the white circles indicate the optimal penalty value for a given chain length. The top and bottom three panels show the EP and QAC with EM strategies, respectively. The optimal γ increases proportionally to the problem scale α . A higher γ opt is required for the EP case, where we perform no decoding. The success probability depends strongly on N, γ , and α . 133 0 0.2 0.4 0.6 0.8 1 20 40 60 80 100 0 0.5 1 AFMchainlengthN Penaltystrengthγ (a) α =0.4 EP 20 40 60 80 100 0 0.5 1 AFMchainlengthN Penaltystrengthγ (b) α =0.5 EP 20 40 60 80 100 0 0.5 1 AFMchainlengthN Penaltystrengthγ (c) α =1.0 EP 20 40 60 80 100 0 0.5 1 AFMchainlengthN Penaltystrengthγ (d) α =0.4 QAC 20 40 60 80 100 0 0.5 1 AFMchainlengthN Penaltystrengthγ (e) α =0.5 QAC 20 40 60 80 100 0 0.5 1 AFMchainlengthN Penaltystrengthγ (f) α =1.0 QAC Figure 3.18: Optimalγ for the [3, 1, 3] 1 code on the DW2-ISI device. The optimal γ increases proportionally to the problem scale α in the EP case, but remains fairly constant for the QAC case, in agreement with Ref. [86]. The success probability depends strongly on N and α , but not as strongly on γ . 134 0 0.2 0.4 0.6 0.8 1 20 40 60 80 0 0.5 1 AFMchainlengthN Penaltystrengthγ (a) α =0.4 EP 20 40 60 80 0 0.5 1 AFMchainlengthN Penaltystrengthγ (b) α =0.5 EP 20 40 60 80 0 0.5 1 AFMchainlengthN Penaltystrengthγ (c) α =1.0 EP 20 40 60 80 0 0.5 1 AFMchainlengthN Penaltystrengthγ (d) α =0.4 QAC (EM) 20 40 60 80 0 0.5 1 AFMchainlengthN Penaltystrengthγ (e) α =0.5 QAC (EM) 20 40 60 80 0 0.5 1 AFMchainlengthN Penaltystrengthγ (f) α =1.0 QAC (EM) Figure 3.19: Optimalγ for the [4, 1, 4] 0 code on the S6 device. The optimal γ increases proportionally to the problem scale α . A higher γ opt is required for the EP case, where we perform no decoding. The success probability depends strongly on N, γ , and α . 135 0 0.2 0.4 0.6 0.8 1 20 40 60 80 0 0.5 1 AFMchainlengthN Penaltystrengthγ (a) α =0.4 EP 20 40 60 80 0 0.5 1 AFMchainlengthN Penaltystrengthγ (b) α =0.5 EP 20 40 60 80 0 0.5 1 AFMchainlengthN Penaltystrengthγ (c) α =1.0 EP 20 40 60 80 0 0.5 1 AFMchainlengthN Penaltystrengthγ (d) α =0.4 QAC 20 40 60 80 0 0.5 1 AFMchainlengthN Penaltystrengthγ (e) α =0.5 QAC 20 40 60 80 0 0.5 1 AFMchainlengthN Penaltystrengthγ (f) α =1.0 QAC Figure 3.20: Optimalγ on the S6 device for the [3, 1, 3] 1 code. The optimal γ increases proportionally to the problem scaleα in the EP case, but remains fairly constant for the QAC case, again in agreement with Ref. [86] (though note that the latter used the DW2-ISI device). The success probability depends strongly on N and α , but not as strongly on γ . 136 Unprotected(U) Classical(C) [3,1,3] 1 EP [4,1,4] 0 EP [3,1,3] 1 QAC [4,1,4] 0 CTQAC [4,1,4] 0 EnergyMinQAC 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer (a) α =0.1 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer (b) α =0.3 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer (c) α =0.5 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer (d) α =0.7 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer (e) α =0.9 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer (f) α =1.0 Figure 3.21: Performance comparison for the DW2-ISI device. The [4, 1, 4] 0 code starts to outperform the C strategy below α ≈ 0.5 and for sufficiently long chains. 137 Unprotected(U) Classical(C) [3,1,3] 1 EP [4,1,4] 0 EP [3,1,3] 1 QAC [4,1,4] 0 CTQAC [4,1,4] 0 EnergyMinQAC 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer (a) α =0.1 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer (b) α =0.3 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer (c) α =0.5 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer (d) α =0.7 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer (e) α =0.9 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 ChainlengthN/N Probabilityofcorrectanswer (f) α =1.0 Figure 3.22: Performance comparison for the S6 device. The [4, 1, 4] 0 code starts to outperform the C strategy below α ≈ 0.9 and for sufficiently long chains. 138 0.10 0.20 0.30 0.40 0.45 0.50 0.60 0.70 0.80 0.90 1.00 0 20 40 60 80 100 0 2 4 6 × 10 − 4 ChainlengthN Tiesperqubit(t/N) Figure 3.23: Ties in the [4, 1, 4] 0 code. The number of ties per qubit is shown for each value ofα ∈ [0.1, 1.0], minimized over the penalty strength γ . The number of ties generally increases with the chain length N and decreases with α . The largest number of ties per qubit is∼ 5× 10 − 4 . 139 Chapter 4 Finite temperature quantum annealing solving exponentially small gap problem with non-monotonic success probability 4.1 Introduction Quantum annealing (QA) [19–26], also known as the quantum adiabatic algo- rithm [17, 18] or adiabatic quantum optimization [129, 130] is a heuristic quantum algorithm for solving combinatorial optimization problems. Starting from the ground state of the initial Hamiltonian, typically a transverse field, the algorithm relies on continuously deforming the Hamiltonian such that the system reaches the final ground state, typically of a longitudinal Ising model—thus solving the optimization problem—with high probability. In the closed-system setting, the adiabatic theorem of quantum mechanics [15] provides a guarantee that QA will find the final ground state if the run-time is sufficiently large relative to the inverse of the quantum ground state energy gap [125, 126]. However, this does not guarantee that QA will generally perform better than classical optimization algorithms. In fact, it is well known that QA, implemented as a transverse field Ising model, can result in dramatic slowdowns relative to classical algorithms even for very simple 140 optimization problems [130–134]. Generally, this is attributed to the appearance of exponentially small gaps in such problems [16]. A case in point is the ferromagnetic Ising spin chain with alternating coupling strength and open boundary conditions studied by Reichardt [130]. The ‘alternating sectors chain’ (ASC) of lengthN spins is divided into equally sized sectors of sizen of ‘heavy’ couplings W 1 and ‘light’ couplings W 2 , with W 1 >W 2 > 0. Since all the couplings are ferromagnetic, the problem is trivial to solve by inspection: the two degenerate ground states are the fully-aligned states, with all spins pointing either up or down. However, this simple problem poses a challenge for closed-system QA since the transverse field Ising model exhibits an exponentially small gap in the sector size n [130], thus forcing the run-time to be exponentially long in order to guarantee a constant success probability. A related result is that QA performs exponentially worse than its imaginary-time counterpart for disordered transverse field Ising chains with open boundary conditions [135], where QA exhibits an infinite-randomness critical point [136]. As a corollary, we may naively expect that for a fixed run-time, the success probability will decrease exponentially and monotonically with the sector size. While such a conclusion does not follow logically from the adiabatic theorem, it is supported by the well-studied Landau-Zener two-level problem [137–139]. How relevant are such dire closed-system expectations for real-world devices? By varying the sector size of the ASC problem on a physical quantum annealer, we find a drastic departure from the above expectations. Instead of a monotonically decreasing success probability (at constant run-time), we observe that the success probability starts to grow above a critical sector size n ∗ , which depends mildly on the chain parameters (W 1 ,W 2 ). We explain this behavior in terms of a simple open-system model whose salient feature is the number of thermally accessible 141 states from the instantaneous ground state at the quantum critical point. The scaling of this ‘thermal density of states’ is non-monotonic with the sector size and peaks at n ∗ , thus strongly correlating with the success probability of the quantum annealer. Our model then explains the success probability behavior as arising predominantly from the number of thermally accessible excitations from the ground state, and we support this model by adiabatic master equation simulations. Our result does not imply that open-system effects can lend an advantage to QA, and hence it is different from proposed mechanisms for how open system effects can assist QA. For example, thermal relaxation is known to provide one form of assistance to QA [64, 65, 101, 140], but our model does not use thermal relaxation to increase the success probability aboven ∗ . We note that Ref. [101] introduced the idea that significant mixing due to open system effects (beyond relaxation) at an anti-crossing between the first excited and ground states could provide an advantage, and its theoretical predictions were supported by the experiments in Ref. [94]. In Ref. [101] an analysis of adiabatic Grover search was performed (a model which cannot be experimentally implemented in a transverse field Ising model), along with numerical simulations of random field Ising models. In contrast, here we treat an analytically solvable model that is also experimentally implementable using current quantum annealing hardware. We also compare our empirical results to the predictions of the classical spin- vector Monte Carlo model [48], and find that it does not adequately explain them. Our study lends credence to the notion that the performance of real-world QA devices can differ substantially from the scaling of the quantum gap. This chapter is organized as follows. In Section 4.2, we review the alternating sectors chain model that is the work-horse of this project. We discuss the experimental observations in Section 4.3. We explain our observations with theoretical analysis in Sections 4.4 142 W 1 W 1 W 1 W 1 W 1 W 1 W 2 W 2 W 2 Figure 4.1: Illustration of an alternating sector chain (ASC). This example has sector size n = 3, length N = 10 and number of sectors 2b + 1 = 3. Red lines denote the heavy sector with coupling W 1 , blue lines denote the light sector with coupling W 2 <W 1 . to 4.6. We compare these results to the SVMC model in Section 4.7. We conclude in Section 4.8. 4.2 The alternating sectors chain model We consider the transverse Ising model with a time-dependent Hamiltonian of the form: H(s) =− A(s) N X i=1 σ x i +B(s)H ASC , (4.1) where t f is the total annealing time, s = t/t f ∈ [0, 1], and A(s) and B(s) are the annealing schedules, monotonically decreasing and increasing, respectively, satisfying B(0) = 0 and A(1) = 0. The alternating sectors chain Hamiltonian is H ASC =− N− 1 X i=1 J i σ z i σ z i+1 , (4.2) where for a given sector size n the couplings are given by J i =          W 1 ifdi/ne is odd W 2 otherwise (4.3) Thustheb+1odd-numberedsectorsare‘heavy’(J i =W 1 ), andthebeven-numbered sectors are ‘light’ (J i =W 2 ) for a total of 2b + 1 = N− 1 n sectors. This is illustrated in Fig. 4.1. 143 N 174 175 172 173 176 175 176 169 172 171 181 170 183 177 172 181 n 1 2 3 4 5 6 7 8 9 10 12 13 14 16 19 20 Table 4.1: Chain length (N) and sector size (n) for N∼ 175. We briefly summarize the intuitive argument of Ref. [130] for the failure of QA to efficiently solve the ASC problem. Consider the N 1 and n 1 limit, where any given light or heavy sector resembles a uniform transverse field Ising chain. Each such transverse field Ising chain encounters a quantum phase transition separating the disordered phase and the ordered phase when the strength of the transverse field and the chain coupling are equal, i.e., whenA(s) =B(s)J i [141]. Therefore the heavy sectors order independently before the light sectors during the anneal. Since the transverse field generates only local spin flips, QA is likely to get stuck in a local minimum with domain walls (anti-parallel spins resulting in unsatisfied couplings) in the disordered (light) sectors, ift f is less than exponential in n. We note that this mechanism, in which large local regions order before the whole is well-known in disordered, geometrically local optimization problems, giving rise to a Griffiths phase [136]. This argument explains the behavior of a closed-system quantum annealer operating in the adiabatic limit. To check its experimental relevance, we next present the results of tests performed with a physical quantum annealer operating at non-zero temperature. 4.3 Empirical results As an instantiation of a physical quantum annealer we used a D-Wave 2X (DW2X) processor. The D-Wave 2X processor (DW2X) is an 1152-qubit quantum annealing device made by D-Wave Systems, Inc., using superconducting flux qubits [31]. The 144 0 5 10 15 20 10 −4 10 −3 10 −2 n P G 0 5 10 15 20 10 −4 10 −3 10 −2 10 −1 10 0 n 0 5 10 15 20 10 −4 10 −3 10 −2 n 0 5 10 15 20 5 10 15 20 n k ∗ 0 5 10 15 20 0 10 20 n 0 5 10 15 20 0 10 20 n (a) (b) (c) (d) (e) (f) (1.0,0.50, 55) (1.0,0.50,175) (0.5,0.25,175) (0.6,0.30,135) (0.8,0.40,175) (1.0,0.50,200) 0 5 10 15 20 10 2 10 3 10 − 2 10 − 1 10 0 Figure 4.2: Empirical success probability vs k ∗ for the ASC problem on the DW2X processor. k ∗ denotes the number of single-fermion energies that fall belowthethermalenergygapatthepointoftheminimumgaps ∗ . Thelegendentries indicate the chain parameters: (W 1 ,W 2 ,N). The error bars everywhere indicate 95% confidence intervals calculated using a bootstrap [103, 142] over different gauges and embeddings. (a)-(c) Contrary to closed-system theory expectations, the success probability P G is non-monotonic in the sector size n, first decreasing and then increasing, exponentially. Inset (a): The minimum gap (in GHz) of the chains as a function of the sector size n∈{ 1,..., 20}. The solid black line denotes the operating temperature energy scale of the DW2X. (d)-(f) For all chains we studied the ground state success probability has a minimum at the sector size n ∗ where the peak in the number of single-fermion states k ∗ occurs [compare with (a)-(c)]. The rise and fall pattern, as well as the location of n ∗ , are in agreement with the behavior of P G within the error bars. Inset (d): The total number of energy eigenstates that fall below the thermal energy gap as a function of the sector size n. In this case the peak position does not agree with the ground state success probability minimum. 145 particular processor used in this study is located at the University of Southern California’s Information Sciences Institute, with 1098 functional qubits and an operating temperature of 12 mK. The total annealing timet f can be set in the range [5, 2000] μ s. See Section 1.2 for more details about the processor. We generated a set of ASCs with chains lengths centered at N∼{ 55, 135, 175, 200} and with sector sizes n ranging from 2 to 20. Since the chain length and sector size must obey the relation (N− 1)/n = 2b + 1 with integer b, there is some variability in N. Table 4.1 gives the (N,n) pair combinations we used for chain set with mean length 175. We usedt f = 5μ s. For each ASC instance we implemented 10 different embeddings, with 10 gauge transforms each [43]. In total, 10 5 runs and readouts were taken per instance. The reported success probability is defined as the fraction of readouts corresponding to a correct ground state. Figures 4.2(a)-4.2(c) show the empirical success probability results for a fixed annealing time t f = 5μ s. Longer annealing times do not change the qualitative behavior of the results, but do lead to changes in the success probability (we provide these results in A.6). A longer annealing time can result in more thermal excitations near the minimum gap, but it may also allow more time for ground state repopulation after the minimum gap. The latter can be characterized in terms of a recombination of fermionic excitations by a quantum-diffusion mediated process [143]. Unfortunately, we cannot distinguish between these two effects, as we only have access to their combined effect in the final-time success probability. In stark contrast to the theoretical closed-system expectation, the success prob- ability does not decrease monotonically with sector size, but exhibits a minimum, after which it grows back to close to its initial value. The decline as well as the initial rise are exponential in n. Longer chains result in a lower P G and a more pronounced minimum, but the position of the minimum depends only weakly on 146 the chain parameter values (W 1 ,W 2 ) (the value ofn ∗ shifts to the right as (W 1 ,W 2 ) are increased) but not on N. What might explain this behavior? Clearly, a purely gap-based approach cannot suffice, since the gap shrinks exponentially in n for the ASC problem [130] [see also the inset of Fig. 4.2(a)]. However, for all chain parameters we have studied, the temperature is greater than the quantum minimum gap. In this setting not only the gap matters, but also the number of accessible energy levels that fall within the energy scale set by the temperature. In an open-system description of quantum annealing [45, 64, 68, 69, 144, 145], both the Boltzmann factor exp(− β Δ) (β denotes the inverse temperature and Δ is the minimum gap) and the density of states determine the excitation and relaxation rates out of and back to the ground state. As we demonstrate next, the features of the DW2X success probability results, specifically the exponential fall and rise withn, and the position of the minimum, can be explained in terms of the number of single-fermion states that lie within the temperature energy scale at the critical point. 4.4 Fermionization We can determine the spectrum of the quantum Hamiltonian [Eq. (A.1)] by trans- forming the system into a system of free fermions with fermionic raising and lowering operators η † k and η k [141, 146]. The result is [130]: H(s) =E g (s) + N X k=1 λ k (s)η † k η k , (4.4) 147 where E g (s) is the instantaneous ground state energy and{λ k (s)} are the single- fermion state energies, i.e., the eigenvalues of the linear system ~ Φ k (s)(A− B)(A + B) =λ 2 k (s) ~ Φ k (s) , (4.5) where the matrices A and B are tridiagonal and are given in Appendix A.2 along with full details of the derivation. The vacuum of the fermionic system|0i is defined by η k |0i = 0∀ k and is the ground state of the system. Higher energy states correspond to single and many-particle fermionic excitations of the vacuum. At the end of the anneal, fermionic excitations corresponds to domain walls in the classical Ising chain (see Appendix A.3). TheIsingproblemisZ 2 -symmetric, sothegroundstateandthefirstexcitedstate of the quantum Hamiltonian merge towards the end of evolution to form a doubly degenerate ground state. Since any population in the instantaneous first excited state will merge back with the ground state at the end of the evolution, the relevant minimum gap of the problem is the gap between the ground state and the second excited state: Δ(s) = λ 2 (s), which occurs at the point s ∗ = argmin s∈ [0,1] Δ(s). In the thermodynamic limit, this point coincides with the quantum critical point where the geometric mean of the Ising fields balances the transverse field,A(s ∗ ) = √ W 1 W 2 B(s ∗ ) [147, 148]. Henceforth we write Δ≡ Δ(s ∗ ) for the minimum gap. 4.5 Spectral analysis Letk ∗ be the number of single-fermion states with energy smaller than the thermal gap at the critical point, i.e., k ∗ = argmax k λ k (s ∗ )<T . (4.6) 148 As can be seen by comparing Figs. 4.2(d)-4.2(f) to Figs. 4.2(a)-4.2(c), we find that the behavior of k ∗ correlates strongly with the ground state success probability for all ASC cases we tested, when we set T = 12 mK = 1.57 GHz, the operating temperature of the DW2X processor (we use k B = ~ = 1 units throughout). Specifically,k ∗ peaks exactly where the success probability is minimized, which strongly suggests thatk ∗ is the relevant quantity explaining the empirically observed quantum annealing success probability. Longer chains result in a larger value of k ∗ and a more pronounced maximum. Of all the ASC sets we tried, we only found a partial exception to this rule for the case (1, 0.5, 200), where k ∗ peaks at n ∗ = 5 [Fig. 4.2(e)] but the empirical success probability for n = 5 and n = 6 is roughly the same [Fig. 4.2(b)]. We show later that this exception can be resolved when the details of the energy spectrum are taken into account via numerical simulations. In contrast, the total number of energy eigenstates (including multi-fermion states) that lie within the thermal gap [E g (s ∗ ),E g (s ∗ ) +T ], while rising and falling exponentially in n like the empirical success probability in Fig. 4.2(a), does not peak in agreement with the peak position of the latter [see the inset of Fig. 4.2(d)]. Why and how does the behavior of k ∗ explain the value ofn ∗ ? Heuristically, we expect the success probability to behave as P G ∼ 1− e − β Δ d , (4.7) where d is the ‘thermal density of states’ at the critical point s ∗ . Note that the role of the gap here is different from the closed-system case, since we are assuming that thermal transitions dominate over diabatic ones, so that the gap is compared to the temperature rather than the annealing time. Contrast this with the closed system case, where the Landau-Zener formula for closed two-level systems and 149 0 5 10 15 20 10 − 8 10 − 7 10 − 6 10 − 5 10 − 4 10 − 3 10 − 2 n Δ/2 k ∗ (1.0, 0.50, 175) (0.5, 0.25, 175) Figure 4.3: Ratio of the gap to the thermal density of states, as a function of sector size. Two alternating sector chain cases are shown. The position of the minimum is determined by d rather than Δ, as can be seen by comparing to Fig. 4.2(d) where the plot of d = 2 k ∗ alone correlates well with the position of minima in the empirical success probability curves. Hamiltonians analytic in the time parameter (subject to a variety of additional technical conditions) states that: P G ∼ 1− e − η Δ 2 t f , where η is a constant with units of time that depends on the parameters that quantify the behavior at the avoided crossing (appearing in, e.g., the proof of Theorem 2.1 in Ref. [139]). Since then P G =O(η Δ 2 t f ), we expect the success probability to decrease exponentially at constant run-time t f if the gap shrinks exponentially in the system size. Our key assumption is that the thermal transitions between states differing by more than one fermion are negligible. That is, thermal excitation (relaxation) only happens via creation (annihilation) of one fermion at a time (see A.7 for a detailed argument). Additionally, the Boltzmann factor suppresses excitations that require energy exchange greater thanλ k ∗ . Starting from the ground state, all single-fermion states with energy≤ E g +λ k ∗ are populated first, followed by all two-fermion states with total energy≤ E g +λ k ∗ +λ k ∗ − 1 , etc. In all, P k ∗ k=1 k ∗ k = 2 k ∗ − 1 excited states are thermally populated in this manner. Thus d ∼ 2 k ∗ states are thermally accessible from the ground state. 150 For a sufficiently small gap we have 1− e − β Δ ∼ β Δ, so that P G ∼ β Δ/d. As can be seen from Figs. 4.2(d)-4.2(f),k ∗ rises and falls steeply forn<n ∗ andn>n ∗ respectively. For the ASCs under consideration, d varies much faster with n than the gap Δ (see Fig. 4.3). Thus P G ∼ 2 − k ∗ . This argument explains both the observed minimum of P G at n ∗ and the exponential drop and rise of P G with n, in terms of the thermal density of states. In Appendix A.8 we give a more detailed argument based on transition rates obtained from the adiabatic master equation, which we discuss next. 4.6 Master equation model We now consider a simplified model of the open system dynamics in order to make numerical predictions. We take the evolution of the populations ~ p ={p a } in the instantaneous energy eigenbasis of the system to be described by a Pauli master equation [149]. The form of the Pauli master equation is identical to that of the adiabatic Markovian quantum master equation [45], derived for a system of qubits weakly coupled to independent identical bosonic baths. The master equation with an Ohmic bosonic bath has been successfully applied to qualitatively (and sometimes quantitatively) reproduce empirical D-Wave data [42, 44, 91, 150]. However, it does not account for 1/f noise [151], which may invalidate the weak coupling approximation when the energy gap is smaller than the temperature [51]. After taking diagonal matrix elements and restricting just to the dissipative (non-Hermitian) part one obtains the Pauli master equation [149] describing the evolution of the population ~ p ={p a } in the instantaneous energy eigenbasis of the system [69]: ∂p a ∂t = X b6=a γ (ω ba )M ab p b − X b6=a γ (ω ab )M ba p a . (4.8) 151 Here all quantities are time-dependent and the matrix elements are M ab (s) = N X α =1 |ha(s)|σ z α |b(s)i| 2 , (4.9) where we have assumed an independent thermal bath for each qubit α and where the indices a and b run over the instantaneous energy eigenstates of the system Hamiltonian (4.4) in the fermionic representation [i.e., H(s)|a(s)i =E a (s)|a(s)i] and ω ab =E a − E b is the corresponding instantaneous Bohr frequency. Since the basis we have written this equation in is time-dependent, there are additional terms associated with the changing basis [45], but we ignore these terms here since we are assuming that the system is dominated by the dissipative dynamics associated with its interaction with its thermal environment. The rates γ (ω ) satisfy the quantum detailed balance condition [152, 153], γ (− ω ) = e − βω γ (ω ), where ω ≥ 0. In our model each qubit is coupled to an independent pure-dephasing bath with an Ohmic power spectrum: γ (ω ) = 2πηg 2 ωe −| ω |/ω c 1− e − βω , (4.10) with UV cutoff ω c = 8π GHz and the dimensionless coupling constant ηg 2 = 1.2× 10 − 4 . The choice for the UV cutoff satisfies the assumptions made in the derivation of the master equation in the Lindblad form [45]. Note that we do not adjust any of the master equation parameter values, which are taken from Ref. [91]. Details about the numerical solution procedure are given in Methods, and in Appendix A.5 we also confirm that the validity conditions for the derivation of the master equation are satisfied for a relevant range ofn values given the parameters of our empirical tests. 152 Numerically solving the master equation while accounting for all thermally populated 2 k ∗ states is computationally prohibitive, but we can partly verify our interpretation by restricting the evolution of the system described in Eq. (4.8) to the vacuum and single-fermion states. This is justified in A.7, where we show that transitions between states differing by more than a single fermion are negligible. In other words, the dominant thermal transitions occur from the vacuum to the single-fermion states, from the single-fermion states to the two-fermion states, etc. The restriction to the vacuum and single-fermion states further simplifies the master equation (4.8) to: ˙ p 0 = X b γ (λ b )M b p b − p 0 X b γ (− λ b )M b (4.11) ˙ p i =γ (− λ i )M i p 0 − γ (λ i )M i p i , (4.12) where{p b } N b=1 are the single-particle fermion energy populations and{λ b } their ener- gies found by solving Eq. (4.5), andM ab [Eq. (4.9)] becomesM b = P N α =1 |h0|σ z α |bi| 2 . For a better approximation that accounts for more states, we can also perform a two-fermion calculation where we keep the vacuum, the firstk ∗ one fermion states and the next k ∗ (k ∗ − 1)/2 two fermion states. For two-fermion simulations the master equation becomes Eqs. (4.11) and (4.12) along with ˙ p i =γ (− λ i )M i p 0 − γ (λ i )M i p i + X j6=i γ (λ j )M j p ij − p i X j6=i γ (− λ i )M j (4.13) ˙ p ij =γ (− λ i )M i p j +γ (λ j )M j p i − γ (λ i )M i p ij − γ (λ j )M j p ij , (4.14) where all summations run from 1 to k ∗ , and p ij denotes the population in the two-particle fermion energy state η † i η † j |0i. 153 0 10 20 30 40 50 60 0 0.02 0.04 0.06 0.08 |ii P i 0 0.2 0.4 0.6 0.8 1 10 − 1 10 0 t/t f P G n = 1 n = 2 n = 5 n = 10 n = 16 (a) (b) 0.4 0.42 0.44 0.46 0.48 0.03 0.1 0.3 Figure 4.4: Master equation results for the state populations when re- stricting the excited states to single-fermion states. (a) The population in eachsingle-fermionstateatt =t f inaone-fermionsimulation. Thechainparameters are N = 176, W 1 = 1, W 2 = 0.5, t f = 5μ s, and n = 5. With the annealing schedule given in Methods, the quantum minimum gap is ats ∗ =t ∗ /t f ≈ 0.424. At this point we findk ∗ = 18 single-fermion states below the thermal energyT = 12mK (D-Wave processor operating temperature). As expected, in one-fermion simulations, most of the population is found in the firstk ∗ states. A long tail of more energetic single particle states beyond the firstk ∗ retain some population. (b) Evolution of the instantaneous ground state populations for ASCs with the same parameters as in (a), but for different sector sizes n and with two-fermion states. The ground state loses the majority of its population as it approaches the minimum gap point at t/t f =s ∗ . The largest drop is found for n =n ∗ = 5. Inset: Magnification of the region around the minimum gap. Relaxation plays essentially no role. Instead, the population freezes almost immediately. 154 5 10 15 20 10 − 2 10 − 1 n P G 1 ferm. g = 1 2 ferm. g = 1 2 ferm. g = 2 2 ferm. g = 1 2 (a) 0 5 10 15 20 10 − 2 10 − 1 n 0 5 10 15 20 10 − 2 10 − 1 n (b) (c) (1.0,0.50, 55) (1.0,0.50,175) (0.5,0.25,175) (0.6,0.30,135) (0.8,0.40,175) (1.0,0.50,200) Figure 4.5: Master equation results for the ground state population when restricting the excited states to single and two fermion states. (a) The result of simulating the ASC problem with parameters (1, 0.5, 175) via the adiabatic Pauli master equation (4.8), restricted to the vacuum + single-fermion states, and vacuum + single-fermion + two-fermion states. Also shown is the dependence on the system-bath coupling parameter g in the two-fermion case; doubling it has little impact, whereas halving it increases the success probability somewhat for n< 14. The position of the minimum at n ∗ = 5 matches the empirical result seen in Fig. 4.2(a), except when g = 1/2, i.e., the position is robust to doubling g but not to halving it. Panels (b) and (c) show additional 2-fermion master equation results with g = 1. Note that for the (1, 0.5, 200) chain, these simulations exhibit better agreement with the DW2X data than the simple k ∗ analysis plotted in Figs. 4.2(d)-4.2(f). This is because the simulations also keep track of the Boltzmann factor. 155 We can now numerically solve this system of equations. We solve the coupled differential Eqs. (4.11) to (4.14) using a fourth order Runge-Kutta method given by Dormand-Prince [154] with non-negativity constraints [155]. We compute the transition matrix elements via Eq. (A.37) and the bath correlation term via Eq. (4.10). As seen in Fig. 4.4(a), where we plot the final populations in the different single particle fermion states at t =t f for one-fermion simulations, only the firstk ∗ single- fermion levels are appreciably populated. This agrees with our aforementioned assumption that states with energy greater thanλ k ∗ are not thermally populated. In Fig. 4.4(b) we plot the population in the instantaneous ground state as a function of time for two-fermion simulations. The system starts in the gapped phase where the ground state population is at its chosen initial value of 1. The ground state rapidly loses population via thermal excitation as the system approaches the critical point, after which the population essentially freezes, with repopulation via relaxation from the excited states essentially absent (see inset). Thus, it is not relaxation that explains the increase in ground state population seen in Fig. 4.2(a)-(c) for n>n ∗ . Instead, we find that the ground state population drops most deeply forn =n ∗ . This, in turn, is explained by the behavior ofk ∗ seen in Fig. 4.2(d)-(f), as discussed earlier. We show in Fig. 4.5 the predicted final ground state population under the one and two-fermion restriction. This minimal model already reproduces the correct location of the minimum in P G . It also reproduces the non-monotonic behavior of the success probability. It does not correctly reproduce the exponential fall and rise. However, including the two-fermion states gives the right trend: it leads to a faster decrease and increase in the population without changing the position of the minimum, suggesting that a simulation with the full 2 k ∗ states would recover the 156 empirically observed exponential dependence of the ground state population seen in Fig. 4.2(a)-4.2(c). 4.7 Comparison to the classical SVMC model The spin vector Monte Carlo (SVMC) model [48] was proposed as a purely classical model of the D-Wave processors in response to earlier work that ruled out simulated annealing [44] and other work that established a strong correlation between D- Wave ground state success probability data and simulated quantum annealing [43]. The SVMC model was found to not always correlate well with D-Wave empirical data; for example, deviations were observed for the SVMC model in the case of ground state degeneracy breaking [91], excited state distributions [42], quantum annealing correction experiments [50] and the dependence of success probability on temperature [51]. But it has generally been successful in predicting the success probability distributions of D-Wave experiments. The SVMC model thus provides a sensitive test for whether anything other than classical effects are at play in a fixed-temperature measurement of the ground state success probability. Details of the “noise-free” SVMC model can be found in Appendix A.4. The SVMC model has two parameters: N s , the number of sweeps and the fixed inverse temperatureβ . In order to realistically emulate the D-Wave processor, we added random Gaussian noiseN (0,σ 2 ) to each J i [49]. The SVMC model then has three free parameters: {N s ,β,σ }, which we used to calibrate it against the DW2X data. Toward this end we used the chain with parameters (1, 0.5, 175) and performed an extensive search in the{N s ,β,σ } parameter space. As shown in Fig. 4.6(a), we obtain a close match for N s = 120× 10 3 , β = 0.75 (GHz) − 1 and σ = 0.05. This is the best fit we found for this particular chain. In general, we found that the SVMC 157 0 5 10 15 20 10 − 4 10 − 3 10 − 2 10 − 1 n P G 0 5 10 15 20 10 − 4 10 − 3 10 − 2 10 − 1 n SVMC DW2X 0 5 10 15 20 10 − 4 10 − 3 10 − 2 10 − 1 n 0 5 10 15 20 10 − 4 10 − 3 10 − 2 10 − 1 n P G 0 5 10 15 20 10 − 1 10 0 n 0 5 10 15 20 10 − 3 10 − 2 10 − 1 n (a) (1,0.5,175) (b) (0.8,0.4,175) (c) (0.5,0.25,175) (d) (1,0.5,200) (e) (1,0.5,55) (f) (0.6,0.3,135) Figure 4.6: Comparison of the SMVC model to the empirical DW2X results. The error bars everywhere indicate 95% confidence intervals calculated using a binomial bootstrap over the different runs of the simulation. (a) The SVMC parameters were optimized to match the empirical DW2X success probability results for the chain with parameters (W 1 ,W 2 ,N) = (1, 0.5, 175). The optimal values found are: N s = 120× 10 3 ,β = 0.75 (GHz) − 1 ,σ = 0.05 [compare to the DW2X’s t f = 5 μ s,β = 0.637 (GHz) − 1 ,σ ∼ 0.03]. (b) With the same optimal SVMC parameter values, but with chain parameters (0.8, 0.4, 175), the SVMC model predicts increased success probability, in contrast to the empirical results. The same trend continues but is more pronounced in (c), with additionally the position of the minimum shifting to the wrong location (n = 7 vsn ∗ = 4). Panel (d) shows that increasing the chain length causes a large deviation in the SVMC results [compare to panel (a)], and also shifts the location of the minimum to the wrong value, but (e) shows showing that reducing the chain length does not degrade the agreement much. (f) Results for another chain parameter set, exhibiting a similar discrepancy to that seen in (c). 158 Chain parameters DW2X k ∗ ME SVMC (1.0, 0.50, 175) 5,6 5 5,6 5 (0.5, 0.25, 175) 3,4,5 4 4,5 5,6,7 (0.8, 0.40, 175) 5 5 5 5,6 (1.0, 0.50, 200) 5,6 5 5,6 4,5,6 (1.0, 0.50, 55) 5,6 5 5,6,8 5 (0.6, 0.30, 135) 4,5 4,5 5,6 5,6 Table 4.2: Locations n ∗ of the DW2X success probability minima vs those found by the fermionic model based on the peak of k ∗ , the master equation model (ME), and the SVMC model. When the location of the minimum is ambiguous within our 95% confidence interval we list all values ofn ∗ that overlap to within one σ . The best agreement is obtained by the master equation model. parameters can be tuned to reproduce the location of the minimum in success probability for any sector size. We were also able to tune the parameters such that the minimum disappears completely and have the success probability increase or decrease monotonically. We were not able to find parameters that give rise to an inverted curve, i.e., a maximum in the success probability. Most of these features can be seen by tuning β and keeping the other parameters fixed. To avoid fine-tuning, we next used the same parameters to compute the success probability of the SVMC model for other chains. As shown in Figs. 4.6(b) and 4.6(c), the SVMC model has the wrong trend with decreasing (W 1 ,W 2 ): it exhibits a higher success probability as the coupling energy scale is lowered. The same happens with increased chain length [ Fig. 4.6(d)], though to a lesser degree with decreased chain length [ Fig. 4.6(e)]. Moreover, as summarized in Table 4.2, it does not agree as well with the location of the minimum of the success probability as the other models. We emphasize that while we performed an extensive search, we cannot rule out the possibility of another set of parameters (or the inclusion of other parameters) that allow SVMC to reproduce the DW2X results for all chain lengths and energy scales. 159 5 10 15 20 0.85 0.9 0.95 n ¯ s SVMC DW2X Figure 4.7: Spin boundary correlation function computed using the SVMC model. The spin boundary correlation function for the same chain and SVMC parameters as in Fig. 4.6(a). The SVMC model does not correctly capture the empirical results despite providing a close match to the success probability in Fig. 4.6(a). As a further test we also considered the spin boundary correlation, defined as the sum of spin correlations over all boundary qubits in the chain, where the boundary qubits are the qubits at the right edge (r) of the heavy sector and left edge (l) of the light sector: ¯ s = 1 |Q| X Q s l s r , (4.15) where Q is the set of boundary qubits and s l ,s r ∈{ 0, 1}. Thus ¯ s = 1 represents perfect alignment (a ground state), while ¯ s < 1 represents the occurrence of an excited state due to misalignment of the different sectors. Fig. 4.7 shows the results for the same set of optimized parameters that provided strong agreement with the ground state success probability in Fig. 4.6(a). It can be seen that the SVMC model predicts the wrong location for the minimum of ¯ s and rises too fast. Unfortunately, 160 master equation simulations for ¯ s are numerically prohibitive, so we cannot assess whether this discrepancy of the SVMC model is fixed by a quantum model. 4.8 Discussion A commonly cited failure mode of closed-system quantum annealing is the expo- nential closing of the quantum gap with increasing problem size. It is expected, on the basis of the Landau-Zener formula and the quantum adiabatic theorem, that to keep the success probability of the algorithm constant the run-time should increase exponentially. As a consequence, one expects the success probability to degrade at constant run-time if the gap decreases with increasing problem size. Our goal in the work of this chapter was to test this failure mode in an open-system setting where the temperature energy scale is always larger than the minimum gap. We did so by studying the example of a ferromagnetic Ising chain with alternating coupling-strength sectors, whose gap is exponentially small in the sector size, on a quantum annealing device. Our tests showed that while the success probability initially drops exponentially with the sector size, it recovers for larger sector sizes. We found that this deviation from the expected closed-system behavior is quali- tatively and semi-quantitatively explained by the system’s spectrum around the quantum critical point. Specifically, the scaling of the quantum gap alone does not account for the behavior of the system, and the scaling of the number of energy eigenstates accessible via thermal excitations at the critical point (the thermal density of states) explains the empirically observed ground state population. Does there exist a classical explanation for our empirical results? We checked and found that the spin vector Monte Carlo (SVMC) model [48] is capable of matching the empirical DW2X results provided we fine-tune its parameters for 161 each specific chain parameter set{W 1 ,W 2 ,N}. However, it does not provide as satisfactory a physical explanation of the empirical results as the fermionic or master equation models, which require no such fine-tuning. Our work demonstrates that care must be exercised when inferring the behavior of open-system quantum annealing from a closed-system analysis of the scaling of the gap. It has already been pointed out that quantum relaxation can play a beneficial role [64, 65, 94, 101, 140]. However, we have shown that relaxation plays no role in the recovery of the ground state population in our case. Instead, our work highlights the importance of a different mechanism: the scaling of the number of thermally accessible excited states. Thus, to fully assess the prospects of open-system quantum annealing, this mechanism must be understood along with the scaling of the gap and the rate of thermal relaxation. Of course, ultimately we only expect open-system quantum annealing to be scalable via the introduction of error correction methods [79, 156]. 162 Chapter 5 Conclusion In this dissertation, we have discussed the behavior of an open system quantum annealer in many different regimes, with problems as small as eight qubits and as large as 175 qubits. For all these problems, we found that an open system master equation holds more predictive power about the behavior of the annealer than one of the many classical models. We also showed that in many respects, these proposed models can be thought of as a restricted classical limit of the quantum master equation. We also demonstrated the crucial problem with the classical models: they don’t have a consistent set of parameters that can predict the performance of the annealer for different problems. On the other hand, the master equation model has one free parameter, the system-bath coupling constant. We can calibrate this value for instances of small size and the same value can explain the behavior of the annealer for problems with hundreds of qubits. We conclude by summarizing the novel aspect of our work. In Chapter 2, we build upon the earlier work of Boixo et al. [44] and showed how the detailed analysis of a toy problem can rule out all the classical models of the D-Wave annealers that were proposed to date. At the same time, we noticed that the classical models were able to qualitatively predict the cumulative ground state populations. We went a step further in Chapter 4. Many of the earlier works, including that in Chapter 2, on detecting and quantifying the quantum behavior of the D-Wave annealer used toy problems with a quantum minimum gap either much larger or comparable to the thermal energy scale imposed by the external environment. In 163 such setting, the environment does not significantly alter the performance of the annealer from its closed system prediction. While the results from the annealer show a good agreement with the open system quantum adiabatic master equation, they also show good agreement to intuitive arguments based on closed system analysis. In this dissertation, we studied the ASC problem where unlike the closed system prediction, the success probability of the D-Wave annealer does not correlates with the quantum minimum gap. Instead, the success probability changes in a non-monotonic fashion against a monotonically decreasing gap. We then showed that this behavior seen of the annealer is consistent with an open system description of the annealer, and the success probability correlates well with the number of states near the ground state close to the critical point. The system-bath interaction restricts this thermal excitation to some particular states near the critical point, and in this fashion the annealer retains the quantum behavior even in a regime where the thermal energy scale is much larger than the quantum minimum gap. We tried to fit the experimental data to a classical rotor model of magnetic needles with no mutual coherence attached with a thermal bath. We were not able to explain our results with this model without extensive fine tuning of physical parameters. Thus, this experiment provides another method to distinguish the behavior of the D-Wave annealer from a classical device. As we demonstrated in our work, the D-Wave annealer is strongly affected by the environment, to an extent that closed system prediction of performance are no longer useful. Ultimately, any quantum system will need to apply quantum error correction to recover this performance loss. We discussed and analyzed the two proposed quantum annealing correction codes in Chapter 3. Although the two codes use same amount of physical resources (viz. 4 physical qubits for 1 logical qubit), their performance was found to be substantially different from each other. 164 On the other hand, they provide different degree of connectivity. Depending on the structure of the problem Hamiltonian, one graph might be preferable to the other. We showed that the biggest performance gains obtained by these codes can be explained a simple overall energy scaling argument, and that post-processing of the physical state vector provides marginal improvement above this energy scale-up. 5.1 Open questions and future work In this section, we propose future work that can further advance the field of quantum testing and quantum error correction. In Chapter 2, we showed that no classical model proposed to date can explain the behavior of the D-Wave device. Further research has conclusively ruled out such classical models in many different situations [42, 50, 51, 91]. However, ruling out a few specific classical models does not foreclose the possibility of another classical model that can explain the results of all the black box experiments. The current black box nature of these devices does not allow for a Bell’s like inequality test. Nevertheless, with enough control, even a classical observer can verify the quantum nature of two black boxes [93]. What is the minimum amount of control needed over the device by the end user to successfully quantify the quantum nature of the device with a Bell’s like inequality? In Chapter 3, we tested the performance of two quantum annealing correction codes. While these codes help in recovering some performance, they nevertheless are limited by their overall energy scale-up. The goal of any error correction scheme is to recover closed system behavior in the presence of external noise. What is the threshold of noise at which these codes can completely recover from any given error? In circuit model of quantum computation, the threshold theorem [82, 157–161] guarantees perfect error correction provided that the external noise is below a given 165 threshold. Can such a theorem be derived for quantum annealing? Even without such a threshold theorem, promising scalable error correction has been demonstrated on the D-Wave device with the use of nested quantum annealing [156, 162, 163]. The [4, 1, 4] 0 code can be similarly nested by contracting a square into a nested logical qubit on each “layer” of the code. Can such nesting also achieve the positive results seen in Ref. [156]? In Chapter 4, we discussed the ASC problem. These problems have a quantum minimum gap that is inversely proportional to an exponential function of the system size. Clearly, this is an easy problem and the apparent “hardness” of this problem arises due to the particular rigid nature of quantum annealing via Hamiltonian (1.2). What modifications can be made to this simple quantum annealing model to remove the exponentially closing gap? What explains the different behavior of the spin vector Monte Carlo algorithm on this problem? Can an error correction scheme recover the closed system behavior of this problem at any arbitrary scale? Hopefully, solving these and other problems will finally put AQC on equal footing with the circuit model of computation. 166 Appendix A Appendix A.1 Chain generation algorithm To generate a chain of required length, we pick a logical qubit at random and run a modified Depth First Search (DFS) algorithm [164]. Our modified version visits each node in the graph in an iterative fashion, starting from a randomly selected node. In every iteration of the algorithm, the next qubit to visit is selected at random. The order of visitation of qubits under DFS yields a chain, which we save if it is longer than the desired length. We describe the procedure in detail with the pseudo code in Algorithm A.1. Since the algorithm is probabilistic, any particular run may or may not find the chain of required length. We ran the algorithm a fixed large number of times and stopped when we got the desired number of distinct embeddings. A.2 Jordan-Wigner transform The Jordan-Wigner transform can be used to map a one-dimensional transverse field Ising Hamiltonian to a Hamiltonian of uncoupled (free) fermions. A system of dimension 2 N × 2 N is thus effectively reduced to a system of dimension N, which is essential for our simulations involving N as large as 200. We briefly summarize the approach found in the classic work of Lieb et al. [146]. 167 Require: n≥ 1 be the length of required chain. 1: Let S and B be two empty stacks. 2: Mark all nodes on graph as not visited. 3: Randomly select a node on graph, and push it to top of stack S, and mark it as visited. 4: while S is not empty do 5: g← S.top. 6: ngbrs← neighbors(g) 7: if ngbrs has no un-visited nodes then 8: S.pop 9: else 10: t← a random, not yet visited node from ngbrs 11: Mark t as visited. 12: S.push(t) 13: end if 14: if size(S) > size(B) then 15: B← S 16: end if 17: if size(B)≥ n then 18: return(B) 19: end if 20: end while 21: Re-run the algorithm if no suitable chain is found. ALGORITHM A.1: Chain generation algorithm We start by defining fermionic raising operatorsa † i = 1 2 ⊗ i− 1 j=1 σ x j (σ z i +iσ y i ) (i = 1,...,N) and their corresponding lowering operators a i . It is easy to check that the fermionic canonical commutation relations are then satisfied:{a i ,a j } = {a † i ,a † j } = 0 and{a † i ,a j } =δ ij . Rewriting Eq. (4.1) as H =− Γ N X i=1 σ x i − N− 1 X i=1 J i σ z i σ z i+1 (A.1) and substituting σ x i = 1− 2a † i a i , σ z i σ z i+1 = (a † i − a i )(a † i+1 +a i+1 ) gives H =− NΓ + X ij a † i A ij a j + 1 2 (a † i B ij a † j +a i B ji a j ) , (A.2) 168 or H =− NΓ + ( ~ a † ) T A~ a + 1 2 h (a † ) T Ba † +~ a T B T ~ a i (A.3) where ( ~ a † ) T = (a † 1 ,...,a † N ),~ a T = (a 1 ,...,a N ), and A =                2Γ − J 1 0 − J 1 2Γ − J 2 0 − J 2 . . . . . . 0 . . . 2Γ − J N− 1 ··· − J N− 1 2Γ                , (A.4) B =                0 − J 1 J 1 0 − J 2 J 2 . . . . . . 0 . . . 0 − J N− 1 ··· J N− 1 0                . (A.5) The Hamiltonian in Eq. (A.2) is not fermion number conserving (it contains terms such as a † i a † j , which means that P i σ x i is not conserved), but it can be diagonalized by a Bogoliubov transformation [146, Appendix A] in terms of a new set of fermionic operators{η i ,η † i }: H =E g + N X i=1 λ i η † i η i . (A.6) 169 where the λ i are the single-fermion energies of the system. The new fermionic operators are real linear combinations of the old ones: η † i = X k g ik a † k + X k h ik a k η i = X k g ik a k + X k h ik a † k . (A.7) From Eq. (A.6) we get [η k ,H] =λ k η k . (A.8) Substituting Eqs. (A.2) and (A.7) in Eq. (A.8) and setting the coefficients of every operator{a i ,a † i } to zero gives λ k g ki = X j g kj A ji − h kj B ji λ k h ki = X j g kj B ji − h kj A ji (A.9) Let φ ik =g ik +h ik and ψ ik =g ik − h ik . Plugging this into Eq. (A.9), we get two coupled equations: ~ Φ k (A− B) =λ k ~ Ψ k (A.10) ~ Ψ k (A + B) =λ k ~ Φ k , (A.11) where ~ Φ k = (φ 1k ,...,φ Nk ) and ~ Ψ = (ψ 1k ,...,ψ Nk ). Eliminating ~ Ψ k gives us the decoupled equation, ~ Φ k (A− B)(A + B) =λ 2 k ~ Φ k . (A.12) Solving the eigensystem given by Eq. (A.12) gives the eigenvalues{λ i } in the Hamiltonian (A.6). 170 We can find the ground state energy E g by taking trace of Eqs. (A.2) and (A.6) [146, Appendix A]. From Eq. (A.2), we have Tr(H) = 2 N− 1 X i A ii − 2 N NΓ (A.13) and from Eq. (A.6) we have, Tr(H) = 2 N− 1 X k λ k + 2 N E g . (A.14) As the trace is invariant under a canonical transform, we get E g =− NΓ + 1 2 ( X i A ii − X k λ k ) =− 1 2 X k λ k . (A.15) If we require that the ~ Φ k are orthonormal, then the transformation given by Eq. (A.7) is a canonical transformation. Solving Eq. (A.10) gives the corresponding ~ Ψ k . Finding matrices Φ and Ψ (and hence g ik and h ik ) gives the forward transform connecting the undiagonalized fermions to the diagonalized fermions. The inverse transform can be defined as a † i = X k ¯ g ik η † k + X k ¯ h ik η k a i = X k ¯ g ik η k + X k ¯ h ik η † k , (A.16) such that a † i +a i = X k ¯ φ ik η † k +η k a † i − a i = X k ¯ ψ ik η † k − η k , (A.17) 171 where ¯ φ ik = ¯ g ik + ¯ h ik and ¯ ψ ik = ¯ g ik − ¯ h ik . Since these transforms are canonical, ¯ Φ = Φ T and ¯ Ψ = Ψ T . A.3 Fermionic domain-wall states The eigenstates of the ASC Hamiltonian (4.1) can be rewritten as many-fermion states. For example,|0i denotes the vacuum which is the ground state of the Hamiltonian,|a b ci =η † a η † b η † c |0iisathree-fermionstatewithenergyE g +λ a +λ b +λ c and|γ i =η † {γ } |0i is another state with|γ | fermions. In this notation|a b ci means a state with a single fermion in each of the positionsa,b, andc. What do these states look like in the computational basis? We can gain some intuition by considering the special case with zero transverse field. For Γ = 0, Eq. (A.12) becomes: ~ Φ k                0 4J 2 1 4J 2 2 . . . 4J 2 N− 1                =λ 2 k ~ Φ k . (A.18) 172 For simplicity, assume J 1 ≤ J 2 ≤ ... ≤ J N− 1 . This immediately yields the eigenvalues λ 1 = 0, λ 2 = 2J 1 , λ 3 = 2J 2 , ..., λ N = 2J N− 1 and eigenvectors Φ =                1 1 1 . . . 1                . (A.19) Using Eq. (A.11) we find: Ψ =                0 0 0 ... 1 − 1 0 0 ... 0 0 − 1 0 ... 0 0 . . . 0 0 0 ... − 1 0                . (A.20) If the J i are not ordered the result remains the same up to a permutation of rows of Φ and Ψ, where the λ i are still arranged in ascending order. Now consider the operator σ z i σ z i+1 = (a † i − a i )(a † i+1 +a i+1 ) in terms of the diagonalized fermionic operators. Using Eqs. (A.17), (A.19) and (A.20) gives σ z i σ z i+1 = X kk 0 ¯ ψ ki ¯ φ i+1,k 0(η † k − η k )(η † k 0 +η k 0) = X kk 0 (− δ k,i+1 )(δ k 0 ,i+1 )(η † k − η k )(η † k 0 +η k 0) = 1− 2η † i+1 η i+1 . (A.21) 173 Thus, the many-fermion states are also the eigenstates of the operators σ z i σ z i+1 with eigenvalue− 1 if there is a fermion occupying level i + 1, and eigenvalue 1 otherwise. In the spin picture, eigenvalue +1 denotes a satisfied coupling and− 1 denotes an unsatisfied coupling. Thus, the presence of a fermion in leveli + 1 can be thought as a domain wall in couplingi of the ferromagnetic chain. The fermionic states|0i and|1i satisfy all the couplings and hence are linear combinations of the all-0 and all-1 states. This is reflective of theZ 2 symmetry inherent in the Ising Hamiltonian, which also manifest itself as λ 1 = 0. Thus, for given state|ai with energy E a =E g +λ a , the state|a 1i has the same energy E g +λ a +λ 1 =E a . Additionally, (1− 2η † i+1 η i+1 )|ai = (1− 2η † i+1 η i+1 )|a 1i ∀ i≥ 1 since the fermion occupying the state|1i is not affected by this operation. Ergo, the states|2i and|2 1i corresponds to a domain wall at the location of the weakest coupling, |3i and|3 1i corresponds to a state with a domain wall at the second weakest coupling, etc. Once the couplers are arranged in alternating sectors such that J 1 =J 2 =... =W 1 >J n+1 =J n+2 =... =W 2 <..., the domain walls first occur in the light sectors, followed by the heavy sectors, etc. A.4 Spin Vector Monte Carlo (SVMC) algo- rithm In this section, we will describe the spin vector Monte Carlo (SVMC) model [48] algorithm. This algorithm is also know as the Smolin-Smith-Shin-Vazirani (SSSV) algorithm after the initials of its authors. 174 In the SVMC model each qubit is replaced by a classicalO(2) rotor parametrized by one continuous angle viaσ x i 7→ sinθ i andσ z i 7→ cosθ i , so that the Hamiltonian 1.2 becomes H(t) =A(t) X i sinθ i +B(t)   X i h i cosθ i + X i,j J ij cosθ i cosθ j   . (A.22) In addition, the angles θ i undergo Metropolis updates every discrete time-step of size 1/N s , where N s is the number of sweeps, at a fixed inverse temperatureβ . Angle updates are performed as follows. We first divide the dimensionless time s into steps of size δs = 1/N s , where N s is the number of sweeps performed during the course of one run, and initialize to the state θ i (s = 0) =π/ 2 for all spins i. At each such time step, we pick a random permutation of the set{1, 2,...,N} where N is the number of qubits in the chain. One by one, we select a random angle for each qubit in this permuted list, changing it from θ i to ˜ θ i where ˜ θ i is picked randomly from [0,π ]. We calculate the energy change ΔE for this move and the new angle is accepted with probability p i = min [1, exp(− β ΔE)] , (A.23) where β is the inverse annealing temperature for the SVMC algorithm. The final state is obtained by projecting the O(2) rotor spins to the computational state at s = 1, setting spin i to be 1 (− 1) if cosθ i > 0 (< 0). We repeat this process many times in order to estimate the success probability of the algorithm. Frequently, this algorithm has been used to model the behavior of the D-Wave device. In such cases, modifications are made to account for calibration and cross- talk errors of the device. See Chapters 2 and 4 for details of such modifications. 175 2 4 6 8 10 12 14 16 18 20 10 3 10 4 10 5 10 6 10 7 10 8 10 9 n Time scale (in ns) max s || ˙ H(s)||/Δ 2 (s) max s |h0| ˙ H(s)|1 2i|/Δ 2 (s) t a = 5 μs Figure A.1: Test of the adiabatic condition. The solid line is the annealing time used in our experiments. Symbols represent the quantity appearing in two versions of the adiabatic condition [for ASC parameters (1, 0.5, 175)] that should be smaller than the annealing time in order for the adiabatic condition to hold. A.5 Test of the adiabatic condition for ASC Here we test the validity conditions assumed for the derivation of the adia- batic Markovian master equation [45], and in particular that the adiabatic ap- proximation is satisfied. To this end we test the ‘folklore’ adiabatic condition t f max s |hg| ˙ H(s)|ei|/Δ 2 (s), where g and e are the ground and first (relevant) excited states, respectively, for ASC parameters (1, 0.5, 175). The result is shown in Fig. A.1 (red circles), and it can be seen that the condition is satisfied forn≤ 14. The relevant first excited state is the two-fermion state|1 2i. Also shown is the more conservative condition given in terms of the operator norm (blue crosses). The adiabatic condition with the operator norm is not satisfied for any value ofn, but it has recently been shown that this condition, which involves an extensively growing operator norm, must be replaced by a condition involving local operators [165]. 176 0 5 10 15 20 10 − 4 10 − 3 10 − 2 n P G 5μs 200μs 500μs 1000μs Figure A.2: Dependence of the success probability on the annealing times. We show the results for the ASC with parameters (1.0, 0.5, 175). The error bars everywhere indicate 95% confidence intervals calculated using a bootstrap over different gauges and embeddings. The location of the minimum is unchanged as the annealing time is varied. A.6 ASC at different annealing times In A.5 we discussed the validity of the “folklore” adiabatic condition for the ASC problem. We expect that the adiabatic condition will also be satisfied if we make small changes in the annealing time compared to the vertical scale of Fig. A.1. Also, neither the gap Δ nor the number of single fermion states k ∗ changes with an increase in the annealing time. Hence we expect that the qualitative nature of the success probability curve, including the location of minima, will be independent of small changes in the annealing time. Since the total thermal transition rate depends on the amount of time system spends near the quantum minimum gap point s ∗ , we do expect to see changes in the value of the success probability. In Fig. A.2, we show the change in success probability as we vary the annealing time on the D-Wave device. As expected, the location of the minima remains unchanged. 177 We do find that the success probability varies depending on the annealing time and sector size. A.7 Computing master equation matrix ele- ments for ASC In order to determine whether thermal transitions between states can occur, we need to calculate the matrix element ofσ z between the two states, where we assume that the dominant interaction term between the system and environment is given by a pure dephasing interaction of the form H SB = P i σ z i ⊗ B i [42, 51, 91, 91, 166]. We we would like to estimate the transition matrix elementhγ 1 |σ z i |γ 2 i between states|γ 1 i and|γ 2 i. We shall show that these transitions are most prominent when the states differ by only a single fermion. In terms of the fermionic operators, the operator σ z can be written as σ z i =   i− 1 Y j=1 (a † j +a j )(a † j − a j )   (a † i +a i ) (A.24) anditcaninturnbewrittenintermofthe{η k }and{η † k }operatorsusing Eq.(A.17). Now consider the matrix element between the vacuum state|0i and the state |γ i. For the operator σ z 1 we find that hγ |σ z 1 |0i =hγ |a † 1 +a 1 |0i =hγ | X k ¯ φ 1k (η † k +η k )|0i = X k ¯ φ 1k hγ |ki. (A.25) 178 This can only be non-zero if|γ i is a single fermion state. Thus, the operator σ z 1 connects the vacuum to single-fermion states. For σ z 2 , we have hγ |σ z 2 |0i =hγ |(a † 1 +a 1 )(a † 1 − a 1 )(a † 2 +a 2 )|0i = X k,l,m ¯ φ 1k ¯ ψ 1l ¯ φ 2m hγ |(η † k +η k )(η † l +η l )(η † m +η m )|0i. (A.26) For the above term to be non-zero, the state|γ i must be either a three-fermion or a single-fermion state. The one-fermion case can be computed in a similar manner to the previous case involving σ z 1 , so we focus on the case when|γ i =|a b ci is a three-fermion state. We have ha b c|σ z 2 |0i = X klm ¯ φ 1k ¯ ψ 1l ¯ φ 2m ha b c|η † k η † l η † m |0i = X klm ¯ φ 1k ¯ ψ 1l ¯ φ 2m ha b c|k l mi = det ¯ φ 1a ¯ ψ 1a ¯ φ 2a ¯ φ 1b ¯ ψ 1b ¯ φ 2b ¯ φ 1c ¯ ψ 1c ¯ φ 2c . (A.27) We find numerically for our problems that the matrix element associated with the three-fermion states is much smaller than that for single-fermion states. For example, for a chain of length N = 176 with parameters n = 5, W 1 = 1 and W 2 = 0.5, we find ats =s ∗ thath3|σ z 1 |0i = 0.12 andh2 3 4|σ z 2 |0i = 1.2× 10 − 8 . Similarly, the matrix element involving σ z 3 requires|γ i to be a state with 1, 3 or 5 fermions, etc. We find that the excitation to the 5-fermion states will be even smaller than that to the 3-fermion states, since they contain terms that are the 179 product of five terms of the type ¯ φ ¯ ψ ¯ φ ¯ ψ ¯ φ . Thus, our numerical results indicate that the vacuum state couples predominantly to single-fermion states. The above analysis can be generalized. Let us consider the matrix element θ ij =h0|σ z i |ji. Let Θ = [θ ij ] and ¯ Ψ = [ ¯ ψ ij ], and consider ˜ G = Θ ¯ Ψ T . We have: ˜ G ij = X m θ im ¯ Ψ T mj (A.28) = X m ¯ ψ jm h0|σ z i |mi (A.29) =h0|σ z i X m ¯ ψ jm (η † m − η m ) ! |0i (A.30) =h0|σ z i (a † j − a j )|0i (A.31) =h0|A 1 B 1 ...A i− 1 B i− 1 A i B j |0i , (A.32) where we have definedA i = (a † i +a i ) andB i = (a † i − a i ). Since the operatorsA i and B j in Eq. (A.32) anticommute when they have different indices, we can simplify the expression using Wick’s theorem [146, 167]. For a set of anticommutating operators {O 1 ,O 2 ,...,O 2n }, Wick’s theorem states that h0|O 1 O 2 ...O 2n |0i = X possible pairings (− 1) p Y all pairs h0|O i 1 O i 2 |0i . (A.33) where the sum is over all possible pairings of the operators{O 1 ,O 2 ,...,O 2n }, the product is over the two-point expectation value of all pairs, and (− 1) p is the sign of the permutation that is required to bring the paired terms next to each other. For 180 an odd number of operators, the expectation value vanishes. Applying the theorem to Eq. (A.32), we can make the following simplifications: h0|A i A j |0i = X k ¯ φ ik ¯ φ jk =δ ij , h0|B i B j |0i =− X k ¯ ψ ik ¯ ψ jk =− δ ij , h0|A i B j |0i = X k ¯ φ ik ¯ ψ jk = ¯ Φ ¯ Ψ T ij ≡ G ij . (A.34) Ifi<j, allthetermsinEq.(A.32)willhavedifferentindices. Thenon-zerotermsare pairs of the formhA k B k 0i. An obvious pairing ishA 1 B 1 i...hA i B j i =G 11 G 22 ...G ij and all other permutations can be obtained by keeping theA’s fixed and permuting the B’s around them. The signature of the permutation will be the signature of the permutations of B’s. The sum over permutations p is then given by ˜ G ij = X p (− 1) p G 1p 1 G 2p 2 ...G ip i = det G 11 G 12 ... G 1j G 21 G 22 ... G 2j . . . G i1 G i2 ... G ij . (A.35) If i ≥ j, we need to further simplify Eq. (A.32) so that it only contains anticommuting terms; hA 1 B 1 ...A j B j ...A i B j i =−h A 1 B 1 ...A j B j B j ...A i i =−h A 1 B 1 ...A j (− 1)...A i i =hA 1 B 1 ...A j ...A i− 1 B i− 1 A i i . (A.36) 181 In Wick’s expansion of this equation, any possible permutation will contain a pair of formhA k A k 0i = 0. Thus, ˜ G ij = 0 for i>j. This gives us a method to calculate the matrix elements of ˜ G, and we can in turn compute the desired transition element matrix via Θ = ˜ G ¯ Ψ . (A.37) Matrix element between arbitrary states can be computed in a similar fashion. For example: hγ 1 |σ z i |γ 2 i =hγ 1 |A 1 B 1 ...A i− 1 B i− 1 A i |γ 2 i =hγ 1 |A 1 B 1 ...A i− 1 B i− 1 A i η † {γ 2 } |0i. (A.38) Since A i = P j ¯ φ ij (η † j +η j ) and B i = P j ¯ ψ ij (η † j − η j ) we have {A i ,η † j } = ¯ φ ij , {B i ,η † j } =− ¯ ψ ij . (A.39) We can therefore anticommute the creation operator ofη † {γ 2 } from the left hand side to the right hand side. We find that the additional terms appearing because of the anticommutation relation in Eq. (A.39) are small relative to the term proportional tohγ 1 |η † {γ 2 } A 1 B 1 ...A i− 1 B i− 1 A i |0i, so we focus only on this term: hγ 1 |σ z i |γ 2 i (A.40) ∼ (− 1) (2i− 1)|γ 2 | hγ 1 |η † {γ 2 } A 1 B 1 ...A i− 1 B i− 1 A i |0i = (− 1) (2i− 1)|γ 2 | hγ 1 − γ 2 |A 1 B 1 ...A i− 1 B i− 1 A i |0i = (− 1) (2i− 1)|γ 2 | hγ 1 − γ 2 |σ z i |0i , 182 where (− 1) (2i− 1)|γ 2 | is an overall phase term associated with anticommuting the set η † {γ 2 } to the left and|γ 1 − γ 2 i =η {γ 2 } |γ 1 i. For this term to be non zero, the state |γ 1 i should contain all the fermions found in the state|γ 2 i. The remaining quantity hγ 1 − γ 2 |σ z i |0i is a matrix element connecting the vacuum, which we have already argued is largest when the state|γ 1 − γ 2 i is a single-fermion state. Therefore, the matrix elements connecting|γ 1 i and|γ 2 i are largest when the two states differ only by one fermion. From the above analysis of the coupling matrix elements, we find that the ground state couples principally with the single-fermion states, and the single-fermion states couple principally to two-fermion states and so on. The transition energies are the single-fermion energies{λ i } found by solving the eigensystem of Eq. (A.12). A.8 Exponentialdependenceofthesuccessprob- ability onk ∗ In this section we provide a more detailed argument than given in the main text for why the success probability exhibits an exponential dependence on k ∗ . Rather than providing a counting argument based on the thermal density of states d, as in Eq. (4.7), we consider the transition rates appearing in the Pauli master equation. Our argument provides a justification for why, if it were numerically feasible, we would expect the master equation simulations to reproduce the exponential dependence on k ∗ seen in our empirical results. GiventheformofthePaulimasterequation[Eq.(4.8)], ˙ p a = P b6=a γ (ω ba )M ab p b − P b6=a γ (ω ab )M ba p a , where[Eq.(4.9)]M ab = P α |ha|A α |bi| 2 , weexpectthesuccess probability to be inversely related to the overall excitation rate. Let M 0b ≡ M b be the matrix element involving a transition between states via the creation or 183 annihilation of a fermion with energy λ b , where λ b ≤ λ k ∗ . Let the number of such connected states be # b = k ∗ b and let M min (M max ) be the minimum (maximum) matrix element within the set{M 1 ,M 2 ,...,M k ∗}. The total transition rate τ is can be estimated as τ ∼ M 1 × # 1 +M 2 × # 2 +... +M k ∗ × # k ∗ , (A.41) so that M min 2 k ∗ .τ .M max 2 k ∗ , (A.42) where we used P k ∗ b=0 # b = 2 k ∗ . This bound is meaningful if the matrix elements do not differ by orders of magnitude. Indeed, we found numerically that the matrix elements are within an order of magnitude of each other. 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Abstract (if available)

Abstract Quantum computing is rapidly progressing from an academic field to one with commercially available quantum computers. We now have access to quantum annealers with thousands of qubits. The end user only has a very limited access to the inner working of such devices. Often, they present themselves as a black box: the user has control over some dials to program her problem, and the machine returns a classical solution at the end of its computation. How can one say that the black box presented is truly solving the problem with a quantum process? In this dissertation, we try to characterize the behavior of commercially available quantum annealers in different regimes and we show that the currently accepted quantum master equation models adequately explain the behavior of these annealers on problems with hundreds of qubits. Not surprisingly, these black box quantum annealers suffer from decoherence and errors due to their interaction with the external environment. As an example, we will discuss an extreme case where the behavior of the device deviates significantly from the noise-free closed system prediction. Quantum error correction techniques must be used to recover the theoretical closed system scaling advantage. We discuss the performance of the proposed error correction technique for quantum annealing and discuss the relative advantages of the two proposed codes.

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Demonstration of error suppression and algorithmic quantum speedup on noisy-intermediate scale quantum computers

Asset Metadata

Creator Mishra, Anurag (author)

Core Title Error correction and quantumness testing of quantum annealing devices

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 07/27/2018

Defense Date 06/12/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,quantum annealing,quantum computing,quantum error correction,quantum testing, D-Wave,quantum thermal annealing

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lidar, Daniel A. (committee chair), Brun, Todd Andrew (committee member), Haas, Stephan Wolfgang (committee member), Spedalieri, Federico (committee member), Takahashi, Susumu (committee member)

Creator Email agmishra.iitk@gmail.com,anuragmi@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-37987

Unique identifier UC11670350

Identifier etd-MishraAnur-6531.pdf (filename),usctheses-c89-37987 (legacy record id)

Legacy Identifier etd-MishraAnur-6531.pdf

Dmrecord 37987

Document Type Dissertation

Format application/pdf (imt)

Rights Mishra, Anurag

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA