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Error suppression in quantum annealing
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Error suppression in quantum annealing
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Error Suppression in Quantum Annealing by Humberto Munoz Bauza 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 2023 Copyright 2023 Humberto Munoz Bauza Acknowledgements This journey would not have been possible without the many intellectual and interpersonal en- counters that occurred during this time. I am thankful for the members of my committee: Daniel Lidar, Todd Brun, Itay Hen, Giacomo Nannicini, and Aiichiro Nakano, as well as those previously present for my qualifying exam, Lorenzo Campos Venuti, and Paolo Zanardi. I received mentorship or academic exchanges from all of them at one point, which has been truly valuable and formative. I was fortunate to have Daniel as my advisor and have his guidance and sense for the vast and growing research in quantum computation during my PhD. I am also thankful for each one of my colleagues, past and present, and especially those with whom I started around the same time and shared many of the same experiences: Haimeng, Vinay, Bibek, Jenia, and Nic. In addition, I am especially thankful for all of the guidance and wisdom from Huo and Tameem when I was starting out. I am reaching to the extent I can thanks to my family: my brother and mother, my sister, and my late father. Finally, I feel grateful to have had Mojgan by my side through these past few years. There is still much more to come, and I hope what follows in this dissertation will be dwarfed in time by a lifetime of learning and new experiences. ii Table of Contents Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2: Theory of Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Principles of Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Bosonic and Ohmic Environments . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Non-Markovian Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Nakajima-Zwanzig Master Equation . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 Time-Convolutionless Master Equation . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Markovian Master Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.1 Redfield Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.2 Time-Independent Lindblad Equation – Rotating Wave Approximation . . . . 10 Chapter 3: The Adiabatic Master Equation and its Applications . . . . . . . . . . . . . . . . 13 3.1 Closed System Quantum Adiabatic Approximation . . . . . . . . . . . . . . . . . . . 13 3.1.1 Adiabatic Closed System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1.2 Diabatic Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1.3 Diabatic Error Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Adiabatic Redfield Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Rotating Wave Approximation in the Adiabatic Limit . . . . . . . . . . . . . . . . . 22 3.3.1 The Adiabatic Master Equation (AME) . . . . . . . . . . . . . . . . . . . . . 23 3.3.2 Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 Quantum Annealing and Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Chapter 4: Hamiltonian Error Suppression in the Born-Markov Approximation . . . . . . . . 28 4.1 Stabilizer Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1.1 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Hamiltonian Error Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.1 Born Master Equation with Error Suppression . . . . . . . . . . . . . . . . . 33 4.2.2 Error-Suppressed Master Equation . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2.3 Suppression of Uncorrelated Pauli Errors . . . . . . . . . . . . . . . . . . . . 37 4.2.4 A Simplified Pauli Master Equation over Irreps . . . . . . . . . . . . . . . . . 38 4.2.5 Example: Repetition Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.3 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 iii Chapter 5: Approximate Optimization Advantage with Quantum Annealing . . . . . . . . . . 44 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2.1 Parallel Tempering Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.4 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Chapter 6: Error Suppressed Quantum Annealing through Boundary Cancellation . . . . . . 54 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.2.1 Review of the BCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.2.2 Adiabatic Master Equation for Quantum Annealing . . . . . . . . . . . . . . 58 6.2.3 A Modified BCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2.4 Freezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.2.5 Boundary Cancellation with a Ramp at the End . . . . . . . . . . . . . . . . 62 6.2.6 An Adiabatic Error Bound that Combines Everything . . . . . . . . . . . . . 64 6.2.7 Anomalous Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.3.1 Boundary Cancellation Protocol (BCP) . . . . . . . . . . . . . . . . . . . . . 65 6.3.2 8 Qubit Gadgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.3.3 Encoding with Quantum Annealing Correction . . . . . . . . . . . . . . . . . 69 6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.4.1 Linear Control Schedule (k = 0) . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.4.2 BCP Schedules with k 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.4.3 High Precision BCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.4.4 QAC-Encoded FM Gadget . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.4.5 Comparison of the BCP to the Pause-Ramp Protocol . . . . . . . . . . . . . . 78 6.5 Discussion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.6 Methods and Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.6.1 Proof of Degeneracy when A(t) = 0 . . . . . . . . . . . . . . . . . . . . . . . . 81 6.6.2 Proof of Proposition 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.6.3 Proof thath0j z j jni =O (A q ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.6.4 Determination of the Freezing Point . . . . . . . . . . . . . . . . . . . . . . . 87 6.6.5 Schedule Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.6.6 Crosstalk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.6.7 Data Collection and Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.6.8 Results for Alternative D-Wave QPUs . . . . . . . . . . . . . . . . . . . . . . 91 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Chapter 7: A Double-slit Proposal for Quantum Annealing . . . . . . . . . . . . . . . . . . . 92 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7.2 Closed System Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.2.1 Adiabatic Interaction Picture for Two-level System Quantum Annealing . . . 94 7.2.2 Magnus Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.2.3 LZ Problem (Linear Schedule) . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.2.4 Strong Quantum Interference Pattern via Gaussian Angular Progression . . . 97 7.2.5 Physical Origin of the Oscillations . . . . . . . . . . . . . . . . . . . . . . . . 97 7.2.6 Role of the Angular Progression . . . . . . . . . . . . . . . . . . . . . . . . . . 99 iv 7.3 Open system analysis and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.3.1 Redfield Master Equation in the Adiabatic Interaction Picture . . . . . . . . 100 7.3.2 Rotating Wave Approximation (RWA) . . . . . . . . . . . . . . . . . . . . . . 100 7.3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.4 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.5 Methods and Computational Detals . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.5.1 Dyson and Magnus Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.5.2 Error Analysis of Gaussian Angular Progression Schedules . . . . . . . . . . . 105 7.5.3 Second Order Term of the Magnus Expansion . . . . . . . . . . . . . . . . . . 107 7.5.4 Magnus Expansion Convergence and Error Bounds . . . . . . . . . . . . . . . 108 7.5.5 Double-slit Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.5.6 Interference Oscillations in the Double-slit Experiment Imply Quantum Coherence in the Computational Basis . . . . . . . . . . . . . . . . . . . . . . 112 7.5.7 Derivation of the Adiabatic-frame TCL2/Redfield Master Equation . . . . . . 114 7.5.8 Necessary Convergence Criterion . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.5.9 Rotating Wave Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.5.10 Derivation of the Semi-empirical Eq. (7.23) . . . . . . . . . . . . . . . . . . . 121 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 v Abstract Quantum annealing technologies have seen rapid progress in scale and applications. Nevertheless, a critical part of the utility of emerging quantum technologies is their computational advantage over classical algorithms. In this dissertation, we investigate avenues of enhancing the successful use of quantum annealing through error suppression and correction. We first review introductory aspectsofopenquantumsystems: thederivationofnon-MarkovianandMarkovianmasterequations from Hamiltonian descriptions, the adiabatic master equation, and the effects of decoherence in quantum annealing. In our first study, we carry out a randomized benchmark using Ising spin glasses defined on the logical graph induced by the quantum annealing correction (QAC) embedding on the D-Wave Advantage quantum processing unit (QPU). We find evidence of an approximate optimization advantage for quantum annealing over classical Markov chain Monte Carlo algorithms through a systematic analysis of optimal annealing times. In our second study, we investigate the implementation of quantum annealing schedules inspired by the boundary cancellation theorem. Whilewearenotablereproducethepredictedscalingbehavioroftheboundarycancellationtheorem (BCT)foropensystems,wedemonstratetheabilitytousetheseschedulestosuppressdiabaticerrors in quantum annealing, enhanced further when used in conjunction with QAC. Finally, we present an analysis of a “double-slit” model for quantum annealing with diabatic transitions and discuss the prospects of using diabatic shortcuts to enhance the standard adiabatic algorithm. vi Chapter 1 Introduction The past decade has seen a dramatic technological growth and investment in the development of quantum computers [1–6]. They have very quickly evolved from devices of academic settings to broadly accessible and potentially commercializable computational tools now often called quantum processing units (QPU). For most QPUs, the eventual goal is to achieve universal fault tolerant quantum computation (FTQC). The most immediate impacts of FTQC would include polynomial- time algorithms for integer factorization and the simulation of quantum systems [1, 7]. It would also include an exponential speedup for solving sparse linear equations [8], and in turn open up avenues of quantum machine learning [9, 10], although this is subject to stringent state preparation assumptions and has been an avenue for “dequantized” classical algorithms [11]. Quantum algorithms are almost universally developed in the “gate model” of quantum computa- tion, in which the progress of the algorithm over time is represented by the application of blocks of one-local and two-local (and higher, if necessary) unitary operations. This is the natural language of algorithms for which the input is a quantum state that has encoded (e.g. binary) information, and hence also the natural language for the analysis of quantum error correcting codes [1]. On the other hand, simulations of the continuous time evolution of quantum systems need not be performed on states that are intended to have some digital structure. In this case, the gate model is a less appropriate tool; instead, it is natural to consider the Hamiltonian operator that generates the time evolution. This Hamiltonian model of quantum computation has merit as a separate language for quantum algorithms, the most significant and influential of which is the adiabatic quantum algorithm [12, 13]. This dissertation will proceed as follows: In Chapters 2 and 3, we will review the principles of open quantum systems and the derivation of non-Markovian and Markovian master equations from first principles, and furthermore give a detailed and pedagogical derivation of the adiabatic master equation. We introduce quantum annealing in the context of this master equation as a model for adiabaticity in open systems. In Chapter 4, we introduce a scheme for simulating systems under Hamiltonian error suppression in the regime of the Born-Markov approximation, and discuss the implications of our formalism independent of the encoded computational Hamiltonian. In our first study, in Chapter 5, we perform a randomized benchmarking study using Ising spin glasses defined 1 on the logical graph induced by the quantum annealing correction (QAC) embedding on the D- Wave Advantage QPU. We find evidence of an approximate optimization advantage for quantum annealingoverparalleltemperingwithclustermovesanddiscussthesignificanceofthesefindingsfor approximating the ground state energy of finite-dimensional spin glasses. In Chapter 6, we present the first set of results of this dissertation on the implementation of quantum annealing schedules inspired by the boundary cancellation theorem. While we are not able reproduce the predicted scaling behavior of the BCT, we demonstrate the ability to use these schedules to suppress diabatic errors in quantum annealing, enhanced further when used in conjunction with QAC. Finally, in Chapter 7, we present an analysis of a “double-slit” model for quantum annealing with diabatic transitions and discuss the prospects of using diabaticity in Hamiltonian quantum algorithms. 2 Chapter 2 Theory of Open Quantum Systems No physical experiment is truly isolated from its environment, and this is all the more-so true for implementations of quantum computing, where unwanted interactions with the environment interfere with the intended unitary control of the system. The theory of open quantum systems deals with the time-evolution and properties of such systems, and has been encapsulated under a variety of physical contexts and applications, including quantum thermodynamics, quantum optics, nanoscale devices, and quantum computation, just to name a few [1, 14–22]. In this chapter, we will followapresentationstartingfromafirst-principlesdefinitionofaHamiltonianoperatorfortheopen system, agnostic to the system Hilbert space to the extent it is feasible. Naturally, our application of central interest is quantum computing, formulated under a Hamiltonian or gate model, for which the Hilbert space is generally assumed to be finite-dimensional. Hence, we make no attempt to accommodate unbounded quantum operators on the system in this chapter. 2.1 Principles of Open Quantum Systems The total (generally time-dependent) joint Hamiltonian H(t) of a system interacting with an envi- ronment (i.e. a heat bath) is written as H(t) =H S (t) +H B +H SB ; (2.1) where H S (t) is the system Hamiltonian, which may be controlled and hence is time-dependent, H B is the Hamiltonian of the environment, i.e. a heat bath, which is generally assumed to be time- independent, andH SB istheinteractionbetweenthesystemandtheenvironment. Theenvironment is assumed to hold a steady state, typically the Gibbs state at an inverse temperature , B = 1 Z e H B ; Z = Tr[e H B ]: (2.2) The system-bath interaction is assumed, without loss of generality, to have the form H SB = X A B ; (2.3) 3 where A are the system operators and B are the bath operators of the interaction. For our purpose, it is sufficient to assume that the Hilbert space of the system is finite-dimensional. Hence, the system operators are always bounded, so we will assumekA k = 1. Bath operators need not be bounded, in general. However, we can assume the interactions with the environment are described by a set of two-time correlation functions B (t;t 0 ) = B (t)B (t 0 ) B = Tr[B (t)B (t 0 ) B ]; (2.4) whereB (t) are the bath operators in the Heisenberg picture with respect toH B . It directly follows for a bath in the Gibbs state that the correlation function is time-translationally invariant, so we can writeB(t;t 0 ) =B(tt 0 ). To quantify the strength of the interaction between the system and the bath as well as the suitability of master equations for different systems, we require two characteristic timescales: the decoherence time SB and the correlation function decay time B . These times are defined by 1 SB = max ; Z 1 0 jB ()jd (2.5a) B = SB max ; Z T 0 tjB ()jd; (2.5b) where T is a long-time cutoff. The rate 1 SB is the largest possible rate that the spectral density (defined in Section 2.3.2) can take, and hence is an upper on the rate of decoherence that the system is subject to. The timescale B measures the expected time for the correlation function to decay. If the bath decay is exponential, then B will be precisely the exponential decay time of the bath. 2.1.1 Bosonic and Ohmic Environments A bosonic environment is one whose Hamiltonian can be written as a sum of independent quantum harmonic oscillators H B = X k ~! k a y k a k (2.6) wherek is an independent mode or wave-vector of the environment,! k is its angular frequency, and the creation and annihilation operators are a y k and a k , subject to canonical commutation relations. We will suppress explicitly writing~ and prefer energy-time units where~ = 1. As the modes of the environment are independent, the Gibbs state of a bosonic environment is simply B = 1 Z Y k e ! k a y k a k ; (2.7) where the partition function is Z = Y k 1e ! k 1 : (2.8) 4 Various forms for the bath coupling operator are possible depending on the physical situation. We outline frequent general categories of interactions and some example physical systems where they arise. Linear Coupling: In the simplest case, the system is coupled to the environment through a single linear combination of its bosonic modes, which must be Hermitian. That is, we have the single bath operator B = X k g k a k +g k a y k ; (2.9) which, expressed in the interaction picture, becomes B(t) = X k g k e i!t a k +g k e i!t a y k : (2.10) It is straightforward to show thathBi = 0 and that the correlation function is B() =hB()B(0)i = X k jg k j 2 1e ! k e i! k +e i! k ! k : (2.11) Energy exchange coupling: When the system is a single qubit, an energy exchange interaction can take the form H SB = + B + B y (2.12) where the bath operator is B = X k g k a k (2.13) and g k are the coupling constants of the modes. In the notation of the preceding section, we have B 1 =B andB 2 =B y . Note that, due to canonical commutation,hB 1 (t)B 1 (t 0 )i =hB 2 (t)B 2 (t 0 )i = 0. Furthermore,duetothecyclicpropertyofthetrace,hB 1 (t)B 2 (t 0 )i = D B y 2 (t)B y 1 (t 0 ) E =hB 2 (t)B 1 (t 0 )i . Thus, the two-time bath correlation functionB is such that the diagonal correlations are zero, and B 12 () =B 21 ()B(). While the energy interaction for a bosonic bath has two bath operators, it is sufficient to describe this interaction with a single two-time correlation B() =hB y ()B(0)i = X k jg k j 2 ha y k ()a k i = X k jg k j 2 e i! k ! k 1e ! k : (2.14) Irreversible dynamics arises in the limit where the modes become tightly-spaced, and hence are well-described by a continuum. If we have a sum over modes weighed by the square magnitude of the coupling constants, and the summand depends on the mode k only through its frequency ! k , the continuum limit is taken through the replacement X k jg k j 2 f(! k )7! Z 1 0 d!J(!)f(!); (2.15) 5 where J(!) is the spectral density function of the environment. Physically, this continuum limit arises when an extensive sum over the modes can be replaced by a continuous integral over wave- vectors 1 V X k 7! Z d n k (2) n ; (2.16) where n is the spatial dimension of the modes. Provided the summand depends on k only through ! k , the integration domain can be transformed to the frequency domain through the density of states D(!) Z d n k (2) n (!! k ): (2.17) An Ohmic environment is one where the spectral density has linear dependence on small fre- quencies, sustained up until some cutoff frequency ! c . This cutoff behavior can be modeled as a hard cutoff, i.e. J(!)/!(! c !), but it is more phenomenologically suitable to impose a hard cutoff. J(!) =!e !=!c : (2.18) 2.1.2 Wick’s Theorem We introduced some examples of two-time correlation functions for a bosonic environment, under different forms of coupling. Wick’s theorem is an important computational result in theoretical physics stating that the one-time and two-time correlations are sufficient to characterize all higher- order moments for linear operators of a free bosonic field. This has important consequences for the calculation of perturbative non-Markovian master equations, which we will elaborate on in the next section. Wick’s theorem is most commonly stated in its formulation for zero-temperature quantum field theories [23] and applied as-is for finite temperature correlation functions. However, its use in the context of finite-temperature bosonic baths is worth elaborating on. We will follow Ref. [24] for the case of a simple free bosonic theory. Let + (t) and (t) be some arbitrary “split” of an interaction Hamiltonian that is linear in the creation and annihilation operators of the bath, in the interaction picture of the free Hamiltonian of the system and the bath. The normal order of any product of these two operators is simply the rearrangedproductsuchthatall + operatorsprecede(aretotherightof)all operators, withthe relative order between each set of operators unchanged. The normal order of any operator expressed in the algebra generated by all split operators is defined similarly, distributing over summation. The contraction between two operators is defined as the difference between the time-ordered product and the normal-ordered product D[ 1 2 ] =T + 1 2 N 1 2 : (2.19) where the time-ordering is applied over the algebra of the split operators. 6 The main computational utility of Wick’s theorem follows when the contraction is a complex number. However, Wick’stheoremisparticularlytractablewhentheexpectationvalueofallnormal- ordered products vanish. Suppose we would like to calculate n-time correlation functions of the general linear coupling bath operator (2.10) using Wick’s theorem with a splitB(t) = + (t)+ (t). The zero-temperature split for this operator would simply segregate the creation and annihilation operators + (t) = X k g k e i!t a k ; (t) = X k g k e i!t a y k : (2.20) However, the expectation value of the normal-ordered operator (t) + (t) only vanishes when the bath is in the vacuum state. At non-zero temperatures, is therefore necessary to use another split. It can be shown that the split + (t) = X k g k (1f k )e i!t a k +g k F k e i!t a y k ; (t) = X k g k (1F k )e i!t a y k +g k f k e i!t a k ; (2.21) where f k and F k are the real numbers f k =n k + p n k (n k + 1); F k =n k p n k (n k + 1); (2.22) indeed yields hN[B(t 1 )B(t 2 )]i = Tr[ 2 + 1 + 1 + 2 + 1 1 + + 1 + 2 ) B ] = 0: (2.23) The evaluation of time-ordered multi-time correlation functions such ashT + B(t 1 )B(t 2 )B(t 3 )B(t 4 )i in terms of two-time contractions therefore becomes tractable with Wick’s theorem. 2.2 Non-Markovian Quantum Dynamics 2.2.1 Nakajima-Zwanzig Master Equation LetP be a projective superoperator. Given a fixed reference state B for the bath (e.g. the Gibbs state) the standard projector for the system state given a joint system-bath state is P = Tr B () B : (2.24) This projector has the interpretation of “forgetting” correlations with the environment and resetting it to its reference state. We will use it here for concreteness and to derive “standard” master equations, but it is not necessary to assume this projector. A different choice is possible, for example, if only energy populations are to be kept. 7 Starting from the exact Liouville-von Neumann equation for the state of the joint system-bath stat, a formally exact integro-differential equation can be derived forP. Define the shorthand notation for projected and orthogonally projected operators ^ X =PX XQX: (2.25) ByasimpleexpansionoftheidentitysuperoperatorintoP+Q, theLiouville-vonNeumannequation is equivalent to the system @ t ^ = ^ L^ + ^ L (2.26a) @ t = ^ L^ + ^ L (2.26b) We eliminate (t) using its formal solution, (t) =G(t;t 0 ) (t 0 ) + Z t t 0 G(t;t 0 ) L(t 0 )^ (t 0 )dt 0 ; (2.27) whereG(t;t 0 ) is the time-ordered propagator for the homogeneous part of G(t;t 0 ) =T + e R t t 0 L(t 0 )dt 0 : (2.28) Thus, we can describe ^ with the master equation @ t ^ (t) = ^ L(t)^ (t) + ^ L(t)G(t;t 0 ) (t 0 ) + Z t t 0 ^ L(t)G(t;t 0 ) L(t 0 )^ (t 0 )dt 0 : (2.29) The first term is the contribution due driving from the bath, and vanishes provided PLP = 0: (2.30) The second term is an inhomogeneity due to an initial state not completely within the range of the projector. This can also be made to vanish provided Q(t 0 ) = 0: (2.31) Thus, under both conditions (2.30) and (2.31), we arrive at the Nakajima-Zwanzig master equation (NZME) @ t ^ (t) = Z t t 0 K(t;t 0 )^ (t 0 )dt 0 ; (2.32) where the memory kernel is K(t;t 0 ) = ^ L(t)G(t;t 0 ) L(t 0 )P; (2.33) 8 which includes a leadingP for clarity. This equation is formally exact and follows for the standard projector under the following two conditions: The expectation value of bath operator B in the reference state vanishes hB i B = 0; (2.34) and the initial joint system-bath state has the factored form (t 0 ) = S (t 0 ) B : (2.35) Therearefewsituationswherethememorykernelisexactlycomputed. Amoretractableapproachis toperformaperturbativeDysonseriesexpansionofG(t;t 0 ). Infact, thezero-thorderapproximation G(t;t 0 )I itself leads to the non-trivial memory kernel K 0 (t;t 0 ) = ^ L(t) L(t 0 ) =PL(t)L(t 0 )P; (2.36) where we expandedQ = IP and used (2.30). Alternative non-perturbative approaches for calculating the memory kernel also exist, such as formulating an integral equation in Volterra form [25, 26] or expressing it in terms of the reduced system propagator [27]. 2.2.2 Time-Convolutionless Master Equation The NZME is not a time-local equation even with a perturbative expansion, which complicates its analysis. However, for sufficiently short times, it can be formulated in a time-local form know as the time-convolutionless master equation (TCL-ME) Since the full von Neumann equation is formally unitary, the joint system-bath state at any previous time t 0 <t can be expressed as (t 0 ) =U (t 0 ;t)(t): (2.37) Using this back-propagated form of the state and a similar derivation as the NZME using projector superoperators, we find that @ t ^ (t) =K (TCL) (t)(t) (2.38) where the TCL superoperator is K (TCL) (t) = ^ L(t)(I (t)) 1 P; (2.39) and (t) is the back-propagated NZME kernel (t) = Z t t 0 G(t;t 0 ) L(t 0 ) ^ U (t 0 ;t)dt 0 : (2.40) The TCL-ME requires that the (I (t)) operator is invertible. Since (t)! 0 as t! 0 + , the TCL-ME is valid and formally equivalent to the NZME for a sufficiently short amount of time. 9 Nevertheless, as with the NZME, computing (t) exactly is not feasible in general, and hence we take the usual perturbative approach to derive tractable time-local master equations. 2.3 Markovian Master Equations 2.3.1 Redfield Equation The TCL-ME enabled writing down time-local master equations valid for short time. The lowest- order for centered bath operators is known as the Redfield equation. Written out in terms of the interaction Hamiltonian, @ t (t) = Z t t 0 dt 0 Tr B H B (t); H B (t 0 );(t) B : (2.41) Traditionally, the Redfield equation is derived from the Born equation via the direct Markov ap- proximation (t 0 ) (t). As it is also the lowest order of the TCL expansion, it is possible to estimate the short-time error of the Redfield equation with the next order, i.e. TCL4. Expanding out the interaction Hamiltonian, the Redfield equation is also written as @ t (t) = X Z t t 0 d 0 B ( 0 ) A (t);A (t 0 )(t) + h:c:: (2.42) 2.3.2 Time-Independent Lindblad Equation – Rotating Wave Approximation While time-local, it is famously known that the Redfield equation (and indeed any truncation of the TCL-ME) generates a map that is not guaranteed to be completely positive. Time-local master equations that guarantee complete positivity were characterized by Lindblad [28] and have the general form @ t =i[H S ;] + X ij a ij F j F y i 1 2 fF y i F j ;g ; (2.43) wherefF i g is a basis set of operators forB(H) and a ij is a positive matrix. Such a matrix can be diagonalized with a unitary operator, which transforms the Lindblad equation into its diagonal form @ t =i[H S ;] + X i i L i L y i 1 2 fL y i L i ;g ; (2.44) where thefL i g operators are called the Lindblad operators. We can easily bring the Redfield equation into Lindblad form for a time-independent Hamilto- nian with an assumption called the rotating wave approximation (RWA). For a time-independent Hamiltonian, the system interaction operatorsA (t) has a decomposition in terms of the frequencies of the system A (t) = X ! e i!t A ;! ; (2.45) 10 where A ;! = X E j E i =! i A j ; (2.46) where i is the projector to theE i energy eigenspace of the Hamiltonian. Now, when the correlation function decay time B is sufficiently small (consistent with the Markov approximation), we can extend the upper limit of the integral in the Redfield equation @ t (t) = X X !;! 0 e i(!+! 0 )t Z 1 0 de i! B () A ;! ;A ;! 0(t) + h:c:; (2.47) where we sett 0 = 0 without loss of generality for a time-independent Hamiltonian. With the RWA, we assume that terms such that ! +! 0 6= 0 are very rapidly oscillating and hence can be dropped for the dynamics of the state, as the oscillations will self-cancel on average. This finally yields @ t (t) = X X ! Z 1 0 de i! B ()[A ;! ;A ;! (t)] + h:c:: (2.48) We will now show that this master equation can be brought to Lindblad form. Define the one-sided Fourier integral of the bath (!) = Z 1 0 de i! B (): (2.49) It is a well-known result in complex analysis that this integral has a decomposition into real and imaginary parts (!) 1 2 (!) +iS(!) (2.50) where (!) is the Fourier transform (!) = Z 1 1 de i! B (); (2.51) and S (!) is a principal value known as the Lamb shift frequency S (!) 1 2 Z 1 1 P 1 !! 0 (! 0 )d! 0 : (2.52) We note that both andS are Hermitian in their indices. It therefore becomes a short exercise to write @ t (t) =i X X ! S (!)[A ;! A ;! ;(t)] + X X ! A ;! (t)A ;! 1 2 fA ;! A ;! ;(t)g : (2.53) When all system operators are Hermitian, A ;! =A y ;! , we have @ t (t) =i X X ! S (!) h A y ;! A ;! ;(t) i + X X ! A ;! (t)A y ;! 1 2 fA y ;! A ;! ;(t)g ; (2.54) 11 so this equation is in Lindblad form, as required, and is called the RWA Lindblad equation. If not all system operators are Hermitian, there will nevertheless be conjugate pairs, as the entire Hamiltonian must be Hermitian, and Eq. (2.53) can again be put in Lindblad form. 12 Chapter 3 The Adiabatic Master Equation and its Applications Theadiabaticmasterequation(AME)wasderivedwiththeaimtoexplainopensystemeffectsonthe adiabatic quantum algorithm [14, 29], and is the workhorse for first-order computational simulations of quantum annealing in the Markovian-adiabatic limit [30]. In this chapter, we will present a detailed derivation of the AME with emphasis on its analogy to the time-independent rotating wave Lindblad equation and a mathematically accessible discussion of diabatic perturbations. We conclude with a discussion of its application to quantum annealing. 3.1 Closed System Quantum Adiabatic Approximation The quantum adiabatic approximation is a simplification of general time-dependent dynamics valid inthelimitofaslowlychangingHamiltonian. Inthislimit,thetime-dependentSchrödingerequation starts to resemble its time-independent counterpart. Thus, if the system starts out in an energy eigenstate at time t = 0 of a time-dependent (and non-degenerate) Hamiltonian H(t), then under the adiabatic approximation it will remain the eigenstate of H(t) that is connected to the initial one. More precisely, if we write the Hamiltonian in the diagonalized form H S = X a " a (t)j" a (t)ih" a (t)j ; (3.1) where a denotes the energy level and the eigenvalues and eigenvectors are continuous functions of time, then an initial statej"(0)i will be propagated toj"(t)i in the adiabatic limit after time t. Such adiabatic evolution is an ideal situation where the Hamlitonian is allowed to change in- finitely slowly, and is not in general achievable over a finite period of time. However, we can quantify how close a time-dependent Hamiltonian will be to the adiabatic limit. Here, we will focus on a class of Hamiltonians whose time evolution can be compactly parametrized as H(t) =t f H(s); (3.2) where t f is the total evolution time, and the Hamiltonian evolves in terms of the rescaled time parameter st=t f ; s2 [0; 1] : (3.3) 13 Therearetwolimitingenergyscalesintheadiabaticlimit: the“intrinsic” diabaticity(non-adiabaticity) h max s;a6=b jh" a (s)j@ s Hj" b (s)ij ; (3.4) and the energy gap min s;a 0. The energy gap between eigenstates is the “barrier” that separates them from each other. Therefore this simplification holds provided the rate of change h in H S (t) is sufficiently slow compared to and provided the total evolution time t f is large compared to 1= (the timescale set by ). From the intuitive combination mentioned above, that both t f and =h should be as large as possible in order to suppress transitions between energy eigenstates, one can propose the standard adiabatic condition: h 2 t f 1: (3.6) In what follows, we present the main concepts of adiabatic perturbation theory and derive the adiabatic condition for a closed system. In Section 3.2 we generalize to the dynamics of an open system and derive the adiabatic master equation. 3.1.1 Adiabatic Closed System We consider a generally time-dependent Hamiltonian H(t) (we drop the subscript S for simplicity until we need it again in Section 3.2) and the Schrödinger equation for its propagator U(t;t 0 ) @ t U(t;t 0 ) =iH(t)U(t;t 0 ) ; U(t 0 ;t 0 ) =I : (3.7) Furthermore, we assume that the Hamiltonian H(t) is diagonalizable for all t into non-degenerate instantaneous eigenvalues and eigenvectors H(t)j" a (t)i =" a (t)j" a (t)i ; (3.8) so that it can be written as H(t) = X a " a (t) a (t) ; a (t) =j" a (t)ih" a (t)j : (3.9) We define the adiabatic intertwiner as the unitary operator W (t;t 0 ) = X a a (t;t 0 ) ; a (t;t 0 ) " a (t) " a (t 0 ) : (3.10) 14 a (t;t 0 ) is a time-translation operator (it takes the a’th energy eigenstate from time t 0 to time t), which reduces to the projector a (t) when t = t 0 . For future reference let’s also introduce the equal-time transition operator ab (t)j" a (t)ih" b (t)j : (3.11) Differentiating, we see that W (t;t 0 ) satisfies the differential equation @ t W (t;t 0 ) =iK(t)W (t;t 0 ) ; (3.12) where K(t) =i X a j@ t " a (t)ih" a (t)j (3.13) is called the intertwiner Hamiltonian. By the identity resolution I = P a j" a (t)ih" a (t)j, we see that 0 =i@ t I =K(t)K y (t), so K(t) is Hermitian. In terms of the adiabatic intertwiner, the time dependent Hamiltonian may be written as H(t) =W (t;t r ) X a " a (t) a (t r ) ! W y (t;t r ) ; (3.14) for an arbitrary reference time t r from the duration of the time evolution. This means that we can “undo" the time-dependence of the projectors by transforming to the adiabatic picture, defined by applying a frame transformation with W (t;t r ) W (t). More specifically, we define the adiabatic picture operators U A (t;t 0 ) =W y (t)U(t;t 0 ) (3.15a) H A (t) =W y (t)H(t)W (t) = X a " a (t) a (t r ) (3.15b) K A (t) =W y (t)K(t)W (t) =i X ab h" a (t)j@ t " b (t)i ab (t r ) : (3.15c) Using Eq. (3.7) and the Hermitian conjugate of 3.12, we arrive at: i@ t U A (t;t 0 ) =i[@ t W y (t)]U(t;t 0 ) +iW y (t)[@ t U(t;t 0 )] (3.16a) =W y (t)K(t)U(t;t 0 ) +W y (t)H(t)U(t;t 0 ) (3.16b) = [H A (t)K A (t)]U A (t;t 0 ) : (3.16c) While t r can be arbitrary for this transformation, it’s typical to set t r = t 0 . However, this is not necessary for numerical computation purposes, where using a mid-point t r can be more accurate or convenient. Suppose that the Hamiltonian H(t) depends only on s = t=t f 2 [0; 1]. We thus write the Hamiltonian as H(s), which is independent of t f , and likewise for the eigenvalues and eigenvectors 15 " a (s) andj" a (s)i. Changing the time variable to s results in the adiabatic frame Schrödinger equation _ U A (s;s 0 ) =i[t f H A (s)K A (s)]U A (s;s 0 ) ; (3.17) where K A (s) =i X ab h" a (s)j _ " b (s)i ab (s r ) ; (3.18) where from now on the dot denotes a derivative with respect tos. Note the crucial point thatK A (s) does not get multiplied by t f . This is becauseK A (t) includes a derivative with respect to t [3.15c], so it scales inversely with t f , and this cancels with the t f that arises from the change from from t to s on the left-hand side. Equation (3.17) shows that the adiabatic limit allows us to reduce a time-dependent Hamilto- nian H(t) with nondegenerate eigenvectors into an effective diagonal Hamiltonian t f H A (s) plus a perturbationK A (s) that is comparatively small in the relevant limit as t f becomes large. With the setup of the adiabatic frame, we can apply the usual techniques of time-dependent perturbation theory to perform analytical or numerical calculations. The next step is to find an expression for the matrix elementsh" a (s)j _ " b (s)i. When a = b we have a (s) =ih" a (s)j _ " a (s)i : (3.19) This phase is directly determined by the phase of the eigenvector pathj" a (s)i and is called the Berry connection. It’s locally unobservable and can be made to vanish in most cases. However, the net phase change over a cyclic adiabatic evolution is the well-known Berry phase, which can be related to physical observables of the system. Correspondingly, we define G(s) X a a (s) a (s r ) ; (3.20) which represents the diagonal components of K A (s). For a6= b, we focus for simplicity on the case where all energy gaps are nonzero. Taking the s-derivative of Eq. (3.8) yields: _ H _ " a j" a (s)i + (H(s)" a (s))j _ " a (s)i = 0 : (3.21) Hence, if b6=a, h" b (s)j _ " a (s)i = 1 ! ba (s) _ H ba ; ! ba (s)" b (s)" a (s) : (3.22) Thus, defining J A (s) =i X a6=b 1 ! ab (s) _ H ab ab (s r ) ; (3.23) 16 we can split K A (s) into diagonal and off-diagonal components: K A (s) =J A (s) +G(s) : (3.24) 3.1.2 Diabatic Perturbations Writing the adiabatic frame Schrödinger equation (3.17) using Eq. (3.24), _ U A (s;s 0 ) =i[t f H A (s)G(s)J A (s)]U A (s;s 0 ); (3.25) it’s natural to associate the “free” evolution of the adiabatic frame with the propagator fort f H A (s) G(s) (the term that corresponds to the evolution that does not mix eigenstates). This propagator satisfies the Schrödinger equation _ V 0 (s;s 0 ) =i[t f H A (s)G(s)]V 0 (s;s 0 ) ; (3.26) whose solution is V 0 (s;s 0 ) = X a e ia(s;s 0 ) a (s r ) ; (3.27) where a (s;s 0 ) = d a (s;s 0 ) + g a (s;s 0 ) (3.28a) d a (s;s 0 )t f E a (s;s 0 ) ; E a (s;s 0 ) Z s s 0 ds 0 " a (s 0 ) (3.28b) g a (s;s 0 ) Z s s 0 ds 0 a (s 0 ) ; (3.28c) where d a (s;s 0 ) is the dynamic phase and g a (s;s 0 ) is the geometric phase, which contains no de- pendence ont f . We may also interpretV 0 (s;s 0 ) as being generated by a new effective Hamiltonian ~ H(s;s 0 ) — with a parametric dependence on s — that is diagonal in the instantaneous eigenbasis of H(s), namely: V 0 (s;s 0 ) =e it f ~ H(s;s 0 ) ; ~ H(s;s 0 ) X a [E a (s;s 0 ) + g a (s;s 0 )=t f ] a (s r ) : (3.29) At the same time we can think of the off-diagonal termJ A as a diabatic perturbation responsible for transitions between eigenstates (it contains the transition operatorsj" a (s r )ih" b (s r )j for a6= b). We thus make a second transformation into the interaction picture of V 0 (s;s 0 ), which transforms the adiabatic frame Schrödinger equation (3.17) into: _ V D (s;s 0 ) =i ~ J(s;s 0 )V D (s;s 0 ) ; (3.30) 17 where we defined the diabatic propagator V D (s;s 0 )V y 0 (s;s 0 )U A (s;s 0 ) ; (3.31) and the diabatic perturbation ∗ ~ J(s;s 0 )V y 0 (s;s 0 )J A (s)V 0 (s;s 0 ) (3.32a) =i X a6=b 1 ! ab (s) _ H ab e i ab (s;s 0 ) ab (s r ) ; (3.32b) where, using Eq. (3.28): ab (s;s 0 ) a (s;s 0 ) b (s;s 0 ) (3.33a) =t f Z s s 0 ds 0 ! ab (s 0 ) Z s s 0 ds 0 [ b (s 0 ) a (s 0 )] : (3.33b) Combining all the transformations above, the Schrödinger picture evolution is expressed as the product of three unitaries: U S (s;s 0 ) =W (s;s r )V 0 (s;s 0 )V D (s;s 0 ): (3.34) The fully adiabatic limit is reached when there are no diabatic transitions, i.e., when V D I. We thus define the adiabatic evolution propagator as: U (Ad) S (s;s 0 ) =W (s;s r )V 0 (s;s 0 ) = X a e ia(s;s 0 ) a (s;s r ) : (3.35) This tells us that in the strict adiabatic limit each eigenstate evolves independently while accu- mulating a phase that is a sum of the dynamic and geometric phases. Moreover, in analogy to Eq. (3.29), we can interpret U (Ad) S (s;s 0 ) as being generated by an effective adiabatic Hamiltonian that is diagonal in the instantaneous eigenbasis ofH(s) and has a parametric dependence ons, i.e.: U (Ad) S (s;s 0 ) =e it f ~ H (Ad) (s;s 0 ) ; ~ H (Ad) (s;s 0 ) X a [E a (s;s 0 ) + g a (s;s 0 )=t f ] a (s;s r ) : (3.36) Recallthats r isarbitrary, andwemay inparticularsetit equaltos. Withthis choice, ~ H (Ad) (s;s 0 ) is simply the instantaneous spectral decomposition of H(s), with eigenvalues equal to the cumulative dynamical and geometric phases in the interval [s 0 ;s]. ∗ The somewhat unusual dependence of the generator ~ J on two time arguments will not present a problem. 18 3.1.3 Diabatic Error Bound The deviation from strict adiabaticity is given by U S (s;s 0 )U (Ad) S (s;s 0 ). To quantify it we define the error of the adiabatic approximation via the operator norm of the difference between the actual and perfectly adiabatic evolutions: E Ad k[U S (s;s 0 )U (Ad) S (s;s 0 )]k =kW (s;s r )V 0 (s;s 0 )[V D (s;s 0 )I]k (3.37a) =k[V D (s;s 0 )I]k ; (3.37b) where in the second line we used the fact that norm is unitarily invariant. To compute V D (s;s 0 )I we can use the Dyson series solution of Eq. (3.30). The Dyson series yields, to first order: V D (s;s 0 )Ii Z s s 0 ds 1 ~ J(s 1 ;s 0 ) =I + X a6=b q ab (s;s 0 ) ab (s r ) ; (3.38) where q ab (s;s 0 ) Z s s 0 ds 1 1 ! ab (s 1 ) _ H ab e i ab (s 1 ;s 0 ) : (3.39) To compute this quantity, note first that since a contains no dependence on t f , it is negligible compared to the dynamic phase in the large t f limit of interest to us, and we can approximate Eq. (3.33b) as: ab (s;s 0 )t f Z s s 0 ds 00 ! ab (s 00 ) : (3.40) Let dv = exp it f R s 1 s 0 ds 00 ! ab (s 00 ) ds 1 , so that v = 1 it f ! ab (s 1 ) exp it f R s 1 s 0 ds 00 ! ab (s 00 ) . Letting u = _ H ab (s 1 )=! ab (s 1 ) we thus find, using integration by parts and Eq. (3.32b): q ab (s;s 0 ) X a6=b Z s s 0 udv = uvj s s 1 =s 0 Z s s 0 vdu (3.41a) =i X a6=b _ H ab (s 1 ) t f ! 2 ab (s 1 ) e it f R s 1 s 0 ds 00 ! ab (s 00 ) s s 1 =s 0 Z s s 0 ds 1 e it f R s 1 s 0 ds 00 ! ab (s 00 ) d ds 1 _ H ab (s 1 ) t f ! 2 ab (s 1 ) ! : (3.41b) Thenewintegralcanagainbeintegratedbypartsbyoncemoreidentifyingdv =e it f R s 1 s 0 ds 00 ! ab (s 00 ) ds 1 , but now u = d ds 1 _ H ab (s 1 ) t f ! 2 ab (s 1 ) . This leads via uv to the two boundary terms H ab t 2 f ! 3 ab and _ H ab t 2 f ! 2 ab _ ! ab , both of which are more suppressed in the adiabatic limit of large t f than _ H ab t f ! 2 ab . Thus, we can focus just on the boundary term in Eq. (3.41b). With this, the error in approximating V D (s;s 0 ) by I is, up to the leading term, E Ad X b6=a 2 sup s 00 2[s 0 ;s] _ H ab (s 00 ) t f ! 2 ab (s 00 ) k ab (s r )kO h t f 2 ; (3.42) 19 where we usedk ab (s r )k = 1 (since ab (s r ) is a rank one operator connecting two normalized states). The big-O notation hides the fact that the sum over b6= a includes a number of terms that scales as the square of the Hilbert space dimension. However, the vast majority of the matrix elements _ H ab is expected to be very small for eigenstates that are far apart in energy, i.e., for which in addition ! 2 ab is large. Also, if (as is typical in many applications), the system starts in an eigenstate at s = 0 (e.g.,j" a 0(0)i), then we can considerk[U S (s;s 0 )U (Ad) S (s;s 0 )]j" a 0(0)ik instead ofE Ad and carry out the same calculation as above, expect that we then apply ab (s r = 0) instead of usingk ab (s r )k = 1, which eliminates the sum over a. We conclude that the error in the adiabatic approximation is sufficiently small provided the evolution satisfies the adiabatic condition in Eq. (3.6). 3.2 Adiabatic Redfield Approximation We now consider the total system-bath Hamiltonian of Eq. (2.1), but whereH S (s) generates an adi- abatic evolution depending only on the dimensionless times =t=t f . While the system Hamiltonian is now time-dependent, we can nevertheless move the system operators into the interaction picture, i.e., A (s; 0)U y S (s; 0)A U S (s; 0) as before, but using s instead of t and writing the propagator’s initial time t 0 = 0 in addition. Changing only the differential time parameter (the bath correlation time is the natural time scale for the integral time parameter), we can write the Redfield equation [Eq. (2.41)] as _ ~ =g 2 t f X Z t f s 0 dB ()[A (s; 0);A (s=t f ; 0)~ (s)] + h:c: ; (3.43) where the dot denotes differentiation with respect to s. In the time-independent case, it was possible to simply resolve A (t) into frequency compo- nents. This is no longer possible in the general time-dependent case. However, the most important consequence of the adiabatic condition is that it allows us to replace the arbitrary time-dependent propagatorU S (s;s 0 ) with the simpler adiabatic propagatorU (Ad) S (s;s 0 ) [Eq (3.35)]. This provides us with a natural time-dependent spectral decomposition for the system operator using the adiabatic approximation: A (s; 0)A (Ad) (s; 0) =U (Ad)y S (s; 0)A U (Ad) S (s; 0) (3.44a) = X ab e i[ b (s;0)a(s;0)] j" a (s r )ih" a (s)jA j" b (s)ih" b (s r )j (3.44b) = X ab e i ba (s;0) a (s r ;s)A b (s;s r ) ; (3.44c) where the time-translation operator a (s r ;s) =j" a (s r )ih" a (s)j = y a (s;s r ), which was defined in Eq. (3.10), satisfies a (s r ;s) b (s;s r ) = ab a (s r ). 20 Let’s use Eq. (3.28) to write ba (s; 0) = b (s; 0) a (s; 0) = [ d b (s) d a (s)] + [ g b (s) g a (s)] (3.45a) =t f [E b (s)E a (s)] + g ba (s) ; (3.45b) where in writing the dynamic and geometric phases d (s) and g (s) we suppressed the initial time argument for simplicity. HereE b (s)E a (s) plays a role analogous to an integrated Bohr frequency [recall that E a (s; 0) = R s 0 ds 0 " a (s 0 )], so we may rewrite Eq. (3.44c) as A (s; 0) = X (s) e it f (s) A ( (s)) (3.46a) = X (s) e it f (s) A y ( (s)) ; (3.46b) where the sum is over all the integrated Bohr frequencies up to s =t=t f , and where A ( (s)) = X E b (s)Ea(s)= (s) e i g ba (s) a (s r ;s)A b (s;s r ) =A y ( (s)) : (3.47) Furthermore, because of the fast decay of the bath, it’s reasonable to assume that the adiabatic evolutionisslowenoughthattheenergyeigenstatesdonotchangeappreciablyduringthebathdecay time, i.e., B =t f 1. This additional fact allows us to simplify the integral using the approximation U S (s=t f ; 0) =U S (s=t f ;s)U S (s; 0) =U y S (s;s=t f )U S (s; 0) (3.48a) e iH S (s) U (Ad) S (s; 0) = X a e i"a(s) e ia(s;0) a (s;s r ) : (3.48b) It’s clear that using Eq. (3.48b) gives us a consistent spectral decomposition for A (s=t f ; 0) as well, which modifies Eq. (3.44c) to A (s=t f ; 0) X ab e i[" b (s)"a(s)] e i ba (s;0) a (s r ;s)A a (s;s r ) (3.49a) = X (s) e it f (s) A 0 ( (s)) ; (3.49b) where A 0 ( (s);) = X E b (s)Ea(s)= (s) e i[ g ba (s)! ba (s)] a (s r ;s)A b (s;s r ) ; (3.50) 21 and where ! ba (s) = " b (s)" a (s) is the instantaneous Bohr frequency. Thus, up to an errorE suppressed by the adiabatic condition and the smallness of B =t f , using Eqs. (3.46b) and (3.49b) we can approximate the integrands appearing in Eq. (3.43) as: A (s; 0)A (s=t f ; 0)~ (s) X (s); 0 (s) e it f [ 0 (s) (s)] A y ( 0 (s))A 0 ( (s);)~ (s) (3.51a) A (s=t f ; 0)~ (s)A (s; 0) X (s); 0 (s) e it f [ 0 (s) (s)] A 0 ( (s);)~ (s)A y ( 0 (s)) : (3.51b) The Redfield equation (3.43) then becomes: _ ~ =g 2 t f X ;; (s); 0 (s) e it f [ 0 (s) (s)] Z t f s 0 dB ()[A y ( 0 (s));A 0 ( (s);)~ (s)] + h:c: (3.52) 3.3 Rotating Wave Approximation in the Adiabatic Limit At this point we may apply the RWA: terms with (s)6= 0 (s) are rapidly oscillating when t f j (s) 0 (s)j 1 . The RWA amounts to setting (s) = 0 (s), i.e., equating the integrated Bohr frequencies and neglecting all other terms, so that: _ ~ =g 2 t f X X (s) Z t f s 0 dB ()[A y ( (s));A 0 ( (s);)~ (s)] + h:c: (3.53a) =g 2 t f X X (s) X E b (s)Ea(s)= (s) Z t f s 0 dB ()e i! ba (s) e i g ba (s) [A y ( (s)); a (s r ;s)A b (s;s r )~ (s)] + h:c: ; (3.53b) where in the second line we used Eq. (3.50). Note that (! ba (s)) =g 2 Z 1 0 de i! ba (s) B () (3.54) is the usual (but now time-dependent) one-sided Fourier transform of the bath correlation function. After taking the upper integration limit to t f s!1 in the only integral involving , justified by the previous assumption that B =t f 1, we can approximate Eq. (3.53a) as: _ ~ =t f X X (s) X E b (s)Ea(s)= (s) (! ba (s))e i g ba (s) [A y ( (s)); a (s r ;s)A b (s;s r )~ (s)] + h:c: (3.55) In order to make further progress we apply a stronger form of the RWA: rather then equating the integrated Bohr frequencies [ (s) = 0 (s)], we equate the instantaneous ones. Namely, rather then setting E b (s)E a (s) = E d (s)E c (s) for all pairs of indicesfa;b;c;dg [recall again that E a (s; 0) R s 0 ds 0 " a (s 0 )], we set " b (s)" a (s) = " d (s)" c (s), i.e., ! ba (s) = ! dc (s) = (s). This 22 allows us to replace (! ba (s)) by (!(s)) and pull it out of the sum, so that we can rewrite Eq. (3.55) as: _ ~ =t f X X (s) (!(s))[A y ( (s));A ( (s))~ (s)] + h:c: ; (3.56) where A ( (s)) = X " b (s)"a(s)= (s) e i g ba (s) a (s r ;s)A b (s;s r ) =A y ( (s)) (3.57) replaces Eq. (3.47), and becomes the adiabatic, interaction-picture Lindblad operator. 3.3.1 The Adiabatic Master Equation (AME) Finally, we transform back to the Schrödinger picture: (s) =U S (s; 0)~ (s)U y S (s; 0), so that _ (s) =it f [H S (s);(s)] +U (Ad) S (s; 0) _ ~ U (Ad)y S (s; 0) ; (3.58) where,consistentwiththeadiabaticapproximation,weapproximatedU S (s; 0)byU (Ad) S (s; 0)[Eq.(3.35)]. We thus need to apply U (Ad) S (s; 0)U (Ad)y S (s; 0) to Eq. (3.56). When we then compute, e.g., U S (s; 0)A y ( (s))U y S (s; 0)U (Ad) S (s; 0)A y ( (s))U (Ad)y S (s; 0) ; (3.59) this cancels the geometric phases: U (Ad) S (s; 0)A y ( (s))U (Ad)y S (s; 0) = X ab e i ba (s;0) b (s;s r ) X " b 0(s)" a 0(s) (s) e i g b 0 a 0 (s;0) b 0(s r ;s)A a 0(s;s r ) a (s r ;s) (3.60a) = X " b (s)"a(s)= (s) e i[ ba (s;0) g ba (s;0)] b (s;s)A a (s;s) (3.60b) = X " b (s)"a(s)= (s) e i d ba (s;0) b (s)A a (s) (3.60c) =e it f R s 0 ds 0 (s 0 ) L y ;! (s) ; (3.60d) where we used Eq. (3.28), wrote a (s;s) = a (s) =j" a (s)ih" a (s)j as per Eq. (3.10), and defined the adiabatic Schrödinger picture Lindblad operators L ;! (s) X " b (s)"a(s)=! a (s)A b (s) = X " b (s)"a(s)=! A ;ab (s)j" a (s)ih" b (s)j ; (3.61) 23 where A ;ab (s) =h" a (s)jA j" b (s)i. ∗ The remaining dynamic phase factors [like e it f R s 0 ds 0 (s 0 ) in Eq. (3.60d)] cancel as well when we combine the Lindblad operators. Aswiththetime-independentRWA,weproceedtoseparate (!(s))intoitsrealandimaginary components. We thus arrive at the adiabatic master equation (AME): _ (s) =it f [H S (s) +H LS (s);(s)] (3.62) +t f X !(s); (!(s)) L ;! (s)(s)L y ;! (s)) 1 2 fL y ;! (s)L ;! (s);(s)g ; where (!(s)) =g 2 Z 1 1 e i (s) B ()d (3.63) and the Lamb shift is: H LS (s) = X !(s); S (!(s))L y ;! (s)L ;! (s) (3.64a) S (!(s)) = 1 2 Z 1 1 (! 0 )P 1 (s)! 0 d! 0 =S (!(s)) ; (3.64b) which should be compared with the corresponding expressions in Section 2.3.2: the difference is that! and the Lindblad operators have acquired ans-dependence. Since the AME is manifestly in Lindblad form with a positive matrix, it generates a CP map. Working with the AME requires knowledge of the instantaneous energy eigenstatesfj" a (s)ig and their eigenvaluesf" a (s)g, since they are needed to construct the Lindblad operators and the ( (s)) and S ( (s)) quantities. This is similar to the time-independent case, where one also needs to find these eigenvectors and eigenvalues, but the difference is that now these are needed for each s, not just once. The AME can be thought of as a natural generalization of the time-independent RWA-LE case, where the Hamiltonian varies sufficiently slowly relative to the bath so that at any instant in time we simply have a copy of the time-independent master equation. In fact, it is clear that we recover the time-independent result by replacing H S (s) by a time-independent system Hamiltonian. 3.3.2 Timescales Let’s now summarize the relations between the different timescales that have arisen in our derivation of the AME, which will tell us the conditions for its validity. First, we have the closed system adiabatic condition t f h= 2 . Second, the validity of the weak coupling and the Markov approximation requires g B 1 andt B [14]. The first of these ∗ You may wonder why we dropped the s-dependence on in writing Eq. (3.61). The reason is to economize on the notation; we could have written L ;!(s) (s), but this is cumbersome. Still, except when " b (s) = "a(s), it should be remembered that ! really does depend on s. Also, note that it really is necessary to make the s-dependence of L;! explicit since these Lindblad operators depend on s not just through! but also through the eigenprojectors a and b . 24 is a weak-coupling approximation, and the second states that we do not expect our approximation to be accurate for times shorter than the bath correlation time. Since by definition t f > t, we can immediately convert the latter to a condition involving the total evolution time: t f B . In addition, we should require that changes in the system Hamiltonian are slow on the scale set by the bath correlation time: h B 1. Finally, the coupling should remain weak also relative to the smallest energy scale set by the system, i.e., g . We can combine all of the above into the following compact set of validity conditions for the AME: h B t f min B ; 1 B (3.65a) g B min 1; g : (3.65b) 3.4 Quantum Annealing and Decoherence The adiabatic limit of the Schrödinger equation forms the basis for the most influential form of Hamiltonian quantum computation: the adiabatic algorithm, sometimes interchangeably called quantum annealing [3, 12, 13, 31, 32]. Suppose the HamiltonianH(s) is parametrized to evolve from an initial Hamiltonian H 0 , whose ground state is easy to prepare, to a more complex Hamiltonian H f whose ground state is of interest. If the interpolation is linear, for example, H(s) = (1s)H 0 +sH f ; s2 [0; 1]: (3.66) While stated as a physical process, this evolution can be considered a genuine algorithm with a computation time T that, on the basis of the adiabatic condition, scales as T 1= 2 . The final Hamiltonian H f may encode the cost function of an NP-complete optimization problem, which is hence solved with a successful execution of the algorithm. The time complexity of the adiabatic algorithm therefore depends directly on the behavior of the minimum gap as the system size in- creases, which becomes a question of the nature of the quantum phase transition of H(s). There are few results on the character of the minimum gap outside of simple toy models. However, it is believed that a certain class of Hamiltonian operators, the stoquastic class, is efficient to simulate classically [33]. As only stoquastic Hamiltonians have been implemented and scaled in present-day quantum annealers, this poses a limitation on near-term prospects for realizations of algorithmic quantum speedup. Inevitably, realization of AQC will be subject to decoherence due to the presence of a finite- temperature environment. We specifically refer the open-system model of AQC and its implemen- tation in hardware as quantum annealing (QA). We must therefore work within the framework of open quantum systems to understand how the adiabatic algorithm performs in practice, and the AME plays a significant role in this understanding [14, 29]. 25 Inexperimentsonmostquantumcomputinghardware(includingquantumannealers), theeffects of the environment resulting in the loss of relative phases as well as entanglement (i.e. decoher- ence) is very often characterized using two Markovian processes: dephasing and thermal relaxation. Dephasing results from the “scattering” of phase between different computational states and arises from coupling with system operators A = z for each individual qubit . Thermal relaxation, or generalized amplitude damping, occurs when the system exchanges energy with an environment in thermal equilibrium, for which the bath operators that satisfy the detailed balance condition (!) =e ! (!); (3.67) also known as the frequency-space statement of the Kubo-Martin-Schwinger (KMS) condition [34]. In a noise model where decoherence occurs independently on each qubit, each qubit is coupled to its own bath through both + and operators as in Eq. (2.12). The rate of population decay due to thermal relaxation is called the T 1 time, while the rate of coherence loss due to dephasing is called the T 2 time. These are the most critical summary descriptors of the quality of a qubit, and increasing them is key to achieving FTQC and coherent quantum annealing. However, decoherence does not necessarily need to be environment induced. It can arise due to uncontrolled interactions between system elements, called crosstalk. These interactions can be mitigated using the same error-suppression techniques as environment-induced decoherence, one notable example in the gate model being dynamical decoupling [35]. The D-Wave quantum annealers implement the transverse field Ising model using superconduct- ing flux qubits, with programmable ferromagnetic and anti-ferromagnetic interactions on a fixed graph of available couplings [3]. That is, the Hamiltonian has the form H(s) = A(s) 2 H X + B(s) 2 H Z ; (3.68) where A(s) and B(s) are hardware-defined schedules of the annealing parameter s = t=t f , H X is the driver Hamiltonian, implementing with one-local transverse fields H X = X i x i ; (3.69) and H Z encodes a classical two-local Ising Hamiltonian H Z = X i h i z i + X i<j J ij z i z j ; (3.70) where non-zero couplings J ij are subject to the restrictions of the quantum annealer graph. The dominant source of decoherence for a quantum annealing Hamiltonian is independent de- phasing in each qubit. Qualitatively, this can be seen when H Z encodes a difficult problem where the lowest energy levels may be a large distance away from each other, and hence would be difficult 26 to mix with one-local spin flips. WhenH X perturbsH Z , the low-energy eigenstates are well approx- imated by linear combinations of the eigenstates ofH Z . Therefore, dephasing in the computational basis results will result in significant non-zero L ;! (s) operators mixing the low-energy eigenstates. An extension of this analysis will be given in Chapter 6 in the context of boundary-canceling sched- ules. 27 Chapter 4 Hamiltonian Error Suppression in the Born-Markov Approximation The formalism of stabilizer codes is perhaps the most common and useful tool for the design of quan- tum error correcting codes, and has formed the foundation of a large amount of modern research into such codes and the prospects for fault tolerant quantum computation [16]. While generally formulated for gate model computation, which involves active measurement and correction oper- ators, in Hamiltonian error suppression stabilizer codes are adapted to passively suppress logical errors due to decoherence. This approach was motivated as a way to protect adiabatic quantum computation from one-local and two-local Markovian noise [36–38] and has been analyzed for gen- eral Hamiltonian quantum computation [39, 40]. In this chapter, we initiate an extension of the open system analysis of Hamiltonian error suppression and develop a general formalism for quantum master equations to simulate error-suppressed systems, which goes beyond the earlier Markovian master equation results. We first briefly review necessary mathematical aspects of stabilizer codes in the first section. Our main result is a master equation of hybrid Redfield-Lindblad form we call the error-suppressed master equation, which treats excitations from the code space in the weak-coupling Markovian limit without assuming a particular treatment for logical errors beyond the Born-Markov approximation. Under certain assumptions, a Pauli equation of the error-syndrome populations can also be obtained, independent of the logical computation that is implemented, which entails that the strength of the penalty must scale logarithmically with the number of stabilizers. Finally, we present a brief error analysis showing that this master equations is consistent with the non-Markovian Born equation in the limit of strong error-suppression. 4.1 Stabilizer Codes Let us recall the standard theorem on subsystem decompositions of a Hilbert space: 28 4.1.1 Irreducible Representations Theorem 4.1 (Subsystem Decomposition). [41] LetA be ay-closed algebra of operators on a Hilbert spaceH. Then there exists an isomorphism betweenA and a block-diagonal decomposition A' M J I n J M d J ; (4.1) whereJ is a label over irreducible representations ofA,M d is the algebra ofd-dimensional complex matrices and I n . The commutant ofA is the algebra of operators of the form ZA' M J M n J I d J ; (4.2) Note that the isomorphism of the Hilbert space decomposition is only unique up to automor- phismsoftheC n J andC d J Hilbertspaces. Thisisusuallyofnoconcernforquantumerrorcorrection as we are only concerned with implementing specified logical operations on a privileged irrep label J = 0. Here, we will be concerned with the dynamics of the state in non-code irreps, all of which are isomorphic to the code irrep. Hence it is necessary and most convenient to explicitly fix the isomor- phism into subsystems such that a logical operator on the code space has the same representation for all J. Let us illustrate this point concretely with the classical [3; 1; 3] repetition code. A natural way to label the irreps may be through the position i of the single-qubit error X i and map each logical state to X i . J = 0 : fj000i;j111ig J = 1 : fj001i;j110ig J = 2 : fj010i;j101ig J = 3 : fj100i;j011ig: However, the eigenvectors of the logical operator Z =Z 1 Z 2 Z 3 are not preserved within each irrep. In order to ensure Z is represented identically in each J, we need to swap the representation of the logical state in J = 1; 2; 3, resulting in J = 0 : fj000i;j111ig J = 1 : fj110i;j001ig J = 2 : fj101i;j010ig J = 3 : fj011i;j100ig: 29 As we have given Z a block-diagonal representation, where each block is identical, we have naturally given it a tensor product representation Z' 1 0 0 1 ! I J ; (4.3) where I J is the identity operator on the irrep factor. Additionally, detectable Pauli errors will also have a tensor product form under this same representation X 1 =X (j1ih0j +j0ih1j +j2ih3j +j3ih2j) (4.4) where the irrep factor operator is simply a permutation matrix and the logical effect is a bit-flip error. These arguments are straightforwardly generalized in the propositions that follow, which are minor generalizations of well-known characterizations of stabilizer groups and detectable errors [16, 41–43]. (See, in particular, Sec. 6.3 of [16].) Proposition 4.1. LetS =hS i i i=1;:::;nk be a [[n;k;d]] stabilizer code (i.e. an abelian subgroup of the n qubit Pauli group G n with nk generators not containingI n ), and supposeL = h X i ; Z i i i=1;:::;k G n are the chosen logical operators of the code encoding k qubits. There exists a unique representation ZS'L S such that the logical operators are represented as Z i ' (I 1 ::: Z i ::: I k ) I nk ; X i ' (I 1 ::: X i ::: I k ) I nk : (4.5) In addition, the stabilizer generators are represented as S i =I k (I 1 ::: Z i :::I nk ) : (4.6) We call this the factor representation. Proof. A complete orthonormal basis for the Hilbert space, and in particular its subsystem decom- position, can be formed using the simultaneous eigenvectors ofS[f Z i g. The basis states can be written asjs 1 ;:::;s nk ;b 1 ;:::b k i, where s i 2f0; 1g is the bit of the i-th stabilizer and b i 2f0; 1g is the bit of the i-th logical Z operator, forming n orthonormal basis states in total. Taking the matrix elements of the stabilizers in this basis, we have Eq. (4.6), and likewise the matrix elements of the logical Z operators are Z i ' M J I 1 ::: Z i ::: I k ' (I 1 ::: Z i ::: I k ) I nk ; (4.7) where Z i is the standard Pauli Z matrix. To obtain a corresponding representation for X i , it is only necessary to apply an appropriate unitary transformation on each block to change the relative 30 phases of the eigenstates of Z i , resulting in a unique orthonormal basis (up to global phase) such that, in addition to (4.7), X i ' (I 1 ::: X i ::: I k ) I nk : (4.8) The algebra generated by the logical operators therefore has the desired factor representation. Proposition4.2. LetS =fS i g i=1;:::;nk be an [[n;k;d]] stabilizer code and supposeA2G n . Then A can be represented as a product of stabilizers, logical operators, and irrep swaps i =I k (I 1 ::: X i :::I nk ) : (4.9) Proof. Clearly X i , Z i , S i , and i form a complete set of generators of the Pauli group G n in the factor representation, so this follows by applying the appropriate group automorphism. We thus observe a Pauli error is detectable if and only if it includes an irrep swap as one of its factors, since then it anticommutes with one of the stabilizer generators by Eq. (4.6). Let J be a projective operator on any irrep J (not just the code space). We have therefore demonstrated that a Pauli error A is detectable if and only if J A J = 0; for all J2Z nk 2 (4.10) The factor representation is identical for subsystem codes, where the stabilizer groupS is simply extended by a subgroupA of the original logical operators to form a non-abelian gauge groupG. We simply transfer some of the qubits in the logical factor to form a new factor spaceL A S, withA being the gauge factor. 4.2 Hamiltonian Error Suppression Suppose that the system Hamiltonian is encoded as H S (t) using the logical operators of an error detecting stabilizer codeS =hS i i i=1;:::;nk enforced by a penalty HamiltonianH P consisting of the generators of the stabilizer group. That is, H P = P 2 nk X i=1 S i : (4.11) where P is the frequency, or energy gap, of the penalty Hamiltonian. Thus H P commutes with H S (t). The total system and bath Hamiltonian is replaced by H tot = H S (t) +H P +H SB +H B : (4.12) 31 The chosen stabilizer code will partition the Hilbert into a direct sum of 2 nk irreps, each of size 2 k . As shown above, the Hilbert space of states under the algebra of ZS can be written as a tensor product decomposition H(t) = H S (t) +H P (4.13a) = M J H (J) S (t) + X J E (J) P J (4.13b) =H S (t) I n J +I S E P;j ; (4.13c) whereH S is the logical factor of the encoded system Hamiltonian andE P;j is a diagonal matrix with the eigenvalues of H P for each irrep. Every irrep J directly corresponds to a syndrome bitstring corresponding to the eigenvalues of the stabilizers in J, and hence the penalty energy levels are E (J) P = P 2 (k 2jJj) (4.14) wherejJj is the weight of the syndrome string of the irrep. We will treat the syndrome strings inter- changeably with its irrep label from now on, with the convention that the code space corresponds to J = 0. Since [H P ;H S (t)] = 0 we can write U(t) = T + exp i R t 0 H(t 0 )dt = U S (t)U P (t), which is a product of the two commuting unitary operators U S (t) =T + exp i Z t 0 H S (t 0 )dt 0 ; U P (t) =e iH P t = X J e iE (J) P J : (4.15) The reduced system state in the H(t)-interaction picture is I (t) = U y (t)(t)U(t), and A I (t) = U y (t)A U(t). Let us suppose the system operators are Pauli operators, and hence have aL S decomposition A = L S ; (4.16) where L is the logical operator, S is a product of stabilizers, and is the irrep swap operator. Since we requireA to be a Hermitian operator, each factor L andS must both be Hermitian or anti-Hermitian. This is immediately characterized by whether S and commute or anti- commute. That is, (S ) y = S =c S ; (4.17) where c 8 < : 1 fS ; g = 0 1 [S ; ] = 0 : (4.18) It therefore follows that L y =c L ; (4.19) 32 and since L is itself in the Pauli group, c L 2 = L y L = I: (4.20) Denote the logical error in the interaction picture by L (t) = U y S (t) L U S (t), which is not generally decomposable for an arbitrary time-dependent Hamiltonian. However, since the penalty Hamiltonian is time-independent, the irrep swap admits an interaction picture frequency decompo- sition (t) = X ! P E (J 0 ) P E (J) P ;! P e i! P t JJ 0 (4.21) where! P isa transitionfrequencyofthe penaltyHamiltonianand JJ 0 = J J 0. The projectors commute with the S . Hence, we write the system operators in the frequency decomposition A (t) = X ! P e i! P t A ! P (t); (4.22) where A ! P (t) = X J;J 0 E (J 0 ) P E (J) P ;! P A JJ 0(t) (4.23) and we write A JJ 0(t) = L (t) S JJ 0: (4.24) 4.2.1 Born Master Equation with Error Suppression We will generalize previous approaches to Hamiltonian error suppression using the NZME with a suitable choice of projection operator. Large energy penalties relative to the system frequencies will result in the suppression of states in a coherent superposition across different irreps. Hence, we will choose the NZME projection operator P = X J J Tr B [] J B ; (4.25) which, inadditiontoresettingthebathstate, removesanycoherencebetweendifferentirreps. Letus analyzethelowest-orderexpansionoftheNZME.Furthermore,letuswrite J (t) = J Tr B [(t)] J (t) for the sub-density matrix corresponding to the irrep J. We should then evaluate the Liouvillian product PL(t)L(t 0 )P = X J;J 0 Tr B [ J A(t) B(t); A(t 0 ) B(t 0 ); J 0 B J ]: (4.26) 33 Using the frequency decomposition, we can write A (t)A (t) J 0(t) = X ! P ! 0 P e i! P e i(! 0 P ! P )t A ! 0 P (t)A ! P (t) J 0(t) (4.27a) J 0(t)A (t)A (t) = X ! P ! 0 P e i! P e i(! 0 P ! P )t J 0(t)A ! P (t)A ! 0 P (t) (4.27b) A (t) J 0(t)A (t) = X ! P ! 0 P e i! P e i(! 0 P ! P )t A ! P (t) J 0(t)A ! 0 P (t) (4.27c) A (t) J 0(t)A (t) = X ! P ! 0 P e i! P e i(! 0 P ! P )t A ! 0 P (t) J 0(t)A ! P (t): (4.27d) Applying theP superoperator to terms of the first and second types will make them vanish except for J 0 = J and leave only the transition frequencies involving the J irrep, so we can reuse the J 0 label for the remaining sum over intermediate irrep transitions, resulting in J A (t)A (t) J (t) J = X J 0 e i! JJ 0 A JJ 0(t)A J 0 J (t) J (t) (4.28a) J J (t)A (t)A (t) J = X J 0 e i! JJ 0 J (t)A JJ 0 (t)A J 0 J (t): (4.28b) where ! JJ 0E (J 0 ) P E (J) P . Terms of the third and fourth types are projected to J A (t) J 0(t)A (t) J =e i! JJ 0 A JJ 0 (t) J 0(t)A J 0 J (t) (4.29a) J A (t) J 0(t)A (t) J =e i! JJ 0 A JJ 0(t) J 0(t)A J 0 J (t): (4.29b) Thus, we arrive at the Born master equation with error suppression. Its block diagonal representa- tion is @ t J (t) = X ;J 0 Z t 0 dB () e i! JJ 0 A JJ 0(t)A J 0 J (t) J (t)e i! JJ 0 A JJ 0 (t) J 0(t)A J 0 J (t) X ;J 0 Z t 0 dB () e i! JJ 0 J (t)A JJ 0 (t)A J 0 J (t)e i! JJ 0 A JJ 0(t) J 0(t)A J 0 J (t) : This form is convenient for integrating the master equation in block-diagonal form as it explic- itly shows the decoherence in irrep J due to each other irrep J 0 . The similarity with the Born master equation is apparent by writing out the complete state in the interaction picture I (t) = P J J (t) J = P J J (t) and swapping the J and J 0 summation variables in appropriate terms, yielding @ t I (t) = X ;JJ 0 Z t 0 dB ()e i! JJ 0 h A J 0 J (t);A JJ 0 (t) I (t) i + h:c:: (4.30) 34 Unlike the Born master equation, here we sum explicitly over the eigenspaces of the penalty Hamil- tonian rather than its transition frequencies. This simplification is possible as we are not concerned with off-diagonal coherences between different irreps. Let us now consider the consequences of the error detection condition Eq. (4.10). If all of the system operators are detectable Pauli errors, then all terms with J =J 0 drop out. @ t I (t) = X ;J6=J 0 Z t 0 dB ()e i! JJ 0 h A J 0 J (t);A JJ 0 (t) I (t) i + h:c:: (4.31) This does not mean that all terms with ! JJ 0 = 0 drop out. However, by construction with the penalty Hamiltonian, these zero-frequency channels only affect irreps at energies higher than the code space. 4.2.2 Error-Suppressed Master Equation Upon taking the first Markov approximation I (t)7! I (t), we arrive at the equivalent Redfield equation under error suppression. @ t I (t) = X ;J6=J 0 h A J 0 J (t); ~ A JJ 0 (t) I (t) i + h:c:; (4.32) where ~ A JJ 0 (t) denotes the “filtered” operator ~ A JJ 0 (t) = Z t 0 dB ()e i! JJ 0 A JJ 0 (t): (4.33) At this point, one can apply standard techniques (e.g. coarse-graining [19] or the adiabatic ap- proximation [14]) to arrive at a completely positive Markovian master equation from this Redfield equation. Error suppression has been analyzed in the adiabatic Markovian limit for stabilizer codes [37], and coarse-graining has been thoroughly presented in the generic time-dependent setting [19], so we do not pursue this direction. Instead, let us pursue a generic description of error suppression that can be made compatible with any of these techniques. First, we note that we can rewrite Eq. (4.32) as @ t I (t) = X J J [L 0;BM ( I (t)) +L P;BM ()] J ; (4.34) which contains a sum of a zero-frequency generator L 0;BM () X [A 0 (t); ~ A 0 (t)] + h:c:; (4.35) and a non-zero frequency generator L P;BM () X X ! P 6=0 [A ! P (t); ~ A ! P (t)] + h:c:; (4.36) 35 where A ! P (t) was defined in Eq. (4.23) and ~ A ! P (t) is the filtered A ! P (t) as in Eq. (4.33), i.e., ~ A ! P (t) = Z t 0 dB ()e i! P A ! P (t): (4.37) ToseethatEq.(4.32)andEq.(4.34)areequivalent, werecallthateachA operatoralwaysperforms a one-to-one swap on the irreps, so Eq. (4.23) collapses into a single sum under an irrep projector, i.e., J A ! P (t) = X J 0 E (J 0 ) P E (J) P ;! P A JJ 0(t): (4.38) Thus, when the block-diagonal structure of is enforced in Eq. (4.34) @ t I (t) = X J X X ! P J [A ! P (t); ~ A ! P (t) I (t)] J + h:c: (4.39a) = X J X X ! P J A ! P (t) ~ A ! P (t) I (t) ~ A ! P (t) I (t)A ! P (t) J + h:c: (4.39b) = X X ! P X JJ 0 E (J 0 ) P E (J) P ;! P A JJ 0(t) ~ A J 0 J (t) J (t) ~ A JJ 0 (t) J 0(t)A J 0 J (t) + h:c: (4.39c) = X ;JJ 0 A J 0 J (t) ~ A JJ 0 (t) J 0(t) ~ A JJ 0 (t) J 0(t)A J 0 J (t) + h:c: (4.39d) = X ;JJ 0 h A J 0 J (t); ~ A JJ 0 (t) I (t) i + h:c: (4.39e) = X ;J6=J 0 h A J 0 J (t); ~ A JJ 0 (t) I (t) i + h:c:; (4.39f) where in Eq. (4.39b) we substituted the summation variable ! P ! ! P for the first term, in Eq. (4.39c) we immediately suppressed the sum P J 00 E (J) P E (J 00 ) P ;! P created by the projector on the right by enforcing J 0 =J 00 , and finally in Eq. (4.39d) eliminated the sum over frequencies with the delta function, swapped the J and J 0 summation variables in the first term in Eq. (4.39e), and finally enforced the error detection condition in Eq. (4.39f), which is precisely Eq. (4.32). We can ascribe simple interpretations to these generators: L P;BM takes the computation out of the code space, but at a rate suppressed by frequencies on the order of P , whileL 0;BM mixes irreps with the same penalty Hamiltonian energies, from which the code space is fully protected by definition. An undetectable error may therefore occur when a state is excited out of the code space byL P;BM , scattered byL 0;BM , and finally relaxed back to the code space. The extent of the scattering that is necessary to incur a logical error will depend directly on the type of error and the distance of the code. The logical operators are generally part of a time-dependent Hamiltonian. However, as the penalty Hamiltonian is time-independent and its associated frequencies are much larger than those associated with the logical computation, we immediately suspect thatL P;BM generates, in effect, 36 Markovian dynamics in the weak coupling limit among the irreps. Let us therefore apply the standard procedure to derive a Lindblad generator fromL P;BM : the rotating wave approximation (which is already imposed due to (4.25)) and extending the integration domain in ~ A to [0;1]. We thus obtain L P ()i[H LS;P (t);] + X X ! P 6=0 (! P ) A ! P (t)A ! P (t) 1 2 fA ! P (t)A ! P (t);g ; (4.40) where, as in the time-independent case, we have the rates (!) = Z 1 1 de i! B (); (4.41) and the Lamb shift H LS;P (t) = X ! P 6=0; S (! P )A ! P (t)A ! P (t) (4.42a) S (!) 1 2 Z 1 1 P 1 !! 0 (! 0 )d! 0 : (4.42b) We have thus arrived at the error-suppressed master equation for the state of the system in the interaction picture @ t I (t) = X J J [L 0;BM ( I (t)) +L P ( I (t))] J : (4.43) When the block-diagonal structure is enforced, via the same simplification as Eq. (4.39), X J J L P () J =i[H LS;P (t);]+ X X JJ 0 ;! JJ 06=0 (! JJ 0) A JJ 0 (t)A J 0 J (t) 1 2 fA J 0 J (t)A JJ 0 (t);g : (4.44) 4.2.3 Suppression of Uncorrelated Pauli Errors As it turns out, even if one begins with the standard NZME projector (2.24) rather than (4.25), the blocks J J 0 decouple into diagonal and off-diagonal blocks when the system operators consist of uncorrelated Pauli errors. That is L P X J 0 J 0 J 0 ! = X J J L P X J J J ! J ; (4.45) which implies Eq. (4.43) holds (i.e. the block-diagonal structure is preserved) when has a block- diagonal initial state. The error-suppressed master equation that is obtained from Eq. (2.24) is @ t I (t) =L 0;BM ( I (t)) +L P ( I (t)): (4.46) 37 We present a detailed error analysis in Section 4.3 which derives Eq. (4.46) from the Born equation. Let us show that (4.46) simplifies to (4.43) under uncorrelated Pauli errors. The correlation function decomposes as B () = B (); (4.47) and likewise (!) = (!) and S (!) = S (!). First, we note that since irreps are only coupled by error operators through swaps, the terms A ! P (t) J 0 J A ! P (t) do not couple the block-diagonal with other blocks of the state, i.e. on a diagonal block, X ! P 6=0 (! P )A ! P (t) J 0A ! P (t) = X JJ 0 ;! JJ 06=0 (! JJ 0)A JJ 0(t) J 0A J 0 J (t) (4.48) Next, we note that, by Eq. (4.24), A JJ 0(t)A J 0 J (t) = L (t) 2 S JJ 0S J 0 J (4.49a) =c L (t) 2 JJ 0S 2 J 0 J (4.49b) =d JJ 0 I J (4.49c) where we used [S ; J ] = 0 and Eq. (4.20), and we define d JJ 0hJj J 0 =d J 0 J : (4.50) We therefore observe the simplification A ! P (t)A ! P (t) = X JJ 0 ! JJ 0;! P d JJ 0 J (4.51) so X ! P 6=0 (! JJ 0)A ! P (t)A ! P (t) = X JJ 0 ;! JJ 06=0 (! JJ 0)d J 0 J J 0 = X JJ 0 ;! J 0 J 6=0 (! JJ 0)A J 0 J (t)A JJ 0: (4.52) Thus, with both Eq. (4.48) and Eq. (4.52) L P ( J 0 J 0) = X J;! JJ 06=0 (! JJ 0) (A JJ 0(t) J 0A J 0 J (t) +fA J 0 J (t)A JJ 0; J 0g) = X J J L P ( J 0 J 0) J (4.53) which implies Eq. (4.45) by linearity. 4.2.4 A Simplified Pauli Master Equation over Irreps Let us now completely ignore the logical state of the system and consider only the irrep populations p J Tr[ J I (t)]: (4.54) 38 Furthermore let us assume (and later justify) that for all time t 0, Tr[ J L 0;BM ( I (t))] = 0 (4.55) With uncorrelated Pauli errors, we can derive a Pauli master equation for these populations. Ap- plying the trace on the relevant terms ofL P ( I (t)) and using Eq. (4.49), Tr[A JJ 0(t) I (t)A J 0 J (t)] =d J 0 J p J 0 (4.56a) Tr[A JJ 0(t)A J 0 J (t) I (t)] =d JJ 0p J : (4.56b) Thus, using Eq. (4.55) and Eq. (4.56), @ t p J = Tr[ J @ t I (t)] = Tr[ J L 0;BM ( I (t))] + Tr[ J L P ( I (t))] (4.57a) = X J 0 ( (! JJ 0)d J 0 J p J 0 (! J 0 J )d J 0 J p J ) (4.57b) = X J 0 W J;J 0p J 0; (4.57c) where the transition matrix is W J;J 0 X (! JJ 0)d J 0 J X J 00 (! J 00 J )d J 00 J J 0 J ! : (4.58) This is the Pauli master equation for the irreps (i.e. the error syndrome) populations, which we derived with no assumptions on the computation encoded byH S (t), and is valid even whenL 0;BM is not in Lindblad form. Provided the KMS condition is satisfied and the transition matrix is ergodic, p J is distributed according to the Gibbs state of H P in the long-time limit. We therefore conclude that we have exponential error suppression in P , as the probability that the system is excited out of the code space is polynomial in e P . Let us introduce a final assumption on the uniformity of the errors. Suppose that the error rates are such that the rate matrix is symmetric under the exchange of degenerate irreps (i.e. with ! P = 0). That is, every bit in the syndrome is equally likely to be flipped at any point and the bits are uncorrelated. In this case, population transfer due toL 0;BM is clearly irrelevant since the net flux between any two degenerate irreps is zero, hence justifying (4.55). The probability that any single bit in the stabilizer is zero in the Gibbs state is (1 +e P ) 1 . Hence, the population remaining in the codespace in the long time limit is p 0 = (1 +e P ) (nk) (4.59a) 1 (nk)e P +::: : (4.59b) 39 We thus conclude, regardless of the details of the logical Hamiltonian, P 1 log nk 1p 0 : (4.60) That is, the energy penalty should scale logarithmically with the number of generators of the stabilizer code to preserve a desired code space population at a given temperature as the size of the system increases. 4.2.5 Example: Repetition Code Let us consider the three-qubit repetition code with stabilizer generatorsS 1 =Z 1 Z 2 andS 2 =Z 2 Z 3 to encode a single logical qubit and protect against bit-flip errorsfX 1 ;X 2 ;X 3 g. Assume all three errors are all uncorrelated with an identical correlation functionB(). The penalty Hamiltonian that is enforced is H P = P 2 (Z 1 Z 2 +Z 2 Z 3 ) (4.61) The code space irrep is J = 0, while the two degenerate irreps are J = 1; 2, and finally the highest energy irrep is J = 3. Using the numerical value of the syndrome bits to label the irreps J =b 2 b 1 , the error operators have a factor representation X 1 = X 1 ; X 2 = X 1 2 ; X 3 = X 2 :: (4.62) where i is the operator that flips the i-th bit of the syndrome. In the interaction picture of H P , X 1 = X (e i P t j1ih0j +e i P t j0ih1j) (4.63) X 2 = X (j1ih2j +j2ih1j +e 2i P t j3ih0je 2i P t j0ih3j) (4.64) X 3 = X (e i P t j2ih0j +e i P t j0ih2j): (4.65) Only X 2 causes zero-frequency transitions between irreps 1 and 2, so it is the only operator in L 0;BM : L 0;BM () = h X(t) j1ih2j; ~ X(t) j2ih1j i h X(t) j2ih1j; ~ X(t) j1ih2j i + h:c:; (4.66) where X(t) is in the interaction picture of H S and ~ X(t) is the same operator filtered byB(). Furthermore, sinceX 1 andX 3 have an identical correlation function, the generatorL P is symmetric under swapping irreps 1 and 2. Hence 1 (t) = 2 (t) for all t 0 if the initial condition is also symmetric under this swap. The effect ofL 0;BM on the irrep populations is therefore irrelevant, as Tr[ 1 L 0;BM ((t))] =Tr[ X(t) ~ X(t) 1 (t)] + Tr[ X(t) ~ X(t) 2 (t)] + h:c: (4.67a) = Tr[ X(t) ~ X(t)( 2 (t) 1 (t))] + h:c: (4.67b) = 0: (4.67c) 40 1 2 5 10 20 0 2 4 6 8 10 12 14 Figure 4.1: The expected temperature-scaled penalty energy P required to preserve the code- spacepopulationforasystemwhereeachqubitisencodedwiththe [3; 1; 3]repetitioncode, forwhich the number of stabilizer generators is nk = 2k, according to Eq. (4.60). The exact numerical solution for P in Eq. (4.59a) is not visually distinguishable from Eq. (4.60). Thus, the irrep Pauli equation (4.57) holds for this error suppression scheme. The values of P that are required to enforce the code-space population at a given temperature are illustrated in Fig. 4.1. 4.3 Error Analysis If we do not assume that the state has a block-diagonal structure in the irreps, its non-Markovian dynamics are described, to lowest order in system-bath coupling, not by Eq. (4.30) but by the full Born equation @ t I = X X ! P ! 0 P e i(! 0 P ! P )t Z t 0 dB ()e i! P [A ! 0 P (t);A ! P (t) I (t)] + h:c:; (4.68) where many terms such that ! 0 P 6= ! P are present, resulting in rapidly oscillating terms with frequencies on the order of the error suppression penalty strength. We will consider here the local error estimate of approximating the full Born equation by the error-suppressed master equation. First, it is well-established that taking the Markov approximation and extending the time inte- gration to t!1 both have an associated local error of [19] E 1 =O( B = 2 SB ); (4.69) 41 so we incur this error immediately upon approximating the Born equation with the Redfield equa- tion. Second (suppressing the and indices in what follows) we wish to estimate the error of replacing the filtered system operator ~ A ! P (t) Z t 0 dB()e i! P A ! P (t) (4.70) with its time-local approximation ~ A ! P (t)! (! P )A ! P (t); (4.71) which, with the rotating wave approximation, yields a generator in Lindblad form. As we expect the frequencies ! P to be large compared to the relevant frequencies of the computation, it will be convenient to keep in mind the asymptotic expansion of an oscillatory integral: Proposition 4.3. Let I ! be the functional integral I ! [f] = Z b a f(x)e i!x dx: (4.72) We can write the oscillatory integral as asymptotic expansion I ! [f] = s X k=1 1 (i!) k y k (b)e i!b y k (a)e i!a + 1 (i!) s I[y 0 k ] (4.73) where y k (x) = d k1 dx k1 f(x): (4.74) The error of the time-local approximation is directly related to the magnitude of ~ A ! P (t) (! P )A ! P (t) = Z 1 0 e i! P B() U y S (t; 0)A ! P U S (t; 0)A ! P : (4.75) Let us apply the asymptotic expansion, with the integrand and its first derivative y 1 () =B() U y S (t; 0)A ! P U S (t; 0)A ! P (t) B()A ! P ;t () y 2 () =B 0 () U y S (t; 0)A ! P U S (t; 0)A ! P (t) iB()U y S (t; 0)[H S (t);A ! P ]U S (t; 0) B 0 ()A ! P ; (t) +B()@ A ! P ; (t): We note that y 1 vanishes at 0 and1. We therefore have ~ A ! P (t) (! P )A ! P (t) = i ! P Z 1 0 e i! P y 2 ()d: (4.76) 42 Thus, up to the higher orders we conclude that the local error (under the operator norm) in the short-time approximation is E 2 (t) = ~ A ! 0 P (t) (! P )A ! P (t) 1 P ( 0 (t) + 1 (t)) (4.77) where 0 (t) = Z 1 0 B 0 () kA ! P ; (t)kd (4.78) 1 (t) = Z 1 0 jB()jk@ t A ! P ; (t)kd: (4.79) Since the norm is unitarily invariant, we have kA ! P ; (t)k 1 = A ! P U y S (t;t)A ! P U S (t;t) (4.80) k@ t A ! P ; (t)k =k[H S (t);A ! P ]k: (4.81) Finally, we drop the off-diagonal terms due to their rapid oscillation (i.e. apply a rotating wave approximation onH P ). The local error of this approximation is not generally bounded. However, it is known to be equivalent to the coarse-graining approximation in the limit of a large coarse-graining time [19, 44], and is valid in the limit P 1= B : (4.82) 4.4 Discussion We have expanded the open system analysis of the Hamiltonian error suppression formalism in the regime where only the Born-Markov approximation may be assumed, and derived a master equation that allows for treating error suppression as Markovian in the limit of large penalty energy. Our treatment does not require any additional assumptions on the logical Hamiltonian to reach a master equation of fully Lindblad form (such as adiabaticity or weak-coupling). Under uncorrelated Pauli noise and unbiased error suppression, the penalty energy only needs to grow logarithmically with the number of stabilizer generators to preserve the code space population. We expect a similar result to hold even if the noise is biased towards particular generators. However, our formalism is ill-equipped to handle correlated Pauli errors. Adapting the error-suppressed master equation to subsystem codes is not difficult, provided the logical operators are chosen to be entirely in a separate factor from the gauge operators, so that [H S ;H P ] = 0. As locality is a critical limitation to the types of codes that can be used for Hamiltonian error suppression [45, 46], further work is most merited in implementing simulating schemes for error suppression with subsystem codes. 43 Chapter 5 Approximate Optimization Advantage with Quantum Annealing Abstract Wepresentevidenceforascalingadvantageinapproximateoptimizationforquantumannealingover the top classical heuristic algorithm: parallel tempering with isoenergetic cluster moves (PT-ICM). The advantage is obtained for a class of 2D spin-glass problems with spin-spin interactions designed to minimize ground-state degeneracy. To achieve this advantage, we implemented an embedding of a stabilizer bit-flip code with energy penalties known as quantum annealing correction (QAC), that leverages the connectivity of the D-Wave Advantage quantum annealer to yield over 1,300 error- suppressed logical qubits on a degree-5 interaction graph. We generate random spin-glass instances on this graph and benchmark their time-to-epsilon, a generalization of the time-to-solution metric for low-energy states. Our results show that aided by QAC, quantum annealing exhibits a scaling advantage over PT-ICM at sampling low energy states within residual energy densities of at least 1:0%. This amounts to an approximate optimization advantage with quantum annealing over the best-in-class classical algorithm. 5.1 Introduction Quantumannealing(QA)[47, 48]hasscaledupinrecentyearstotackleincreasinglylargerandmore highly connected discrete optimization and quantum simulation problems [3, 31, 49–54]. Neverthe- less, despite numerous attempts, a computational quantum advantage in exact optimization using QA hardware has so far remained elusive [55–72]. In the quantum simulation context, impressive recentadvanceshavebeenmadeforfast, coherentannealslastingontheorderofthesinglesupercon- ductingfluxqubitcoherencetime[73, 74]. Whilethisdiabaticapproachisconsideredpromising[75], it cannot be expected to scale up without the introduction of error correction or suppression, as decoherence and control errors pose insurmountable challenges to Hamiltonian quantum computa- tion models [29, 76–83], just as they do for gate-model quantum computers. In the absence of a fault-tolerance threshold theorem [84, 85] for quantum annealing, a variety of Hamiltonian error suppression techniques have been proposed and analyzed as ways to reduce the error rates of this 44 computational model and the closely related model of adiabatic quantum computation [37–40, 46, 86–99], providing tools towards scalability. However, despitethesetheoreticaladvances, therearecurrentlypracticallimitationstothetypes and localityof programmable interactions in the Hamiltonians of quantum annealing hardware, even as new devices have started to emerge [100–105]. To address these limitations, quantum anneal- ing correction (QAC) [106] was developed as a realizable error suppression method for quantum annealing, targeting the available and restricted set of control operations in commercial quantum annealers. QAC has been demonstrated to enhance the success probability of quantum annealing and mitigate the analog control problem [106–113]. The QAC method is based on a repetition-code encoding of the Hamiltonian, and does not fully realize a Hamiltonian stabilizer code. Despite this, it has been shown theoretically to increase the energy gap of the encoded Hamiltonian and reduce tunneling barriers, thus softening the onset of the associated critical dynamics as well as lowering the effective temperature [114–116]. Here, by departing from the standard paradigm of using QA for exact optimization, we demon- strate, by incorporating QAC, the first genuine scaling advantage in approximate optimization (low-energy sampling) using a quantum annealer. Even approximate optimization is computation- ally hard unless P = NP [117, 118], so we do not expect the advantage we exhibit to amount to more than a polynomial speedup. However, whereas the advantages reported in previous work were rela- tive to simulated annealing [66, 71], the advantage we find here is over the best currently available general heuristic classical optimization method: parallel tempering with isoenergetic cluster moves (PT-ICM) [119, 120]. This result is enabled by implementing QAC on the D-Wave Advantage quan- tum annealer for the Sidon-set spin glass instance class [57], embedded on the logical graph formed after the encoding. The advantage of quantum annealing over PT-ICM is diminished without QAC, thus highlighting the crucial role played by quantum error suppression. 5.2 Methods Quantum annealing.—The D-Wave quantum processing unit (QPU) uses superconducting flux qubits to implement the transverse field Ising Hamiltonian H(s) =A(s) X i2V x i +B(s)H z ; (5.1) whereV is the vertex set of the Pegasus hardware graph of the QPU, i is the qubit index, x i are Pauli matrices, and A(s) and B(s) are the annealing schedules, respectively decreasing to 0 and increasing from 0 with s : 0! 1. H z is the Ising problem Hamiltonian: H z = X i2V h i z i + X fi;jg2E J ij z i z j ; (5.2) 45 P P P P P P (a) (b) (c) Horizontal Vertical Cluster Short Diagonal Long Diagonal Figure 5.1: Embedding of QAC on the Pegasus hardware graph. (a) Close-up of the embedding of three clusters of two coupled logical qubits each. Physical qubits are denoted by circles, cou- plings by lines. Penalty qubits are marked by a P, and the three penalty couplings (thin lines) to their corresponding physical data qubits. Thick lines indicate the physical couplings between the corresponding physical qubits of different logical qubits. Only a subset of all possible couplings are shown. (b) A zoomed-out view from (a) showing the logical qubits (circles) and the logical couplings induced by the QAC embedding. Thick lines indicate the logical couplings shown in (a), whose type is colored by direction: horizontal/vertical/diagonal (long or short). (c) A zoomed-out view from (b) showing a 4 4 induced logical graph of QAC. The logical graph is equivalent to a honeycomb graph with additional non-planar bonds. The induced logical graph on of the D-Wave Advantage 4.1 QPU has a maximum of 1322 logical qubits; the largest available side length is 15. where h i and J ij are programmable local fields and couplings, respectively, andE is the edge set of the Pegasus graph. Many NP-complete and NP-hard problems can be mapped to H z [121] and minor-embedded onto the Pegasus graph [122]. We performed our QA experiments on the D-Wave Advantage 4.1 QPU accessed through the D-Wave Leap cloud interface [123]. Quantum annealingcorrection.—Weimplementthe [[3; 1; 3]] 1 QACencodingintroducedinRef.[106], which encodes a logical qubit into three physical qubits, each of which is coupled to the same ad- ditional energy penalty qubit with a fixed coupling strength J p ; the logical qubit is decoded via a majority vote on the first three physical qubits. The logical subgraph induced by the QAC encod- ing on the Pegasus graph has a bulk degree of 5 and admits native frustrated loops of length 5. Fig. 5.1 illustrates the encoding and the induced logical graph. The features of this logical graph are a significant change from the one induced on the Chimera topology of the previous generation of D-Wave QPUs, which has degree 3 and no odd-length loops, thus permitting the benchmarking of harder spin-glass instances than possible on Chimera. All previous QAC results [106–113] were obtained using the Chimera topology. Problems on the logical QAC graph can also be encoded directly by setting J p = 0, resulting in three uncoupled and unprotected classical copies of the problem instance. We simply extract all three copies as independent annealing samples, with no additional post-processing, and denote this 46 “unprotected” QA method by U3. We use the U3 method below to test whether QAC has a genuine advantage over simple classical repetition coding. Spin-glass instances.— We generate random native spin-glass instances on the logical graph induced by the QAC embedding. These types of instances have been widely used to benchmark the previous D-Wave QPUs—with the Chimera hardware graph—against classical algorithms [31, 55–57]. We tested three types of spin-glass disorder: binomial, where J ij was randomly selected as 1 with equal probability, Sidon-7 (S7), and Sidon-28 (S28) [57], where J ij was randomly sampled from the setsf5=7; 6=7; 1g andf8=28; 13=28; 19=28; 1g, respectively. In a Sidon set, the sum of any two members of the set gives a number that is not part of the set. Binomial disorder generally admits a degenerate ground state, which simplifies the optimization problem. In contrast, the S7 and S28 disorder can yield instances with a unique ground state [80]. The ground states are robust to small errors in the implementation of theJ ij values in the binomial disorder case, but this is not the case when high precision in implementing theJ ij values is required (as for Sidon disorder), and the latter case is expected to benefit more from QAC than the former [112]. We focus on the S28 case from here on. Binomial and S7 disorder will be discussed in future work. Time-to-rho metric.—The standard performance metric for exact optimization using heuristic solvers is the time-to-solution (TTS), which quantifies the time to reach the ground state at least once with 99% probability, given that the success probability per run is p: TTS =t f R, where each annealing run lasts time t f and R = log(10:99) log(1p) is the expected number of runs [55]. To address approximate optimization, we instead consider the time required to reach a given residual energy density , and define the time-to-rho for an instance with ground state energy E 0 as TTR(N) =t f log(1 0:99) log(1p EE 0 +JN ) (5.3) where N is the system size, J is the energy scale of the Hamiltonian (set to 1 for our purposes), and p EE 0 +" is the probability that the energy E of a sample is within "NJ of E 0 . The TTS is the special case = 0. We define [TTR] q of an instance class as the q’th quantile of TTR over the entire instance class. Here, we focus only on the median quantile, q = 0:5, which is denoted [TTR] Med . For a given disorder, instance size, and -target, we find the annealing time t f (and penalty strength for QAC) that optimizes [TTR] Med . We restrict the penalty coupling strengths to the setJ p 2f0:1; 0:2; 0:3g to reduce resource requirements for parameter optimization, asJ p = 0:2 is the penalty strength that most frequently optimizes the success probabilities of individual instances, and the dependence on J p above 0:2 becomes weak. Fitting the TTR.—Below, we fit the TTR to a power law: TTR(N) = cN , where is the scaling exponent, the quantity we use to quantify the scaling of the different algorithms we compare. The choice of a power law is motivated by the existence of an O(N) classical algorithm for the residual energy for any > 0; we describe such an algorithm below. We hasten to add that this algorithm is completely impractical due to its huge prefactor. However, its existence is sufficient grounds to preclude exponential fits to the TTR. Due to the power law fit, we should account for 47 factors that can modify the scaling exponent. Indeed, we could use allN max qubits of the QPU and embed Nmax N parallel copies of each problem of sizeN, then select the best of these copies. Since, in reality, we work with only one copy due to a small fraction of the qubits and couplers being absent, we multiply the TTR by a factor of N Nmax [55]. The U3 TTR is similarly multiplied by a constant factor of 3=4 since the instance is repeated over only the subgraphs through each of the non-penalty qubits, but theoretically, if sufficiently many couplings were available, it could also be repeated over the penalty qubits, for a total of four copies of the problem per anneal. 5.2.1 Parallel Tempering Algorithm Ourbaselineclassicalalgorithmisparalleltemperingwithisoenergeticclustermoves(PT-ICM)[120]. The runtime of this algorithm has the best scaling with system size known in the task of finding the ground state of various benchmark problems on D-Wave QPUs [63, 64, 67], with the only known exception being certain XORSAT instances for which highly specialized solvers have been developed [69, 124]. We first ran every replica once forN max sw = 500,000 sweeps to determine the ground state energy, which we took to be the lowest energy found by PT-ICM. For the largest sizeL = 15, we determined N (90%) sw , the number of sweeps that were required for 90% of the instances to reach their lowest recorded energy, where we found N (90%) sw 31,000 for S28 disorder. As this was significantly less thanN max sw , we therefore considered the ground states for these instances as “validated” by PT-ICM for the purpose of calculating median quantities over the instances. We finally ran PT-ICM 100 times for each instance, settingN sw =N (90%) sw . This yields an empirical cumulative density function forp EE 0 +" as a function of the runtime of PT-ICM. The TTR is then evaluated for each instance by optimizing over the runtime of the PT-ICM repetition (wheret f is now the time needed to reach the target, rather than annealing time). The scaling of PT-ICM is ideally evaluated using the parameters that best optimize the TTR for each disorder, instance size, and " target. However, a rigorous optimization of the number and choice of replica temperatures for all " targets and system sizes is computationally infeasible. To ensure our results hold for any choice of reasonably optimized temperatures, we repeated the TTR evaluationwiththreetemperaturesetssummarizedinTable5.1whichincludesbothlogarithmically- spaced and feedback-optimized [125] temperatures. The TTR for a given disorder, instance size, and energy target was chosen from the best TTR out of the three temperature sets. The differences between the different temperature sets is not appreciable and are unlikely to affect our conclusions. Our PT-ICM implementation is available as the TAMC software package. ∗ Our implementation of QAC for the D-Wave Advantage graph is available as part of the PegasusTools Python package † . We also implemented and considered the performance of simulated annealing (SA), but do not present its results as this algorithm was not competitive at large instance sizes. Our PT-ICM implementation is a general-purpose solver written in the Rust programming language, targeting ∗ github.com/hmunozb/tamc † github.com/hmunozb/PegasusTools 48 S28 Set N T min max N icm 1 32 0.1 10.0 8 2 24 0.2 10.0 6 3 32 0.2 10.0 8 Table 5.1: Temperature sets used for PT-ICM for S28 instances, where min is the hottest tem- perature, max is the coldest temperature, and N icm is the number of low temperature (largest ) subject to ICM moves. The temperatures are logarithmically spaced in Set 1. The temperatures in sets 2 and 3 were feedback-optimized with initially logarithmically spaced temperatures. CPU execution, and accepts instances with any connectivity. This is in contrast to previous studies that have used SA or the Hamze-de Freitas-Selby algorithm (HFS) [126, 127] solvers that have been specialized for problems defined on the Chimera topology. For a given system size and energy density target, we calculate the median TTR using a three- step Bayesian bootstrap procedure: (1) the success probability for each gauge is resampled from a beta distribution for N samp samples per gauge, (2) the statistical weight of each gauge is sampled from a Dirichlet distribution of length N gauge to take the weighed average success probability for each instance, and (3) the statistical weight of each instance is sampled from a Dirichlet distribution of length N inst to take the weighed median. We performed our quantum annealing experiments with N samp = 1000(QAC)=3000(C3), N gauge = 10, and N inst = 125. We performed N boots = 200 bootstrap samples per size and energy density target pair, and found annealing time and penalty strength that results in an optimal TTR for each bootstrap sample. The distribution of the optimal median TTR values and the distribution of optimal annealing parameters are the two final products of this sampling procedure. Next, we formulate a null hypothesis and compute P values, as follows: the null hypothesis is that the optimal annealing time t f is the minimum accessible t min f = 0:5s, which we physically interpret as a signature that the true optimal annealing time is not above t min f . The P value is the empirical number of bootstrap samples whose optimal t f was 0:5s, out of a total of 200 samples. Filled circles mean thatP < 0:05 for the probability thatt f = 0:5s in the bootstrap sample, while open circles mean thatP < 0:20. The filled circles show which points have the best confidence that the optimal annealing time is not below 0:5s. 5.3 Results Results.—It is well known that the TTS metric generates unreliable results unless the annealing time is optimized for each size N [55, 58, 66]. Briefly, this is because an artificially high TTS at small N results in an overly flat TTS scaling. The same considerations apply to the TTR, so here we find the annealing time t f that minimizes [TTR] Med for each N—denoted t opt f (N)—and report the resulting median TTR and its scaling estimate for QAC and U3 in Fig. 5.2, along with the analogously optimized PT-ICM results. 49 150 300 600 900 1300 10 -2 10 0 10 2 10 4 10 6 10 8 150 300 600 900 1300 0.5 1 3 9 27 91 PT-ICM U3 QAC 150 300 600 900 1300 10 -2 10 0 10 2 10 4 10 6 10 8 150 300 600 900 1300 0.5 1 3 9 27 91 150 300 600 900 1300 10 -2 10 0 10 2 10 4 10 6 10 8 150 300 600 900 1300 0.5 1 3 9 27 91 150 300 600 900 1300 10 -2 10 0 10 2 10 4 10 6 10 8 150 300 600 900 1300 0.5 1 3 9 27 91 Figure 5.2: Time-to-rho scaling for QAC, U3, and PT-ICM, for Sidon-28 (S28) spin-glass disorder. The bottom panels show the optimal annealing times of U3 and QAC. The top panels show the TTR results for the corresponding optimal annealing times, along with optimized PT-ICM results. The straight lines are best fits assuming power law scaling TTR =cN , and the numbers indicated are the corresponding slopes . As indicated in the legends, the target residual energy density increases from left to right, with a corresponding improvement in quantum annealing’s performance: PT-ICM is far better at low , but for = 1%, while U3 is still worse than PT-ICM, QAC already outperformsPT-ICM.At = 1:25%bothQACandU3havebetterscalingthanPT-ICM.Forhigher residual energy targets, the scaling of QAC becomes unreliable since we can no longer guarantee that the optimal t f has been identified. We used t f 2 [0:5; 91]s and N 2 [142; 1322]. Error bars for TTR data points are twice the standard error of the parameter estimate calculated using bootstrapping. Filled (open) circles correspond to a P value of 0:05 (0:20) that t opt f (N)>t min f for the corresponding problem size N. We use only those t f values for whichP = 0:05 to compute the slope. 50 The shortest available annealing time on the D-Wave Advantage QPU accessed via Leap is t min f = 0:5s, and the bottom panels show that as the target residual energy density is increased, progressively larger problem sizes are needed to ensure that t opt f (N) > t min f . We cannot rule out that with access to lower annealing times, one would find t opt f (N)<t min f for all N values where we empirically findt opt f (N) =t min f . We thus formulate a null hypothesis for eachN thatt opt f (N)t min f and compute a P value [128] as the empirical number of bootstrap samples whose t opt f (N) =t min f , out of a total of 200 samples. To compute the TTR scaling, i.e., the slope in a fit to TTR =cN , for each we use only those t opt f (N) values whose P < 0:05 (filled circles in Fig. 5.2). We can thus be confident that the reported slopes reflect the true scaling of U3 and QAC. The one exception is QAC at = 1:5%, for which we are unable to reach large enough problem sizes to ensure that t opt f (N) < t min f ; in this case, we simply fit the available TTR data, starting from N = 372. The resulting slope is not optimized but agrees qualitatively with the trends we describe next. Our first observation is that the QAC scaling is always better than the U3 scaling, which is consistent with previous studies concerning the effect of analog coupling errors (“J-chaos”) on the TTS for spin-glass instances [112]. Such errors are expected for the S28 instances due to the relatively high precision their specification requires. Second, we observe that U3 and QAC both reduce the absolute algorithmic runtime by four orders of magnitude compared to PT-ICM. However, this is not a scaling advantage and since our PT-ICM calculations could be sped up by employing faster classical processors, we do not consider this a robust finding. Third, and most significantly, we observe that as the target residual energy density increases, QA’s scaling overtakes PT-ICM.Notably, at = 1%, QACexhibitsascalingexponentof 1:780:06, compared to PT-ICM’s 2:090:15, and at = 1:25%, U3 exhibits a scaling exponent of 1:890:06, compared to PT-ICM’s 2:06 0:14. The scaling of QAC is not determined above this threshold, as all data points are likely to be optimal at the smallest annealing time. However, the scaling of U3 at optimal annealing times continues to decrease as increases; at = 1:5% (not shown), the scaling exponents for U3 and PT-ICM become 1:57 0:08 and 2:01 0:14, respectively. This is robust evidence of a quantum annealing scaling advantage over the best available classical heuristic optimization algorithm. Given the consistently better scaling of QAC for lower values of , where t opt f (N) > t min f for QAC over a range of problem sizes, it is reasonable to conclude that the QAC scaling would be a further improvement over U3 if its true t opt f (N) could be established for 1:5%; this would require access to shorter annealing times or larger system sizes. 5.4 Discussion and Conclusion We emphasize that it is unsurprising that, given a large enough residual energy density threshold, the D-Wave QPU is nearly guaranteed to return sample energies within that threshold. Similarly, we can expect that for large enough , even simulated annealing or greedy descent will be nearly 51 guaranteed success in near-linear time (we describe such an algorithm below). The significance of our result is that QA reaches near-linear scaling at a smaller residual energy density threshold than PT-ICM. Thus, we refer to this result as an approximate optimization advantage for quantum annealing. We note that Ref. [74] similarly reported a QA optimization advantage for the residual energy density (for 3D spin glasses), but this was done at a fixed problem size (ofN = 5374 spins), and instead concerned the convergence of the residual energy density with the annealing time. We reemphasize that, in contrast, here we are reporting a scaling advantage as a function of problem size, which is the proper context for quantum speedup claims. The approximate optimization of finite-range spin glasses has, in theory, a linear scaling using a simple divide-and-conquer algorithm. Namely, for a given residual energy target , an algorithm exists whose TTR scales linearly in the system size for a sufficiently large size, with a large prefactor depending on the residual energy. For 2D spin glasses on a square lattice, it can be summarized as follows: (i) Partition the size-N =L 2 instance into KK subgraphs G x;y , with x;y2f1;:::;Kg, eachwithconstant sidelengthofL 0 =L=K spins; (ii)Findthelocalgroundstatesforeachsubgraph using an exact or heuristic solver for each L 0 L 0 instance. This step has cost C(L 0 ), potentially exponential inL 2 0 ; (iii) Patch together all subgraph ground state configurations as the approximate ground state for the global Hamiltonian, and return this state’s energy as the approximate ground state energy E (N). This last step requires O(N) time due to the need to sum the energies of K 2 = O(N) patches. The overall complexity is, therefore, C(L 0 )O(N). This algorithm optimizes bulk energies throughout the volume of the spin-glass at the expense of large energy violations over regions scaling with the surface area of the spin-glass, i.e., at the patch boundaries. The boundary excitations become negligible for sufficiently large sizes compared to the bulk energies, with the latter eventually reaching the desired residual energy. More precisely, up to 4L 0 boundary couplings may be violated per volume of L 2 0 , so the residual energy density for this algorithm is upper-bounded by 4=L 0 . Thus, we estimate that the regime of system sizes where this algorithm applies for the targets we examine, e.g., 1%, would require a reliable, efficient solver for instances with at least a sub-problem side length of L 0 400, resulting in a prefactor of 2 400 10 120 , which is completely impractical. Nevertheless, this algorithm could be used as a starting point for parallel and quantum-classical hybrid algorithms for massive, finite-dimensional problems. In summary, by using the largest available quantum annealer to date, we have demonstrated an approximate optimization scaling advantage for quantum annealing on a class of spin-glass problems with low ground state degeneracy and high susceptibility to analog coupling errors. Our demonstration involves up to 1300 logical spin variables and concerns the residual energy density. The advantage is relative to PT-ICM, which is the best classical heuristic algorithm currently known for such spin glass problems and appears at residual energy densities of& 1%. Our results do not imply that finding states within small, constant residual energies (and indeed finding the ground state itself) is easy for quantum annealing, nor do they imply that all spin glass problems are amenable to an approximate optimization scaling advantage via quantum annealing. The fact that theproblemswehavestudiedarefinite-rangeandtwo-dimensionalisanadditionalcaveatthatlimits 52 the range of applications. To achieve an algorithmic quantum advantage in an application setting, the next challenge for quantum optimization is demonstrating a hardware-scalable advantage in densely-connected problems at sufficiently small residual energy densities. Acknowledgements This chapter is based on research currently in preparation for publication. The conceptual design of the time-to-epsilon/residual energy methodology is due to Evgeny Mozgunov. We thank Dr. Victor Kasatkin fordiscussions about the theoreticallower bound of finite-range approximateoptimization. The authors acknowledge the Center for Advanced Research Computing (CARC) at the University of Southern California for providing computing resources that have contributed to the research results reported within this publication. 53 Chapter 6 Error Suppressed Quantum Annealing through Boundary Cancellation Abstract The boundary cancellation theorem for open systems extends the standard quantum adiabatic theorem: assuming the gap of the Liouvillian does not vanish, the distance between a state prepared by a boundary cancelling adiabatic protocol and the steady state of the system shrinks as a power of the number of vanishing time derivatives of the Hamiltonian at the end of the preparation. Here we generalize the boundary cancellation theorem so that it applies also to the case where the Liouvillian gap vanishes, and consider the effect of dynamical freezing of the evolution. We experimentally test the predictions of the boundary cancellation theorem using quantum annealing hardware, and find qualitative agreement with the predicted error suppression despite using annealing schedules that onlyapproximatetherequiredsmoothschedules. Performanceisfurtherimprovedbyusingquantum annealing correction, and we demonstrate that the boundary cancellation protocol is significantly morerobusttoparametervariationsthanprotocolswhichemploypausingtoenhancetheprobability of finding the ground state. 6.1 Introduction Adiabaticquantumcomputing[13]andquantumannealing[12]canbeusedtopreparegroundstates andthermalstatesofquantumsystems, acentraldesideratumofquantumcomputation. Thequality ofgroundstatepreparationandGibbssamplingiscentraltotheperformanceofquantumalgorithms foroptimizationaswellasmachinelearningproblems[129]. Theperformanceofthesecomputational paradigms is quantified by the adiabatic theorem in its different forms, whether for closed or open quantum systems. The theorem, or family of theorems, places a bound on the adiabatic error: the distance between the state that we set out to prepare (usually the solution to a computational problem) and the state the system actually ends up in through the dynamical evolution. In general, if a spectral gap condition is satisfied, the adiabatic error is bounded by C=t f for some constantC, where t f is the total evolution time [130–132]. The boundary cancellation theorem (BCT) shows that an improvement is possible if the schedule from the initial to the final generator (Hamiltonian 54 or Liouvillian) has certain properties. In the closed system case the adiabatic error is bounded by C k =t k+1 f for some constant C k independent of t f , if the time-dependent Hamiltonian H(t) has vanishing derivatives up to orderk at the beginning and end of the evolution [133], a bound that can be improved to exponentially small int f under additional assumptions [134–138]. This theorem has recently been extended to the open system setting for a general class of time-dependent Liouville operatorsL(t), where it can be used to prepare steady states of Liouvillians instead of ground states of Hamiltonians. The theorem can be succinctly stated as follows: For a time-dependent Liouvillian L(t) with a unique steady state(t) separated by a gap at each timet, the adiabatic error is likewise bounded by C k =t k+1 f , but it is sufficient for the derivatives to vanish up to order k only at the end time t f [96]. Boundary cancellation (BC) plays a significant role in the theoretical analysis of error suppression in Hamiltonian quantum computing [29, 139]. In this work we set out to demonstrate the BCT predictions in experiments using the D-Wave 2000Q (DW2KQ) quantum annealer. The DW2KQ implements the transverse field Ising model H(t) =A(t)H X +B(t)H Z , whereH X andH Z are transverse and longitudinal (Ising) Hamiltonians, respectively [3, 140]. It allows limited programmability of the control schedules A(t) and B(t) of the system Hamiltonian, which we exploit to implement boundary cancellation protocols. ToguideourintuitionwemodelthebehavioroftheD-Wavequantumannealerwiththeadiabatic master equation (AME) derived in [14]. This is a time-dependent Davies-like master equation [141] which has been successfully used to interpret several D-Wave experiments, e.g., [142, 143] (though notalways[83]). WhencombiningtheAMEfordephasingwithboundarycancellation, weencounter a problem. Namely, as we explain in detail below, the Liouvillian gap vanishes at t = t f , which prevents us from directly applying the BCT in the form given in Ref. [96]. To circumvent this problem, in this work we generalize the BCT and identify the conditions on the Liouvillian gap under which BC does or does not remain effective. We find that the generalized BCT plays an important role in the D-Wave implementation. There is another significant consideration regarding the implementation of BC on D-Wave: the phenomenon of freezing. In essence, freezing refers to a significant increase in all relaxation timescales well before the end of the anneal (t<t f ), in such a manner that the system is frozen in a state that does not correspond to the steady state ofL(t f ) [14]. However, the frozen state is not truly static but quasi-static [144], i.e., relaxation is not fully switched off but instead proceeds on a timescale that may be much longer than practically realizable anneal times t f . Such glassy-like behavior can take place even in very simple systems that do not have a glassy landscape. We give an explanation of freezing in Section 6.2.4 below, based on the AME. Essentially, it arises due to the fact that the system Hamiltonian commutes with the system-bath coupling whenA(s) = 0, and the fact that the schedule A(t) has nearly vanished long before t = t f (see Fig. 6.1). This is true in particular of currently available commercial quantum annealers manufactured by D-Wave [145], which can be effectively described by longitudinal field coupling to a bosonic bath, which commutes with the system Hamiltonian at sufficiently low transverse fields. The state of the system is insen- sitive to the details of the schedules past the freezing point, so that BC becomes ineffective. This 55 freezing mechanism plays an adverse role in demonstrating the predictions of the BCT, since the experimentally accessible anneal times are significantly shorter than the relaxation time required to reach the true steady state ofL(t f ). In order to overcome both issues, i.e., the gaplessness of the Liouvillian att =t f and freezing at t 0:5t f , we design the programmable DW2KQ schedule so that it flattens (and thus implements BC) at a pointt BC before freezing, followed by a ramp to the final values ofB(t f ) andA(t f ), upon which the system is measured in computational basis (eigenbasis of H Z ). We apply the protocol to two 8-qubit gadgets: 1) a “ferromagnetic chain gadget” (FM-gadget), and 2) a “tunneling gadget” (T-gadget) with a larger tunneling barrier between the ground state and the first excited state around the minimum gap [146]. We estimate the scaling of the adiabatic error for the FM-gadget and compare the experimental results of the boundary cancellation protocol (BCP) with those predicted by open system simulations using the AME. Based on these simulations, we discuss the conditions under which the BCT-predicted scaling may be observed. We also apply the protocol to an error-corrected version of the FM-gadget using quantum annealing correction (QAC) [106]. The QAC encoding mitigates the effects of Hamiltonian programming errors [88, 112] and effectively lowers the temperature of the environment [115, 147]; we find that QAC improves the performance of the BCP. Thestructureofthispaperisasfollows. InSection6.2wefirstreviewtheBCT,andtheformulate a version which accounts for the possibility of the Liouvillian gap closing at the same point as where the boundary cancellation conditions are enforced, which is relevant for our experiments. The rest of the section is devoted to a discussion of freezing, the effect of a ramp after the BC point, and anomalous heating, all of which are phenomena affecting the performance of the BCP due to their presence in our experiments. In Section 6.3 we describe our methods: the BCP we used in detail, the two 8-qubit gadgets used in our experiments, and encoding with quantum annealing correction. Our results are presented in Section 6.4; we start with a standard linear control schedule and then present the results for the experimental BCP implementation. We include simulation results for a higher precision version of the BCP for reference, and then report on experiments from the QAC- encoded version of one of our gadgets. Finally, we compare the performance of the BCP to that of the pausing protocol. We conclude with a discussion and outlook in Section 6.5. Finally, Section 6.6 includes background on the adiabatic master equation, proofs of various Propositions, supporting numerical results, additional information needed to reproduce our experiments and analysis, and experimental results from two other D-Wave processors. 56 6.2 Theory 6.2.1 Review of the BCT We consider a finite-dimensional system whose density matrix t f satisfies the master equation d t f =dt =L t f (t) t f (t), whereL t f (t) is a time-dependent Liouvillian superoperator. IfL t f (t) de- pends on the time t only through the “anneal parameter” :=t=t f , where t f is the total evolution time, then defining () := t f (t) andL() :=L t f (t), the master equation takes the form d d =t f L()() : (6.1) The main result of Ref. [96] is summarized in the following proposition: Proposition 6.1. Assume thatL() is such thatkL()k 1;1 is summable in 2 [0; 1], ∗ L() is differentiable to order k + 2 in a neighborhood of = 1, and generates a trace-preserving and hermiticity-preserving contractive semigroup, i.e., e rL() 1;1 1 for all r 0, 2 [0; 1]. Suppose L() has a unique steady state () with eigenvalue zero separated from the rest of the spectrum by a nonzero gap82 [0; 1]. Let() be the solution of Eq. (6.1) with initial condition(0) =(0). IfL() has vanishing derivatives at = 1 to orderk, i.e., @ (j) L =1 = 08j = 1; 2;:::;k, then there is a constant C k independent of t f such that k(1)(1)k 1 C k t k+1 f : (6.2) We note that the smoothness assumptions of Prop. 6.1 are more relaxed than in Ref. [96]. From here on we focus in particular on the AME, which we briefly review in Section 6.2.2. It was shown in [96] that, under the action of the AME Liouvillian, it is possible to enforce the BCT as required in Prop. 6.1 by controlling just the time-dependent system Hamiltonian H S : Proposition 6.2. Assume the system evolves according to the time dependent Liouvillian in Eq. (6.5a). Assume further that @ (j) t H S (t) t=t f = 0, for j = 1; 2;:::;k. Then @ (j) t L t=t f = 0 8j = 1; 2;:::;k. This means that rather than having to directly check that the entire Liouvillian satisfies the conditions of vanishing derivatives at the end, it suffices to check that the system HamiltonianH S (t) satisfies the same conditions. However, the Prop. 6.1 requirement that the Liouvillian has a finite gap above its zero eigenvalue throughout the evolution window must be checked independently. For the finite dimensional systems that we consider here, this is equivalent to the requirement that the steady state is unique throughout the evolution, or alternatively, that there are no “level crossings” ∗ The superoperator norm we use is the induced 1; 1 norm: kAk 1;1 := sup kyk 1 =1 kAyk 1 for a superoperatorA, where the trace-norm is defined askAk1 := Tr p A y A, for any operator A, i.e., the sum of A’s singular values. A function is summable if its integral is finite on every compact subset of its domain. 57 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 Figure 6.1: Native annealing schedule of the D-Wave 2000Q low noise processor. ofL()’s zero eigenvalue throughout the evolution. ∗ Since the system Hamiltonian is naturally the controllable component of the total Hamiltonian or Liouvillian, this is also a physically sensible controllability condition. Moreover, it was verified numerically in [96] that Eq. (6.2) remains valid for a more general Liouvillian in time-dependent Redfield form, even though it does not satisfy the hypothesis of Props. 6.1 and 6.2, i.e. for which the vanishing of the derivatives of the system Hamiltonian does not imply the vanishing of the derivatives of the Liouvillian. See [96] for details. We next review the AME for quantum annealing problems and discuss a modified form of the BCT, which is tailored to the situation we encounter in our experiments using the D-Wave annealer. 6.2.2 Adiabatic Master Equation for Quantum Annealing We now consider the quantum annealing problem defined by a system Hamiltonian of the form H S (s) = A(s) 2 H X + B(s) 2 H Z ; (6.3) where H X = X i x i ; H Z = X i h i z i + X i<j J ij z i z j : (6.4) and 2 (0; 1] is a scaling factor that controls the energy scale of the computational problem. We refer to A(s) and B(s) as the physical schedules of the anneal. They are fixed properties of the quantum annealing hardware (see Fig. 6.1), but s can be made to depend on = t=t f , in the ∗ In principle one must also check that R 1 0 dskL(s)k 1;1 <1, but this follows automatically if the control functions satisfy R 1 0 dsjA(s)j<1 and R 1 0 dsjB(s)j<1, which is a natural assumption. 58 adjustable way that we describe below for the DW2KQ. We call the function s() : [0; 1]7! [0; 1] the control schedule. We model the anneal according to the AME [14] with the Hamiltonian in Eq. (6.3). The AME is derived from the total Hamiltonian H(t) =H S (t) +H B +H SB , withH SB =g P i A i B i where the A i (B i ) are the dimensionless system (bath) operators, subject to an adiabatic approximation. The resulting Liouvillian is of the time-dependent Davies form L(t) =i [H S (t) +H LS (t);] +D(t) (6.5a) D(t) = X ij! ij (!) A j (!)A y i (!) 1 2 n A y i (!)A j (!); o : (6.5b) HereH LS (t) is the Lamb-shift term, ij (!) is the Fourier transform of the bath correlation function G ij (t 1 ;t 2 ) =hB i (t 1 )B j (t 2 )i =G ij (t 1 t 2 ) ; (6.6) B i (t) = e iH B t B i e iH B t , hXi = Tr( B X) where B is the initial bath state, and !(t) are the Bohr frequencies of H S (t) (to simplify notation we suppress their explicit time-dependence when convenient), i.e., the differences between the instantaneous eigenenergies of H S (t). The Lindblad operators A i (!) that appear in the Davies generator (“dissipator”)D are defined by e irH S (t) A i e irH S (t) = X ! e ir! A i (!) ; (6.7) i.e., A i (!) = X E b Ea=! a A i b ; (6.8) where H S = P a E a a and a are the eigenprojectors of H S . The Lamb shift is given by H LS = X ij! S ij (!)A y i (!)A j (!) ; (6.9) with S ij (!) = Z +1 1 d! 0 ij (! 0 )P 1 !! 0 ; (6.10) whereP is the Cauchy principal value. It commutes with H S . This master equation generates a completely positive map and describes a system with a slowly varying system HamiltonianH S (t), weakly interacting with an infinite bath. MoreoverL(t) has the property that it has the Gibbs state as a steady state, i.e.,L(t)(t) = 0 with(t) =e H S (t) =Z(t), Z(t) = Tre H S (t) . 59 Inoursimulationsweassumelongitudinalfieldcoupling, i.e., thesystemoperatorsinthesystem- bath interaction Hamiltonian are A i = z i with ij (!) = ij (!) (independent coupling) and the bath has an Ohmic spectral density, so that (!) = 2 0 g 2 ! 1e ! : (6.11) 6.2.3 A Modified BCT Unfortunately, wecannotdirectlyapplyProp.6.1toannealerswithA(t f ) = 0(suchastheDW2KQ) because of the following result (proven in Ref. [98]): Proposition 6.3. Assume thatL() is the AME Liouvillian [Eq. (6.5a)] with the Hamiltonian in Eq. (6.3) and longitudinal system-bath coupling (i.e., that commutes with H Z ). Then, at the end of the anneal when A(t f ) = 0, the zero eigenvalue ofL(t f ) is at least d-fold degenerate, where d is the system’s Hilbert space dimension. Since presumably, whenA(t)6= 0 fort<t f the LindbladianL(t) has a unique steady state(t) which corresponds to its zero eigenvalue, Prop. 6.3 means that the Liouvillian gap closes at t f and we cannot use Prop. 6.1. We now generalize the BC result Prop. 6.1 to the situation where one enforces the BC conditions at the same point as where the Liouvillian gap closes, e.g., at t =t f . Proposition 6.4. Assume thatkL()k 1;1 is summable in 2 [0; 1],L() is differentiable to order k + 2 in a neighborhood of = 1, and generates a trace-preserving and hermiticity-preserving contractive semigroup, i.e., e rL() 1;1 1 for all r 0, 2 [0; 1]. SupposeL() has a unique steady state() at each time2 [0; 1] except at the point 0 = 1 where the Liouvillian gap closes as ' v( 0 1) for a linear schedule (i.e., in the absence of a BC schedule). Let () be the solution of Eq. (6.1) with initial condition (0) = (0) and (1 ) := lim r!1 (r). Assume we enforce boundary cancellation at the end, i.e.,L() has vanishing derivatives at = 1 to order k: @ (j) L =1 = 08j = 1; 2;:::;k [note that hereL(s()) is written for simplicity asL()]. Then boundary cancellation is only mildly effective in that the adiabatic error satisfies (1)(1 ) 1 C t f ; (6.12a) with = k + 1 k + + 1 < 1 : (6.12b) The proof is given in Ref. [98] which also discusses the relaxed smoothness assumptions of Prop. 6.1 relative to Ref. [96]. The fact that the gap closes for the Lindbladian modeling the D- Wave annealers was already noted in Ref. [148], and indeed Eq. (6.12b) with k = 0 was derived there. 60 6.2.4 Freezing As the Hamiltonian starts to commute with the system-bath coupling operators near the end of the anneal, the dynamics of the system will begin to slow down. This gives another significant consideration regarding the successful implementation of BC schedules, which is that the system state appears to stop evolving at a point t 0 < t f , a phenonemon of quantum annealing called freezing. To explain freezing, we write the master equation in the instantaneous energy eigenbasis fjnig,n = 0; 1;:::;d1 (we omit the time dependence for clarity), i.e., the basis which diagonalizes the Hamiltonian H S (t). Consider the region of the anneal where A (t) is small but non-zero, and also B(t) > 0. It is then natural to assume that the spectrum of H S (t) is non-degenerate. The density matrix in the energy basis is(t) = P mn mn jmihnj. The diagonal elements of in this basis evolve according to the Pauli master equation [149] (for a modern derivation see, e.g., Ref. [150]). In particular, the ground state probability 00 =h0j(t)j0i evolves according to _ 00 = X n ( nn W 0n 00 W n0 ) (6.13a) = X n>0 W 0n nn 00 e (EnE 0 ) ; (6.13b) where the transition rate matrix W has the following matrix elements: ∗ W mn = X i;j ij (E n E m )hmj z j jnihnj z i jmi: (6.14) On the basis of Eqs. (6.13) and (6.14), freezing is seen to be a consequence of the following argument. Since we are in the adiabatic regime, only the lowest levels are populated, i.e., nn ' 0 for n greater than some n A . Eq. (6.13) is then replaced by _ 00 ' n A X n=1 W 0n nn 00 e (EnE 0 ) (6.15a) 00 d X n=n A +1 W 0n e (EnE 0 ) ; (6.15b) where d is the system’s Hilbert space dimension. When A(t) = 0, the Hamiltonian is diagonal in the computational basis. Let the Hamming distance between the ground statej0i and the excited statesfjnig n A n=1 when A(t) = 0 be at least q (note that q 1). Using perturbation theory around A(t) = 0 it can be shown thath0j z j jni =O (A q ); see Section 6.6.3. Hence, at the time t 0 at which A is sufficiently small (but non-zero) one has W 0n = O(A 2q )' 0 for n = 1; 2;:::;n A , and we can neglect the sum in line (6.15a); we determinet 0 below. As for the term in line (6.15b), the transition rates between higher excited states are typically smaller, and moreover, the terms are exponentially ∗ The matrix W satisfies detailed balance, i.e., Wnme Em = Wmne En , which follows from an analogous equation for (!), which in turns follows from the Kubo-Martin-Schwinger (KMS) conditions on the bath correlation function. 61 suppressed because of the larger gaps (i.e., for sufficiently small temperatures E n E 0 1=, given that nn A + 1 2). As a consequence, Eq. (6.15) becomes _ 00 0, i.e., the ground state population is effectively frozen for tt 0 . The location of the freezing point for the nth excited state is determined by the point where the relaxation time for thenth excited state, given by (n) rel =W 1 0n , becomes longer than the anneal time. As long as just one of these relaxation times, with n2f1;:::;n A g is too long, the system will not thermalize. For the system to freeze, i.e., _ 00 0, we need all transitions to cease. Hence, we define the freezing point as the solution t 0 of min n2f1;:::;n A g (n) rel =t f , or: ∗ t 0 :=fmint2 [0;t f ] s.t. max n2f1;:::;n A g W 0n (t) = 1=t f g : (6.16) In terms of the control schedule s(), we have s 0 := s( 0 ), where 0 = t 0 =t f . In practice, to determine the freezing points 0 we implement a closely related procedure described in Section 6.6.4. Note that, as a consequence of the perturbative argument, (for sufficiently small A) the rates W 0n are decreasing functions of A. So if A(t) is decreasing in t, W 0n can be considered zero for tt . A similar argument works when substituting 0$m, and one finds that the population in the mth excited state, mm , is frozen, albeit with possibly different freezing points than given by Eq. (6.16). Another consequence of this argument is that the phenomenon of freezing should be more pro- nounced (i.e., occur for smaller t 0 and resulting in smaller _ 00 for t>t 0 ) for those problems where the ground state is separated by a large Hamming distance from the excited states, i.e., a larger tunneling barrier such as the T-gadget compared to the FM-gadget, discussed in Section 6.3.2 be- low. These problems are the ones which are harder to simulate with standard classical simulated annealing with single spin flip moves [151] (cluster flip moves [152] would not necessarily be similarly affected). Finally, note that a simulation of a master equation to which the considerations above apply, such as the AME (see Section 6.2.2), is also expected to exhibit freezing. 6.2.5 Boundary Cancellation with a Ramp at the End Since the ground state population does not change past the freezing point, no change in the schedule would be effective if performed after freezing. In view of these considerations we perform BC before freezing sets in, which — for the standard control schedule s() = (see Fig. 6.1) — happens for s 0:55, depending on the problem. Our strategy will be the following. First, evolve from t = 0 to t f with a BC schedule. To avoid freezing, we ensure that A(t f )6= 0 at the end of the BC schedule. Thus, t f does not correspond to the usual total anneal time for which A(t f ) = 0. Right after the BC schedule ends, we perform a linear ramp † of duration t r = 1s until the schedules reach their ∗ A shortcut to deriving Eq. (6.16) is to interpret 1= (n) rel = (EnE0)jh0jVjnij 2 =W0n as a statement of Fermi’s golden rule for the transition rate, with V = P j z j playing the role of the perturbation, and the density of states. † The term “quench” is used in the D-Wave documentation instead of ramp [153]. 62 final values (in particularA(t f +t r ) = 0), after which the system is measured in the computational basis. The state after the entire evolution can be written asE ramp E BC (0), whereE BC (E ramp ) is the evolution through the BC schedule (ramp). However, random local fields and coupler perturbations [integrated control errors (ICE)] result in a Hamiltonian that does not behave as intended in the ideal case [82], and these errors have been well documented in the D-Wave processors [153]. The effect of such random perturbations can be controlled with error suppression and correction [88, 106, 112], which will be employed below for the ferromagnetic chain gadget. Due to this ICE effect, the measured state is better represented by nal :=E J [E ramp E BC (0)] ; (6.17) where we denoted by E J [] the average over the noise on the random couplings J ij and fields h i . Let us now define(t f ) :=E BC (0), while(t f ) is the Gibbs state at the end of the BC schedule [recall that (0) =(0)], corresponding to the Hamiltonian in Eq. (6.3). The adiabatic theorem in its various forms, including with boundary cancellation, provides an upper bound on the pre-ramp distancekk 1 , where :=(t f )(t f ). Defining nal :=E J [E ramp (t f )] ; (6.18) can be related to nal and nal via the following bound: k nal nal k 1 =kE J [E ramp ]k 1 (6.19a) E J kE ramp k 1 (6.19b) E J [kk 1 ] : (6.19c) Here Eq. (6.19b) follows from from Jensen’s inequality [154] and the fact that every norm is convex (implyingkE J [x]k 1 E J [kxk 1 ]), whileEq.(6.19c)followsbecauseE ramp isacompletelypositiveand trace-preserving (CPTP) map [1] (implyingkE ramp k 1 kk 1 ). Note that nal is the empirically measured state, while nal differs from the state that we sought to prepare (t f ), by the presence of the extra operations E J [E ramp ]. The bound (6.19) implies a similar bound for the ground state probabilities. Let us define P GS := Tr [jGSihGSj nal ] (6.20a) P GS := Tr [jGSihGSj nal ] ; (6.20b) wherejGSi is the ground state of the Hamiltonian at the end of the anneal, i.e., the ground state of H Z . Since ∗ jTr [jGSihGSjx]jkjGSihGSjk 1 kxk 1 =kxk 1 ; (6.21) ∗ Herekyk 1 is the operator norm of y, i.e., its maximum singular value, which is 1 for an orthogonal projection. 63 it follows that the adiabatic error defined as D GS :=jP GS P GS j ; (6.22) satisfies D GS k nal nal k 1 : (6.23) The adiabatic error quantifies the difference between the ground state overlaps of the experimentally measured state ( nal ) and the Gibbs state ( nal ). 6.2.6 An Adiabatic Error Bound that Combines Everything Combining Eqs. (6.19) and (6.23) with Props. 6.1 and 6.4 and our numerical evidence, we obtain D GS k nal nal k 1 C t f ; (6.24) where C is now the noise averaged constant (E J []) and depends on the physical assumptions. Namely, = k + 1 if the Liouvillian either has a non-zero spectral gap throughout the anneal or if BC is enforced after the Liouvillian gap has already closed somewhere along the anneal, while = (k + 1)=(k + + 1) if the Liouvillian gap closes at the same point at which BC is enforced (Prop. 6.4). Of course we may reformulate Eq. (6.24) as a bound on P GS : P GS C t f P GS P GS + C t f : (6.25) 6.2.7 Anomalous Heating Our discussion so far has assumed that the effective temperature of the system remains constant as a function of both the anneal parameter = t=t f and the total anneal time t f . However, there is evidence to suggest that the latter is in fact not the case in the D-Wave devices, i.e., the temperature is t f -dependent. The reason is an unintentional but omnipresent high-energy photon flux that enters the D-Wave chip from higher temperature stages through cryogenic filtering, which accumulates over long anneal times and manifests as an effectively higher on-chip temperature [155]. This anomalous heating phenomenon will hinder our ability to test the BCP, since it means that in fact P GS is a function of t f , which complicates testing the BCT prediction as summarized by Eq. (6.25). Indeed, for a Gibbs state final = e H S =Z (with , H S , and the partition function Z all evaluated at t =t f ), expanding H S in its eigenbasis as H S = P i=0 E i jiihij we readily find P GS =hGSj nal jGSi = 1 1 + P i=1 e (t f ) i ; (6.26) wherej0i is the ground state and i :=E i E 0 . 64 Assuming for simplicity that (t f ) = 0 +a=t f , i.e., a temperature that depends linearly on t f with rate a> 0, and that 1(t f ) 1 (t f ) i for i 2, we can write this as P GS 1e (0) 1 e a 1 =t f ; (6.27) i.e., the algebraic scaling with t f of Eq. (6.25) becomes obscured by an exponential scaling due to P GS . However, in reality we do not know the functional form of (t f ), and the assumption 1 (t f ) 1 (t f ) i may not hold. Therefore, in the analysis of our experimental results in Sec- tion 6.4 below, we instead use an ansatz of the form P GS = P GS + C 0 t 0 f ; (6.28) where P GS is a free fitting parameter representing an averaged value of the true (unknown)P GS (t f ), C 0 becomes another fitting parameter which already accounts for noise averaging as explained in Section6.2.6, and 0 playstheroleoftheeffectivescalingexponent, i.e., ourproxyfor inEq.(6.25). 6.3 Methods 6.3.1 Boundary Cancellation Protocol (BCP) We performed most of our experiments with the D-Wave 2000Q low noise (DW-LN) processor accessed through D-Wave Leap. We also performed additional experiments with the D-Wave 2000Q processor at the NASA Quantum Artificial Intelligence Laboratory (DW-NA) as well as the D-Wave Advantage (DWA) processor through D-Wave Leap. In the processor specifications, the hardware- determined schedulesA(s) andB(s) are parametrized by the user-defined control schedules(). In the standard case the control is linear, i.e., s() = but in general s() can be programmed in a piecewise linear manner as a function of time. The processor permits a maximum of 12 points to specify the piecewise linear function s(). We take advantage of this enhanced capacity to approximate a BC schedule. The allowed range of programmable anneal times is t f 2 [1; 2000]s. Even thoughA(s) andB(s) themselves need not satisfy the vanishing derivative requirement of the BCP, it follows from the chain rule thatA(s(t)) andB(s(t)) do, as long as the control schedule s(t) satisfies this requirement andA(s) andB(s) are differentiable to the same order ass(t). To be concrete, we take s() = k () ; =t=t f ; (6.29) 65 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 0 10 20 30 40 Figure 6.2: An example of the D-Wave boundary cancellation protocol with the 2 schedule ramped at s BC = 0:50, with the constructed parametric schedule (top) and the corresponding physical schedules (bottom). For the schedule shown here we chose an anneal time oft f = 100s. The ramp duration ist r = 1s. Contrast with the native schedule of the DW-LN processor shown in Fig. 6.1. The most significant impact is on the B(s) schedule, which is not natively flat, in contrast to the A(s) schedule, which natively approaches 0 in a very flat manner. -1.00 0.75 -0.50 0.25 1 2 3 4 5 6 7 8 -1 -2/3 2/3 -1 -1/3 1 -1 1 (a) (b) Figure 6.3: Illustration of the FM-gadget (a) and T-gadget (b) embedded into the Chimera ar- chitecture as represented by an Ising Hamiltonian H Z [Eq. (6.4)]. Ferromagnetic (blue) and anti- ferromagnetic (dashed purple) couplings all have the same strength of J ij = 1 and J ij = 1, respectively. Local fields h i are indicated by the arrows and their value beside them. The ordering of the qubits in the Chimera architecture is also enumerated in (a). 66 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 (a) 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 (b) Figure 6.4: Spectral gaps of (a) the FM-gadget and (b) the T-gadget (with = 0:5), as the schedule is varied according to typical physical schedules A(s) and B(s) of the D-Wave processor (Section 6.6.5), shown in units of ns 1 (~ = 1). The notation n;n+1 denotes the energy gap between the nth and n + 1th Hamiltonian eigenstates, with n = 0 being the ground state. The 13:5 mK energy scale is also shown, which is the reported dilution fridge temperature of the DW- LN processor. The minimum gap location is s = 0:43 for the FM-gadget and s = 0:44 for the T-gadget. The minimum gap region is marked by the light blue shading, while the estimated start of frozen dynamics at s 0 is marked by pink shading, found by solving Eq. (6.16) (see Section 6.6.4 for details). Note that for both gadgets, the first, second, and third excited states are all initially degenerate in the subspace of a singleji excitation of the transverse-field ground state. The nonzero excited state gaps 12 and 23 of the FM gadget ( 12 only for the T gadget) ats = 1 are the result of symmetry breaking due to a crosstalk term we included in the Hamiltonian, of strength = 0:02 (see Section 6.6.6). The avoided level crossing of the T-gadget is slightly narrower than that of the FM-gadget. 67 where k () := B (1;k + 1) B 1 (1;k + 1) (6.30a) = 1 (1) k+1 (6.30b) is the regularized incomplete beta function of order k [96, 156], with B x (a;b) := Z x 0 y a1 (1y) b1 dy (6.31) being the incomplete beta function. As is apparent from Eq. (6.30b), the function k () has exactly k vanishing-derivatives at = 1, as required by the BCT. We refer to this class of control schedules simply as beta schedules. As we discussed above, in contrast to the theoretical setup of Ref. [96], due to freezing there is no practical advantage to flattening the schedule as s approaches 1. The effectiveness of the BCP is most apparent when the flattening of the schedule occurs after the avoided level crossing of the anneal (ats =s ), but before the dynamics freeze (ats =s 0 ). Thus, rather than using the standard schedulesfA(s);B(s)g with linear s() that freeze out around s' 0:5, the beta schedule needs to be adjusted so that it terminates at an appropriate points BC corresponding tot =t f and satisfying 0 0:04. 77 Lin. 0.5 1.0 1.5 2.0 2.5 3.0 1.6 1.8 2.0 2.2 2.4 2.6 0.5 1.0 1.5 2.0 2.5 3.0 1.6 1.8 2.0 2.2 2.4 2.6 0.5 1.0 1.5 2.0 2.5 3.0 1.6 1.8 2.0 2.2 2.4 2.6 0.5 1.0 1.5 2.0 2.5 3.0 1.6 1.8 2.0 2.2 2.4 2.6 0.5 1.0 1.5 2.0 2.5 3.0 1.6 1.8 2.0 2.2 2.4 2.6 0.5 1.0 1.5 2.0 2.5 3.0 1.6 1.8 2.0 2.2 2.4 2.6 Figure 6.9: Number of tries to solution for the T-gadget, comparing the pausing protocols (empty symbols) and 0 ; 1 BCP (all with a ramp at s BC ), along with the linear anneal without a ramp (denoted “Lin.”; the same data for all panels), for reference. Results shown are for DW-LN, as a function of the total anneal time of each protocol. Different panels have different values of s BC , from 0:45 to 0:55. The pausing protocol is shown for different initial anneal times t 0 of 1,5, and 20s (empty symbols), followed by a pause of varying length t p and a 1s long ramp (so that the paused portion of the schedule is comparable to BCP). For BCP (filled symbols), t a = t f + 1s. For pausing, t a = t 0 +t p + 1s. Both schedules are plotted against t a rather than their natural parameters (t f and t p respectively) for direct comparison. that for the optimal- linear QAC anneal in Fig. 6.8 (for which 0 = 0:69), signaling that freezing has already mostly occurred before s = 0:50, and that the optimal value is s BC -dependent. Our results demonstrate that the combination of BCP and QAC as error suppression methods is more powerful in terms of the scaling of the adiabatic distance than either method alone: while k = 2 BCP has 0 = 1:1 [Fig. 6.6(a)] and optimal- linear(k = 0) QAC has 0 = 0:69 (Fig. 6.8), their combination yields 0 = 1:3 [Fig. 6.6(d)] and the largest P GS of any other schedule for t f = 12s. 6.4.5 Comparison of the BCP to the Pause-Ramp Protocol A different protocol that attempts to exploit slowing down the anneal is the pausing protocol, which interrupts an ordinary linear anneal by a single pause [164–168]. Pausing directly uses thermal relaxation at a single point in the anneal to try to increase the ground state probability. As in the BCP, this point should also be after an avoided level crossing, but before the open system dynamics freeze [165]. We compare the BCP to a variant of the usual pausing schedule [164] referred to here as the pause-ramp (PR) schedule. While the pausing protocol is typically constructed by interrupting a linear anneal with a pause (but no ramp), PR is constructed by making the first linear segment last 78 fort 0 , pausing fort p , and ramping to the end. As with the BCP, we use a ramp that is 1s long in all cases. PR is better suited for comparison with BCP than the usual pausing schedule since they differ only in their equilibrium state preparation and not in their behavior during the frozen phase. As our metric for comparison we use not the ground state probability but rather the number of tries-to-solution (NTS) at a fixed anneal time (with 90% confidence): NTS 90% (P GS ) = log(1 0:90) log(1P GS ) (6.34) This metric is simply another way to study the ground state probability, interpreted as the expected number of tries needed to find the ground state at least once with 90% confidence, at the given anneal time. We use this rather than the standard time-to-solution (TTS) metric, since the latter requires identifying the optimal anneal time [55, 146, 169]. However, neither the FM-gadget nor the T-gadget exhibited an optimal anneal time in our experiments (not shown). Figure 6.9 shows the NTS results for the T-gadget at various values of s BC . It compares the standard schedule (no BCP or pausing), the BCP protocol atk = 0 andk = 1, and the PR schedule with different initial anneal and pause times. Starting at s BC = 0:45 (top left) the gap is small and thermalizationisfast. BCPexhibitstwodistinctoptimalannealtimes(minima)fork = 0andk = 1, both of which result in slightly smaller NTS than pausing. Also, thek = 1 case is optimal at smaller t a than k = 0, which can be interpreted as as advantage of the higher order protocol. This trend persists for most of the s BC values shown. Right at s BC = 0:46, pausing achieves its optimal NTS curve, with a slight advantage over BCP, and the BCP of each order k is very distinct. However, the advantage of pausing is only present at s BC = 0:46, and disappears already at s BC = 0:47, showing that pausing performance is highly sensitive to the pause point. Since the BCP slowdown is smooth, the influence of the optimal thermalization at s BC = 0:46 remains present, although the distinction between BCP orders diminishes ass BC grows. Well befores BC approaches the freeze-out point (s 0 0:51), pausing becomes detrimental for the T-gadget due to excitations (witnessed by the increase in the NTS metric), while BCP mitigates the excitations until the anneal time becomes too long to avoid them. These results paint a picture of the two protocols as follows. The relaxation induced by the BCP depends on the properties of the Liouvillian over a neighborhood (s BC ;s BC ]. If this neighborhood overlaps with the neighborhood of an avoided level crossing, then the BCP has an opportunity for an advantage, as the speed at which the crossing is traversed is minimized and detrimental excitations to the excited state are reduced. In contrast, PR schedules must try to discontinuously stop the anneal as close as possible to the crossing, and the time required by a paused schedule could be increased due to the longer pause time needed to recover the ground state population, unless the precise optimal pause point is quickly found. This suggests an important practical point: optimization of the BCP can be simpler than that of PR schedules, by the need to optimize just one discrete parameter k and one continuous parameter s BC . Moreover, s BC need not be exactly tuned, since even an approximate value can allow the 79 BCP to closely approach its optimal performance. On the other hand, pausing requires s BC to be optimized very precisely, as well as the anneal and pause times t 0 and t p . 6.5 Discussion and Outlook WehaveextendedtheboundarycancellationtheoremforopensystemstothecasewheretheLiouvil- lian gap vanishes at the end of the anneal, and derived the asymptotic scaling of the adiabatic error with the anneal timet f . Armed with the corresponding theoretical expectation of the scaling of the adiabatic error for the gapped and gapless cases, we set out to test the scaling predictions and to im- prove the success probability of quantum annealing hardware by implementing boundary cancelling schedules. The specific functional form of these schedules induces a smooth slowdown in accor- dance with the boundary cancellation theorem. We experimentally tested boundary cancellation protocols for open systems and evaluated their performance and error-suppression characteristics on specifically designed 8-qubit gadgets embedded on the DW-LN annealer. Whileaquantitativeagreementwiththetheoreticalpredictionswasnotobserved, wediddemon- strate that as long as the protocol terminates before the onset of freezing, it can increase the ground state population in the examples studied here beyond what is achievable with simple linear an- neals, and it does so with shorter anneal times. These results are in qualitative agreement with the theoretical scaling predictions of the BCT. In conjunction with quantum annealing correction, the boundary cancellation protocol is also capable of improved adiabatic error scaling over what would be achieved with either method alone. While this does not immediately translate to a ground state solution speedup within the annealing problems studied here, we have shown that BCP-QAC is an error suppression strategy that suc- cessfully combines two complimentary methods: the suppression of environmentally induced logical errors and the promotion of relaxation via boundary cancellation. In contrasting the BCP with the pause-ramp protocol, we found that the BCP is significantly less sensitive to the location of the ramp point, and achieves better performance except at the exact ramp point where the pause-ramp protocol is optimal. With the small system size of 8 qubits used in this work, it was possible to collect a large number of annealing samples as well as validate the protocol behavior against the energy spectrum and open system simulations. Future work will assess the protocol for larger system sizes and the impact it has on the scaling of time-to-solution as a function of problem size. The largest expected improvement in the protocol’s performance, based on our simulations, will arise from an increase in the number and resolution of interpolation points along the annealing schedule, which will allow a more faithful experimental implementation of the ideally smooth annealing schedules demanded by the theoretical protocol. 80 6.6 Methods and Computational Details 6.6.1 Proof of Degeneracy when A(t) = 0 We assume that A i = z i . When A(t) = 0, H S / H Z and commutes with A i , so that Eq. (6.7) yields A i (!) = z i !;0 . Therefore, the dissipator in Eq. (6.5a) becomes Dj A(t)=0 = X ij ij (0) z j z i 1 2 z i z j ; : (6.35) NowletjnibeaneigenstateofH Z , wheren = 1;:::;danddisthesystem’sHilbertspacedimension. Thenjnihnj commutes with z i , so thatD (jnihnj) = 0. At the same time, A i (!) = z i !;0 implies that also the Lamb shift is diagonal in the z basis. Together with the fact thatjnihnj also commutes with the system Hamiltonian this impliesL (jnihnj) = 0, which means that the kernel ofL is at least d-dimensional at A(t f ) = 0, i.e., that the zero eigenvalue ofL is at least d-fold degenerate. 6.6.2 Proof of Proposition 6.4 Here we prove Proposition 6.4, which concerns the case where the gap closes at the end of the anneal and we enforce boundary cancellation at this point. We denote byL(s) the Liouvillian without the BC schedule (i.e., with a linear schedule around s = 1) whileL(s()) [orL() for simplicity] indicates the Liouvillian with the BC schedule. We will use the series developed in [170] and refined in [96] together with ideas from [148, App. J]. The assumption that R 1 0 dskL(s)k 1;1 <1 assures, by Carathéodory’s theorem, that the solution of Eq. (6.1) exists and is unique in an extended sense [171], while the requirement thatL() is k + 2 differentiable atk = 1 implies that one can use the series Eq. (6.43a) below with = 1 andN =k. By assumption, we have a boundary cancelling schedule at s = 1. Such a schedule is a function s =s() where s(1) = 1 and s (j) (1) = 0 for j = 1; 2;:::;k. That is, s() = 1g(1) k+1 +O( k+2 ) ; 1 (6.36) when ! 1 for some constant g. Let L(s) = X j=1 j (s)P j (s) +D(s) (6.37) be the instantaneous spectral decomposition ofL(s) (without the BC schedule), i.e., j (s) are eigenvalues, P j (s) are eigenprojectors, and D(s) is a nilpotent term. We denote by P (s) (without a subscript) the projector related to the zero eigenvalue. The assumption of contractivity implies that D(s) cannot belong to theL(s) = 0 subspace. We denote by () the unique steady state of L(s()), i.e.,L(s())() = 0, Tr(()) = 1 while () is the solution of Eq. (6.1) with the BC schedule and with the initial condition (0) =(0). Note that in the main textL(s()) is written for simplicity asL(). 81 We use a similar approach (and notation) as in Ref. [148]. HereE() :=E(; 0) is the evolution superoperator satisfying @ E(; 0 ) = t f L(s())E(; 0 ) andE( 3 ; 2 )E( 2 ; 1 ) =E( 3 ; 1 ) together withE(;) = 1 I, i.e., E( 2 ; 1 ) =E 1 ( 3 ; 2 )E( 3 ; 1 ): (6.38) Let V () be the adiabatic intertwiner as defined in [148] and, following the same notation therein, W () :=P ()V () =V ()P (0). We start with Eq. (D4) of Ref. [148] which we reproduce here for clarity: E 1 ()V () 1 I P (0) = Z 0 E 1 ( 1 ) _ W ( 1 )d 1 ; (6.39) where the dot denotes differentiation with respect to the argument of the corresponding operator. Hence E 1 (1)V (1) 1 I P (0) = Z 0 + Z 1 E 1 ( 1 ) _ W ( 1 )d 1 (6.40a) = E 1 ()V () 1 I P (0) + Z 1 E 1 ( 1 ) _ W ( 1 )d 1 ; (6.40b) Now applyE(1) :=E(1; 0) from the left and use Eq. (6.38) repeatedly, to obtain: (V (1)E(1))P (0) = (E(1;)V ()E(1; 0))P (0) + Z 1 E(1; 1 ) _ W ( 1 )d 1 (6.41a) =E(1;) (V ()E())P (0) + Z 1 E(1; 1 ) _ W ( 1 )d 1 : (6.41b) Now apply (0) from the right. Note that V () is continuous, being the solution of a differential equation. So V (1)P (0)(0) = lim !1 () =(1 ) and we obtain finally (1 )(1) =E(1;) (()()) + Z 1 E(1; 1 ) _ W ( 1 )(0)d 1 : (6.42) Eq. (6.42) is the starting point of our analysis. Since in Eq. (6.42) < 1 and the system is gapped in [0;], we can use the result of [96] to obtain the behavior of ()(). Assuming that L() is at least N + 2 times differentiable in a neighborhood of = 1, one has the following series [96] with = 1=t f for in the same neighborhood: () =() + N X n=1 n b n () + N+1 r N (;) (6.43a) b 1 () =S() _ P ()() =S() _ () (6.43b) b n+1 () =S() _ b n (); n = 1; 2;:::;N (6.43c) 82 where the remainder is r N (;) =b N+1 ()E()b N+1 (0) Z 0 E(; 0 ) _ b N+1 ( 0 )d 0 (6.44) and S() is the reduced resolvent, defined as: S() := lim z!0 (1 IP ()) (L()z) 1 (1 IP (r)): (6.45) We will keep N free and only fix it at the end when needed. The series (6.43a) cannot be used at = 1, because the assumption of a gapped Liouvillian does not hold there. Hence we must keep < 1 but investigate the kind of divergences that arise when ! 1 . The other assumption is that, without the BC schedule, the Liouvillian gap closes at s = 1 in such a way that 1 (s)'v(1s) ass! 1 . Therefore, using Eqs. (6.36) and (6.37), close tos = 1 the Liouvillian with the BC schedule behaves as L(s())' ~ v(1) ~ P 1 (s()) + X j>1 j (s())P j (s()) +D(s()); (6.46) with ~ v =vg and ~ =(k +1). Accordingly, the most diverging term of the reduced resolventS() behaves as ∗ S() P 1 ~ ; (6.47) where henceforth we do not always explicitly emphasize the dependence on when it does not lead to a divergence. Now, the derivatives of the projectors behave as @ P j (s()) =@ s P j (s)@ s()@ s P j k : (6.48) Hence, we see that when ! 1 , b 1 diverges as b 1 () 1 ~ k : (6.49) When we construct b n according to Eq. (6.43c), at each iteration we gain a derivative and a factor ofS. Thus, defining n viab n () n , we obtain n+1 = n + ~ +1. This has the solution n =n (~ + 1)k 1 (6.50a) =n (k + + 1)k 1: (6.50b) ∗ Here, and analogously in the following, the symbol ’’ in Eq. (6.47) is intended to mean lim !1 kS()k 1 ~ = kP1(1)k 1 . This implies that one can find a positive constant C such thatkS()k 1 C= ~ for sufficiently small . This fact will be repeatedly used in the following. 83 Hence, we can find positive constants A n such that kb n ()k 1 A n n : (6.51) Turning to the remainder, we take the trace norm of Eq. (6.44) and use the triangle inequality: kr N (;)k 1 kb N+1 ()k 1 +kE()b N+1 (0)k 1 + Z 0 kE(; 0 ) _ b N+1 ( 0 )k 1 d 0 : (6.52) Next, we usekExk 1 kEk 1;1 kxk 1 for a superoperatorE and operator x, and the fact that since E( 2 ; 1 ) is a completely positive trace preserving map for 2 1 , we have kE( 2 ; 1 )k 1;1 = 1 ; 2 1 : (6.53) Therefore Z 0 kE(; 0 ) _ b N+1 ( 0 )k 1 d 0 Z 0 k _ b N+1 ( 0 )k 1 d 0 : (6.54) Now using b n () n again we obtaink _ b N+1 ( 0 )k 1 (1 0 ) N+1 1 , which integrated from 0 to implies that Z 0 kE(; 0 ) _ b N+1 ( 0 )k 1 d 0 B N+1 ; (6.55) for some positive constant B. All in all, we obtain for the remainder kr N (;)k A N+1 N+1 +kb N+1 (0)k + B N+1 ; (6.56) Herekb N+1 (0)k is just another constant, independent of . Combining these results we obtain from Eq. (6.43a), after redefining the constants: k()()k N+1 X n=1 n A n n + N+1 A N+2 : (6.57) It remains to estimate the second term on the RHS of (6.42). Using the mean value theorem, _ W = _ PW [148, Eq. (C1)], and W ()(0) =(), we have: Z 1 E(1; 1 ) _ W ( 1 )(0)d 1 =E(1; ~ ) _ P (~ )(~ ); (6.58) where ~ 2 [; 1], and we used = 1. Hence, when taking the norm of Eq. (6.58), using Eqs. (6.48) and (6.53), we obtain, for some positive constant C: Z 1 E(1; 1 ) _ W ( 1 )(0)d 1 C k+1 : (6.59) 84 Using Eqs. (6.57) and (6.59), we finally obtain after taking the norm of Eq. (6.42): (1)(1 ) N+1 X n=1 n A n n + N+1 A N+2 +C k+1 (6.60) =:f(;): In this derivation = 1 is sufficiently close to one but otherwise free. Thus, we can now minimize f as a function of . Differentiating f we obtain: @ f = N+1 X n=1 n n A n n+1 + (k + 1)C k = 0 (6.61a) ,! k @ f = N+1 X n=1 n n A n n(k++1) + (k + 1)C = 0: (6.61b) We now make the ansatz = . We wish to find a such that the solution of Eq. (6.61b) does not depend on . This is satisfied if all the summands in Eq. (6.61b) do not scale with , i.e., are constant, which holds for n n (k + + 1) = 0. This means that = 1= (k + + 1). The value of can be found by solving N+1 X n=1 n A n n(k++1) = (k + 1)C: (6.62) PickingN = 1 in Eq. (6.62) one can show that the resulting quadratic equation in x := (k++1) always has a positive solution, which means that Eq. (6.62) always has a real solution . Substituting = with = 1= (k + + 1) into Eq. (6.60), we see that all terms scale as with the exponent = k + 1 k + + 1 : (6.63) In other words: (1)(1 ) C 0 ; (6.64) as reported in the main text. See Figure 6.10 for numerical simulations confirming Eqs. (6.63) and (6.64). 6.6.3 Proof thath0j z j jni =O(A q ) In the main text we stated thath0j z j jni =O (A q ), and hence that the transition matrix elements W 0n = O A 2q are suppressed for n 1. Generalizing beyond the scenario in the main text, we now allow for degeneracy in the eigenstates of H Z . We assume the transverse field is not zero and investigate the behavior of the rates when A! 0. For this value of A(t)6= 0, let H S = P n E n ~ P n be the spectral decomposition of the system 85 k=0: -0.681 - 0.332 t k=1: -0.635 - 0.399 t k=2: -0.585 - 0.428 t k=3: -0.541 - 0.444 t 6.0 6.2 6.4 6.6 6.8 7.0 -3.6 -3.4 -3.2 -3.0 -2.8 Figure 6.10: Test of the bound given in Proposition 6.4, for the case of a single qubit subject to evolution under the AME. The Hamiltonian is H() = ! x (1 k ()) x +! z k () z with BC schedule k () = B (1;k + 1)=B 1 (1;k + 1) [B is the incomplete beta function, Eq. (6.31)] which has k vanishing derivatives at = 1 and is linear at = 0. The system-bath coupling is via z , which assures that the Liouvillian is gapless at = 1 with a gap-closing exponent = 2 [148]. The bath spectral density is Ohmic [Eq. (6.11)]; the parameters are g = 1; = 1; ! x = ! z =1=2 in arbitrary units. Dots are results of solving for the evolution while the continuous lines are linear fits. The theoretical expectation given by Eq. (6.63) is =f0:333; 0:4; 0:428; 0:444g fork =f0; 1; 2; 3g, respectively, and is an excellent agreement with the the last column of the legend (obtained via a linear fit). Hamiltonian, where ~ P n are the (possibly degenerate) spectral projectors. The Bohr frequencies have the form ! =E b E a and we can label the jump operators A j (!) as A ab;j (t) = ~ P a (t) z j ~ P b (t): (6.65) We omit the time dependence from here on. Let us consider the operators A ab;j perturbatively around A(t) = 0. We denote by P a the (unperturbed) spectral projectors at A(t) = 0, which are diagonal in the computational basis. By assumption, the degeneracy of ~ P a does not change along the anneal, so that the ~ P a are smoothly connected with the P a . We use the Kato perturbation theory formalism [163], and denote the perturbation by V :=H X . Moreover, for level a, we define the reduced resolvent as R a = X n6=a P n E n E a : (6.66) For fixed a6=b we expand the jump operators in powers of the transverse field: A ab;j = ~ P a z j ~ P b := X n=0 A n C n : (6.67) 86 At zeroth order we have (since a6=b) C 0 =P a z j P b =P a P b z j = 0: (6.68) At first order, perturbation theory gives C 1 =P a z j (P b VR b +R b VP b ) (P a VR a +R a VP a ) z j P b (6.69a) =P a z j R b VP b P a VR a z j P b (6.69b) = z j R b P a VP b P a VP b R a z j : (6.69c) Now recall that V = H X = 2S x = (S + +S ) where S x (S ) is the total spin-x operator (ladder operator) and hence V changes the Hamming distance by one. As a consequence, P a VP b = 0 if HD(a;b)6= 1 where we denoted by HD(a;b) the smallest Hamming distance among the representa- tive states of the projectors P a and P b . I.e., if P a = P da =1 jaihaj, P b = P d b =1 jbihbj for some Ising statesjai;jbi, HD(a;b) := min ; HD (jai;jbi): (6.70) At second order we obtain C 2 = (P a VR a +R a VP a ) z j (P b VR b +R b VP b ) + P a z j R b VP b VR b +R b VR b VP b R 2 b VP b VP b + P a VR a VR a +R a VP a VR a P a VP a VR 2 a z j P b : (6.71) The formulas become increasingly complicated, but we see that at orderk,C k is sum of terms of the form O (1) z VO (2) z VO (k+1) z where there are k V’s and we denote by O (j) z any operator diagonal in the same basis as z . Two of theO (i) z (i = 1; 2;:::;k +1) compriseP a andP b . Letd s be the number of V’s sandwiched between P a and P b . Then this term is zero if HD(a;b) > d s . Since at order k, d s is necessarily k, we see that C k is zero if HD(a;b) > k. Alternatively, note that the effect of each O (j) z on computational basis states is simply that of creating a multiplicative constant but otherwise it leaves computational basis states unchanged. So, apart from a multiplicative constant each term in C k has the formjaihajV ds jbihbj. Summarizing, if HD(a;b) =q, A ab;j =O (A q ): Since in the Pauli matrix W ab there appear two Lindblad operators per Bohr frequency (a;b pair) we have just shown that W ab =O A 2q . In particular, this is true if the level a = 0 and b =n are non-degenerate as it is the case in the main text. Figure 6.11 shows that this prediction is borne out in numerical simulations. 6.6.4 Determination of the Freezing Point The resulting transition rates between energy eigenstates in the AME are shown in Fig. 6.13 for both the FM and the T-gadgets. Since the magnitude of available annealing times are limited to 10 3 s, 87 0.05 0.10 0.50 1 5 10 10 -21 10 -11 0.1 Figure 6.11: The transition matrix elementsW 0n of the FM-gadget forn = 1; 2; 3 (solid), for which q = 8; 7; 6, respectively, as a function of A, along with A 2q shifted to most closely align with the W 0n (A) curves (dashed). The agreement is clearly visible for sufficiently small A. The transverse field schedule used is A(s) = 30 ns 1 (1s) while B(s) = 10 ns 1 is held constant. This allows the scaling to be seen clearly for smallA without changing the overall energy scale of the Hamiltonian. 50 100 500 1000 0.80 0.85 0.90 0.95 (a) 5 10 50 100 500 1000 0.65 0.70 0.75 0.80 (b) Figure 6.12: Ground state probabilities and fits for the BCP applied to the FM gadget using (a) DW2KQ at NASA QuAIL and (b) D-Wave Advantage at the appropriate values of s BC where the fits are most distinguishable. 88 0.0 0.2 0.4 0.6 0.8 1.0 10 -11 10 -9 10 -7 10 -5 0.001 0.100 0.0 0.2 0.4 0.6 0.8 1.0 10 -11 10 -9 10 -7 10 -5 0.001 0.100 Figure 6.13: Thermal transition rates of the adiabatic master equation (with 0 g 2 = 5:0 10 4 ) for some pairs of energy eigenstates a!b as the DW-LN schedule progresses linearly for the FM- gadget (top) and the T-gadget (bottom). The horizontal dotted line denotes the rate 1=t f , which is on the order of magnitude of the smallest rate that can be reliably observed within an anneal time of t f = 10 3 s. Below this line, thermal relaxation events rapidly become extremely rare, hence the onset of frozen dynamics during the anneal. The Hamming distances between the final computational states corresponding to the eigenstate transitions are also listed. When an excited state is only a small Hamming distance away from the ground state, the transition freezes later in the anneal since only a much smaller number of qubits need to tunnel through. This is the case for the 2! 1 and 3! 1 transitions in the FM-gadget, and the 3! 0 transition in the T-gadget. we expect that a transition will become too rare to observe once it drops below 1=t f 10 6 GHz. With the majority of the population in the ground state and first excited state, we therefore expect freezing to occur when 10 goes below this value. This intersection determines the freeze-out points highlighted in Fig. 6.4. 6.6.5 Schedule Construction The annealing schedule of the DW-LN used in our main experiments is shown in Fig. 6.1. All simulations are based on this schedule and its operating temperature at the time of sample collection (13.5 mK). First, suppose s () = k () is a beta-schedule terminating at s = 1 and let (s) denote its inverse function. The construction of a piecewise-linear approximation to s () is as follows: 1. The initial point ( 0 ;s 0 ) = (0; 0) and the final point ( 10 ;s 10 ) = (1; 1). 2. A separation point ( 5 ;s 5 ) = ( (s c );s c ) for some choice of s c . 3. Linearly partition s 1 ;:::s 4 , i.e., assign s j =js c =5 and set j = (s j ) for j = 1::: 4. 4. Logarithmically partition s 6 ;:::;s 9 between s c and 1, i.e., assign s j+1 = 1 (1s j )=2) and set j = (s j ) for j = 6::: 9. Thus, for a schedule that terminates at the ramp points BC , we have the following 12-point scheme: 89 1. Construct the first 11 points according to the above procedure, and then rescale each s j ! s BC s j . 2. The final 12th point is ( 11 ;s 11 ) = ( 10 +; 1), where is constrained by the maximum ramp slope of the hardware, i.e., = 1s=t f . Note that when the ramp is included s = 1 corresponds to = 1 + and s BC < 1. Aside from the orderk, there are the three annealing parameters s c ;s BC ; andt f that must be specified for the protocol. We took s c = 0:9 for the construction of the beta schedules. Adjusting thes c parameter to 0:95 to improve the precision of the analytic behavior of the beta schedule did not have a substantial effect on the ground state probabilities of the empirical D-Wave results. It is likely that such a precision exceeds what can be reliably and repeatedly implemented on the hardware. 6.6.6 Crosstalk Itisknownthat,duetothedifficultyofhardwarefabricationoftheD-Waveprocessors,programming the couplings and field to have certain values induces non-zero fields and couplings also on other links of the graph, despite designs meant to minimize such crosstalk [172]. We use the ferromagnetic crosstalk model defined by the following transformations [142]: h i 7!h i X k6=i J ik h k (6.72a) J ij 7!J ij + X k6=i;j J ik J jk (6.72b) where is the crosstalk strength. We assume a strength of = 0:02 throughout. We have checked that optimizing the value of further does not substantially affect our numerical results. This crosstalk has the effect of breaking the degeneracy of the first and second excited states shown in Fig. 6.4. We note that the transformation (6.72) commutes with gauge (or spin-reversal) transformations of the Hamiltonian [173, 174], so population shifts due to crosstalk do not depend on the gauge. 6.6.7 Data Collection and Fitting The probabilities at a given anneal time are obtained from a sample of 20 submissions (10 for QAC), where each submission programs 5 random global gauges of the gadget repeated over 31 unit cells, and samples each gauge 200 times (see Ref. [174] for a review of the gauging procedure and other best practices). To mitigate time-fluctuating systematic biases, submissions for the same anneal time are not consecutively submitted (anneal times are cycled for every schedule). Estimates for the ground state probability are taken from the average of the ground state probabilities of the 20 submissions. Error bars for each measured ground state probabilities are derived from twice the standard deviation over all submissions. Either by non-linear fits on the transformed data, or by 90 linearfitsonthelog-logtransformeddataassumingafixed P GS , wefindtheconstantC andexponent such that the 2 withD GS is minimized. Nonlinear fits are performed as a function of 1=t f . The estimates of 0 for the D-Wave data are obtained by performing fits on 100 bootstrap samples of the collected ground state probabilities. We use the median values of the parameter distributions as the estimators for the parameters as they are robust to the outliers in bootstrapping that occur. The error estimate is derived from half of the range of the 90% confidence interval of the parameter distribution. The non-linear fits are evaluated using the Mathematica function NonlinearModelFit withthemethodNMinimize(NelderMead). Thenon-linear 2 objectivewasweighedbythevariances of the ground state probabilities. 6.6.8 Results for Alternative D-Wave QPUs We performed similar BCP quantum annealing experiments with the DW2KQ at NASA QuAIL (DW-NA) and with the D-Wave Advantage (DWA) annealers, while the main results used the low- noise DW2KQ processor through D-Wave Leap (DW-LN) The scaling behavior of the ground state probabilities at appropriate values of s BC is shown in Fig. 6.12. These results should be contrasted with Fig. 6.6(a), for the DW-LN. All three QPUs have different schedules and thermal properties that affect the qualitative behavior of the BCP, so it is difficult to make a direct comparison between the devices. However, is generally larger in DW-LN compared to either DW-NA or DWA when s BC is in the appropriate region for each annealer. Furthermore, DWA appears to exhibit more significant time-dependent deviations, as can be noted in the 2 schedule at t f = 431 in Fig. 6.12. Acknowledgements Thecontentinthischapterwasoriginallypublishedin[98]. Thestatementandproofofthemodified BCT is due to Lorenzo Campos Venuti. This research is based upon work (partially) supported by the Office of the Director of Na- tional Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) and the Defense Advanced Research Projects Agency (DARPA), via the U.S. Army Research Office con- tract W911NF-17-C-0050. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, ei- ther expressed or implied, of the ODNI, IARPA, DARPA, ARO, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. We thank the Center for Advanced Research Computing (CARC) at the University of Southern California for providing computing resources that have contributed to the research results reported in this chapter. URL: https://carc.usc.edu. 91 Chapter 7 A Double-slit Proposal for Quantum Annealing 7.1 Introduction Feynman famously wrote that the double-slit interference experiment “...has in it the heart of quan- tum mechanics. In reality, it contains the only mystery” [175]. Here we propose a double-slit experiment for quantum annealing (QA). In analogy to Feynman’s particle-wave double-slit, the proposed experiment can only be explained by the presence of interference and would break down upon either an intermediate measurement or strong decoherence. We are motivated by the recent resurgence of interest in quantum annealing using the transverse field Ising model [47, 176], which has led to major efforts to build physical quantum annealers for the purpose of solving optimiza- tion and sampling problems [177–180], and significant debate as to whether quantum effects are at play in the performance of such devices [181, 182]. The mechanisms by which QA might achieve a speedup over classical computing remain hotly contested, and while tunneling is often promoted as a key ingredient [61] and entanglement is often viewed as a necessary condition which must be demonstrated [183, 184], a consensus has yet to emerge. Yet, an explicit example is known where QA theoretically provides an oracle-based exponential quantum speedup over all classical algorithms [185], and other examples are known where QA provides a speedup over classical sim- ulated annealing [186–191]. An essential feature in all these cases are diabatic transitions which circumvent adiabatic ground state evolution to enable the speedup, in the spirit of the idea of short- cuts to adiabaticity [192]. When these transitions result in a coherent recombination of the ground state amplitude (a phenomenon known as a diabatic cascade [188, 193]), the result is a wave-like interference pattern in the ground state probability as the anneal time is varied [194–196]. We thus conjecture that coherent recombination of ground state amplitudes after coherent evolution between diabatic transitions can play a critical role in enabling quantum speedups in QA. The double-slit proposal we formulate and analyze here is designed to test for the presence of quantum interference due to such coherent evolution. Viewed from a different perspective, our double-slit proposal joins a family of protocols designed to probe the dynamics of what Berry called the “simplest non-simple quantum problem” [197], a 92 driven TLS near level crossings [198]. The two-level paradigm was introduced long ago by Landau and Zener (LZ) [199, 200]. The corresponding Hamiltonian for the generalized LZ problem is H S (t) =a(t)Xb(t)Z ; (7.1) where X, Y and Z are the Pauli matrices. In the original protocol which LZ solved analytically, a(t) is constant, b(t) is linear in t, and t runs from1 to1. The problem has since been studied under numerous variations, including Landau-Zener-Stueckelberg interferometry where b(t) is periodic [201, 202], the subject of various experiments [203–205]. Complete analytical solutions were limited until recently to certain particular functional forms of b(t) with constant a(t) [206], a finite-rangelinearscheduleforbotha(t)andb(t)[207],andperiodica(t)andb(t)[208]. Ananalytical solutionforgeneralb(t)butconstanta(t)wasfoundinRef.[209], whichwasthenextendedtogeneral (but implicitly specified)a(t) as well [210, 211]. Here we consider the case of general schedulesa(t) and b(t), and develop a simple to interpret, yet surprisingly accurate, low-order time-dependent perturbation theory approach, that allows us to identify a class of schedules exhibiting “giant” (relative to linear schedules) interference oscillations of the ground state population as a function of the total annealing time. Our proposal should in principle be straightforward to implement using, e.g., flux qubits, and toward this end we also study the effects of coupling to a thermal environment. The structure of this chapter is as follows. In Sec. 7.2 we analyze the TLS quantum annealing problem in the closed system limit. We first transform to an adiabatic interaction picture and perform a Magnus expansion, which allows us to give a simple expression for the ground state probability in terms of the Fourier transform of a key quantity we call the angular progression. We thenanalyzeboththeLZproblem(withalinearschedule)andanew“Gaussianangularprogression” schedule which gives rise to large interference oscillations. We explain how these oscillations can be interpreted in terms of a double-slit experiment generating interference between ground state amplitudes. InSec.7.3weanalyzetheprobleminthepresenceofcouplingtoathermalenvironment. We consider the weak-coupling limit both without and with the rotating wave approximation, and find the range of coupling strengths and temperatures over which the interference oscillations are visible, using parameters relevant for superconducting flux qubits. We find a simple semi- empirical formula that accurately captures all our open-system simulation results in terms of three physically intuitive quantities: the oscillation period, rate of convergence to the adiabatic limit, and damping due to coupling to the thermal environment. We express all three are in terms of the input parameters of the theory. Conclusions and the implications of our results are discussed in Sec. 7.4. A variety of supporting technical calculations and bounds are provided in Sec. 7.5. 93 7.2 Closed System Analysis and Results 7.2.1 Adiabatic Interaction Picture for Two-level System Quantum Annealing We first consider the closed system setting. Consider a two-level system (TLS) quantum annealing Hamiltonian in the standard form (7.1), where the annealing schedules a(t);b(t) 0 respectively decrease/increase to/from 0 with time t2 [0;t f ], where t f is the duration of the anneal. The schedules need not be monotonic, and our analysis thus includes “reverse annealing” [212–216] as a special case. The TLS can be a single qubit or the two lowest energy levels of a multi-qubit system separated by a large gap from the rest of the spectrum. Key to our analysis is a series of transformations designed to arrive at a conveniently reparametrized interaction picture. First, we rewrite Eq. (7.1) in the form H S (s) = 1 2 E 0 [A(s)Z +B(s)Y ]; (7.2) where A(s) = 2a(t)=E 0 and B(s) = 2b(t)=E 0 are dimensionless schedules parametrized by the dimensionless times =t=t f , andE 0 > 0 is the energy scale of the Hamiltonian. We have cyclically permuted the Pauli matrices for later convenience. The ground states of H S (0) and H S (1) arej0i andjii, respectively. Second, we parametrize the annealing schedules in the angular form A(s) = (s) cos(s); B(s) = (s) sin(s) ; (7.3) where(0) = 0and(1) ==2. UnderthisparametrizationtheeigenvaluesofH S (s)areE 0 (s)=2, so the gap is (s) =E 0 (s). Thus, any non-trivial time-dependence of the gap is encoded in the time-dependence of (s), which we refer to as the dimensionless gap. The quantity (s) Z s 0 ds 0 (s 0 ) (7.4) is the cumulative dimensionless gap. Third, changing variables from s to to absorb (s), the system satisfies the Schrödinger equation i d d j i = 1 2 E 0 t f [cos()Z + sin()Y ]j i (7.5) (we work in ~ = 1 units throughout). The Hamiltonian is diagonalized at each instant by the rotationR X () =e iX=2 . Thus, fourth, we change into the adiabatic frame [217, 218] withj ad i = R X ()j i, yielding: i d d j ad i =H ad j ad i; H ad () 1 2 d d XE 0 t f Z : (7.6) We call d d the angular progression of the anneal. 94 Finally, we transform into the interaction picture with respect to the free Hamiltonian H 0 = E 0 t f Z=2 and its propagator U 0 () = e iH 0 . Letting S = (XiY )=2 denote the spin raising and lowering operators we have X I () =U y 0 ()XU 0 () =e iE 0 t f S + +h.c., and obtain i d d j I i =H I ()j I i; H I ()()X I () ; (7.7) wherej I i =U y 0 j ad i and () = 1 2 d d . Therefore, we see that in this adiabatic interaction picture the dynamics of the annealed TLS is a rotation about the time-dependentX I axis with a rate equal to the angular progression. 7.2.2 Magnus Expansion The corresponding time-ordered propagator U I () = T + e i R 0 d 0 H I ( 0 ) can be calculated in time- dependent perturbation theory using the Magnus expansion (reviewed in Sec. 7.5.1) for the Hermi- tian operatorK (N) () = P N n=1 K n (). The resultingU (N) I () = exp iK (N) () converges toU I () uniformly with growing N, and is unitary at all orders [219]. To first order: K 1 () = Z 0 d 1 H I ( 1 ) = (E 0 t f )S + +h.c. ; (7.8) where (!) 1 2 Z 0 d 1 d d 1 e i! 1 : (7.9) To systematically go beyond first order we note that the K n () arenth order nested commutators, and hence closure of the su(2) Lie algebra guarantees that at all ordersK (N) () = (N) ()^ n (N) ()~ , where (N) ()> 0, ^ n (N) () is a unit vector, and~ = (X;Y;Z). It thus follows that U (N) I () =I cos (N) ()i^ n (N) ()~ sin (N) () : (7.10) We will be concerned primarily with the probability of remaining in the ground state at the final time, denoted p 0 0 . Sincej I (s)i =U y 0 ((s))R X ((s))j (s)i, we havej I (0)i =j0i andj I (1)i = ij0i. Thus, to Nth order: p (N) 0 0 = 1p (N) 1 0 =jh0jU (N) ( f )j0ij 2 (7.11a) =j cos (N) ( f )in (n) Z ( f ) sin (N) ( f )j 2 ; (7.11b) where the statesj0i andj1i are the initial ground and excited states, and where f (1). To first order we find (see Sec. 7.5.1 for the explicit form of U (1) ): p (1) 0 0 =jh0je ijjX j0ij 2 = cos 2 (jj) ; (E 0 t f ): (7.12) 95 Figure 7.1: The numerically exact (dotted) and first order Magnus expansion (solid) ground state probabilities of the linear (orange) and two-step Gaussian progression (blue) atE 0 = 0:25 GHz. For the two-step Gaussian we set = 32 and = 101=800. Insert: zoomed-in view of the linear schedule results. Here and in other plots we use parameters compatible with quantum annealing using flux qubits [177–180]. Also shown is the prediction of a simplified double-slit type analysis (dashed, red). Both the latter and the first order Magnus expansion result are in excellent agreement with thenumericallyexactsolution. Theeffectofstrongdephasingintheinstantaneousenergyeigenbasis is shown as well (dashed, black), obtained using a phenomenological noise model with dephasing parameter described in Sec. 7.5.6. In this case the interference oscillations are strongly damped. This conceptually elegant result already indicates that quite generally one may expect the ground state probability to oscillate as a function of the anneal timet f , before the adiabatic limit sets in, a conclusion also reached in Ref. [196] on the basis of either a large-gap (near-adiabatic limit) or very small gap (stationary phase approximation) assumption. Our analysis applies for arbitrary gaps. 7.2.3 LZ Problem (Linear Schedule) Let us first consider the simplest annealing schedule, namely a linear interpolation of the type considered in the original LZ problem [199, 200]: A(s) = 1s andB(s) =s. To evaluate Eq. (7.9) we can change the integration variable to s and approximate (s) f s in the exponent, yielding f (!) = 1 2 R 1 0 ds 1 s 2 +(1s) 2 e i! f s for the first-order Magnus expansion. We compare this to the numerically exact solution in Fig. 7.1, which shows remarkably good agreement. The simplicity of our Magnus expansion approach should be contrasted with the analytical solution for linear schedules in terms of parabolic cylinder functions [207]. Also notable is that while a quantum interference pattern is visible, the oscillations are very weak and not controllable (see the insert of Fig. 7.1). This motivates us to introduce new schedules with strong and controllable quantum interference. 96 7.2.4 Strong Quantum Interference Pattern via Gaussian Angular Progression Our goal is to identify a family of annealing schedules that generate strong interference between the paths leading to the final ground state, such that “giant” oscillations of the ground state probability can be observed. Therefore we now introduce Gaussian angular progressions. Suppose that the angular progression is two-step Gaussian, namely, a sum of two Gaussians centered at f =2 (with < f =2): d d =c e [(( f =2+))] 2 +e [(( f =2))] 2 : (7.13) Note that R f 0 d d d = (1)(0) = 2 , which fixes c. If we assume that 1 then we may approximate R f 0 by R 1 1 (we bound the approximation error in Sec. 7.5.2). Thus c = p =4 and Eq. (7.9) yields f (!) = 4 e i! f =2 e [!=(2)] 2 cos(!). Using Eq. (7.12), to first order the ground state probability is then p (1) 0 0 = cos 2 h 4 e (t f =t ad ) 2 cos(t f =t coh ) i (7.14a) t ad 2=E 0 ; t coh =(E 0 ) : (7.14b) The ground state probability thus approaches its adiabatic limit of 1 on a timescale of t ad (set by the Gaussian width), while undergoing damped oscillations with a period of t coh . The oscillations are overdamped when t ad <t coh . In particular, a single Gaussian ( = 0) can thus not give rise to oscillations. We plot the ground state probabilityp G (t f )p 0 0 in Fig. 7.1, for a two-step Gaussian progres- sion with parameters chosen to represent the underdamped case; the associated annealing schedules are shown in Fig. 7.2(a). The amplitude of the resulting pre-adiabatic oscillations seen in Fig. 7.1 is, as desired, much larger than that associated with the linear schedule. The accuracy of the first- order Magnus expansion is again striking, especially given its simplicity compared to the analytical solution approaches [209–211]. We give a bound on the first-order Magnus expansion approximation error in Sec. 7.5.2. 7.2.5 Physical Origin of the Oscillations What is the origin of the oscillations? The answer is an interference effect between the two paths created by the two-step schedule, which enforces a double-slit or an unbalanced Mach-Zender inter- ferometer scenario, with =4 beam-splitters: see Fig. 7.2(b). The first step is a perturbation that generates amplitude in the excited state, while the second step allows for some of this amplitude to recombinewiththegroundstate. Therelativephasebetweenthetwopathsis =E 0 t f R s + s (s 0 ) ds 0 , which results in oscillations. In Sec. 7.5.5 we derive this result via a simple interferometer-type model that predicts the curve marked DS = 0 in Fig. 7.1, which is in excellent agreement with the numerically exact result. 97 ! " #|1⟩ |0⟩ " ( " ) *|0⟩ + ,-. #|1⟩ (* ) ++ ,-. # ) )|0⟩ + ,-. # ) * # * ) Figure7.2: Top: ExampleannealingschedulesA(s)(blue)andB(s)(orange)foratwo-stepGaussian progression with = 32 and = 101=800, subject to the dimensionless gap (s) = 0:99 cos 2 (2s)+ 0:01, which is shown as well (dashed, green). Bottom: Equivalent interferometer model in the adiabatic interaction picture. The system starts in the ground statej0i. At s 1 0:25 the first Gaussian splits the amplitude, some of which evolves in the excited statej1i, where it acquires a relative phase / t f . The second Gaussian at s 2 0:75 returns part of the excited state amplitude to the ground state, where it recombines. The total ground state amplitude isa 2 +e i b 2 . Each Gaussian acts as an unbalanced (a;b) beamsplitter (purple), where a = cos 8 e (t f =t ad ) 2 , b =i sin 8 e (t f =t ad ) 2 (see Sec. 7.5.5 for details). 98 A natural question is whether the observation of interference oscillations as a function of t f implies the existence of quantum coherence in the computational basis at t f . We give a formal proof that the answer is affirmative in Sec. 7.5.6. An illustration is given in Fig. 7.1, for the case of dephasing in the instantaneous energy eigenbasis, which is equivalent to performing a measurement in this basis between the two Gaussian steps. The final ground state probability is then the sum of classical conditional probabilities through each beam-splitter, and as expected, the oscillations disappear. 7.2.6 Role of the Angular Progression We emphasize that the angular progression d d(s) = B 0 (s)A(s)A 0 (s)B(s) (s) 3 ; (7.15) is the sole quantity needed to determine the ground state probability, per Eqs. (7.9) and (7.12). In particular, per Eq. (7.15), any transformation of A(s), B(s) and (s) that leaves d d invariant will not affect P G in the closed-system setting. Note, furthermore, that specifying the angular progression does not uniquely determine the annealingschedulesA(s)andB(s). Thisisadvantageousforpracticalpurposes, sincesuchschedules are typically implemented via arbitrary waveform generators (AWGs) with bandwidth constraints that can be incorporated into the schedule design process. To determine these schedules we need to specify the dimensionless gap (s) and the angular progression d d . We can determine (s) by solving the differential equation d ds = (s) subject to the boundary condition (0) = 0. Then (s) can be determined by solving the differential equation d ds = (s) d d =(s) ; (7.16) subjecttoappropriateboundaryconditions. Together, (s)and(s)determinetheannealingsched- ules A(s) and B(s) via Eq. (7.3). In the two-step Gaussian case this means integrating Eq. (7.13), which, for a constant gap, yields (s) as a sum of erf functions. A particularly interesting example of a dimensionless gap schedule is one that represents the presence of two avoided level crossings, a significant feature of the glued trees problem [185]. An example is shown in Fig. 7.2(a), representing an example of the procedure outlined above for nu- merical determination of the schedule. It is clear from Eq. (7.15) that the main contribution to the angular progression is the near-vanishing of the gap. In contrast, when (s) is constant, the main contribution to the angular progression is the suddenness of the schedule, i.e., a largeA 0 (s) orB 0 (s). 99 7.3 Open system analysis and results While a phenomenological model of dephasing in the instantaneous energy eigenbasis already shows clearly how the interference pattern disappears under decoherence (Fig. 7.1 and Sec. 7.5.6), this is not a realistic model of decoherence. We thus examine the effect of coupling the TLS to a thermal environment that corresponds more closely to experiments, e.g., with superconducting flux qubits. We consider a dephasing model wherein the total system-bath Hamiltonian isH =H S (t)+H B + gY B, whereB is the dimensionless bath operator in the system-bath interaction, H S (t) is given in Eq. (7.2), H B is the bath Hamiltonian, and g is the coupling strength with units of energy. We assume a separable initial state S (0) B , with B = exp(H B )=Z the Gibbs state of the bath at inverse temperature and partition function Z = Tr[exp(H B )]. We transform to the interaction picture with respect toH B , so thatH7! ~ H(t) =H S (t)+gY ~ B(t), with ~ B(t) =U y B (t)BU B (t), and U B (t) =e itH B . The same series of transformations as those leading to Eq. (7.6) can be summarized as: Y ~ B(t)7! t f Y ~ B(s)7! t f R X ()YR X () ~ B(s) = t f [cos()Y + sin()Z] ~ B(s). After the final transformation to the H 0 -interaction picture, the total Hamiltonian replacing H I () in Eq. (7.7) becomes H tot (s) = 1 2 _ (s)X I (s) +gt f ~ (s)~ ~ B(s) ; (7.17) where~ = (sin cos; cos cos; sin) is a unit vector in polar coordinates, with(s)E 0 t f (s), and henceforth the dot denotes d ds . 7.3.1 Redfield Master Equation in the Adiabatic Interaction Picture The time-convolutionless (TCL) expansion [220] provides a convenient and systematic way to derive master equations (MEs) without requiring an adiabatic or Markovian approximation. With the detailed derivation given in Sec. 7.5.7, the 2nd order TCL (TCL2) ME in the adiabatic-frame can be written as: _ S (s) =i[H I (s); S (s)] (gt f ) 2 [~ (s)~ ; (s) S (s)] +h.c. ; (7.18) where (s) = Z s 0 ds 0 C(s;s 0 )U I (s;s 0 )~ (s 0 )U y I (s;s 0 )~ ; (7.19) and C(s;s 0 ) = Tr[ ~ B(s) ~ B(s 0 ) B ] = C (s 0 ;s) is the bath correlation function. We assume that the bath is Ohmic with spectral density J(!) = !e !=!c . To ensure the validity of the TCL2 approximation—which is also known as the Redfield ME—we derive a general error bound in Sec. 7.5.8, and apply this bound to the Ohmic case. We find the condition t f g 2 , which is always satisfied in our simulations. 7.3.2 Rotating Wave Approximation (RWA) In general, the Redfield ME (7.18) does not generate a completely positive map, which can result in non-sensical results such as negative probabilities [221, 222]. Although this is not necessary for 100 complete positivity [223], a further rotating wave approximation (RWA) is usually performed. The resulting Lindblad-type ME also lends itself to a simpler physical interpretation. As detailed in Sec. 7.5.9, this leads to _ S =i 1 2 _ X I +H LS ; S g 2 t f d ba jbihaj + ab jaihbj (7.20) +g 2 t f t aa e bb jbihbjjaihaj ; where ab =haj S jbi, all quantities exceptg,t f and ares-dependent, and the effective dephasing and thermalization rates d and t , respectively, and the basisfjai;jbig, are given by ja(s)i =U I (s)j (s)i;jb(s)i =U I (s)j + (s)i (7.21a) d (s) = 1 2 t (s) 1 +e (s) ; t (s) = ((s)) : (7.21b) Herej (s)i =U y 0 (s)ji are the instantaneous eigenvectors of H I (s). The Lamb shift is: H LS (s) =g 2 t f (S((s))jbihbj +S((s))jaihaj) : (7.22) The functions (!)=2 andS(!) are the real and imaginary parts of the one-sided Fourier transform ofthebathcorrelationfunction, andareimplicitly-dependent(seeSec.7.5.9, wherewealsodiscuss the validity conditions for the RWA). 7.3.3 Numerical Results The numerical solutions of Eqs. (7.18) and (7.20) are shown in Fig. 7.3 for the two-step Gaussian schedule with parameters as in Fig. 7.1 and for the gap schedule plotted in Fig. 7.2(a). The main message conveyed by this figure is that oscillations are visible over a wide range (an order of magnitude) of temperatures and system-bath coupling strengths. We also note that for these parameter values the Redfield ME produces physically valid solutions, despite the concerns about complete positivity mentioned above. The Redfield ME results in consistently higher ground state probabilities than the RWA. These numerical results are accurately reproduced in terms of a simple semi-empirical formula, also shown in Fig. 7.3, and derived in Sec. 7.5.10: P 0 G (t f ) = P G (t f ) 1 2 e d t f +P E () (7.23) where P 0 G (t f ) and P G (t f ) denote the open and closed system success probabilities, respectively, where d =g 2 Z 1 0 ds 0 d (s 0 ) (7.24) 101 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 0.6 0.7 0.8 0.9 1 Figure 7.3: Ground state probability as a function of total annealing time in the open system setting. Shown are the numerical results of the TCL2 master equation without the RWA [Eq. (7.18), Redfield] and with the RWA [Eq. (7.20), Lindblad], and the semi-empirical Eq. (7.23). The bath is Ohmic with a cutoff frequency ! c = 4 GHZ. Top: g 2 = 2 10 4 for a range of temperatures. Bottom: T = 20 mK for a range of coupling values. TCL2 0 (0) is the caseP E (0), and is an excellent agreement with the RWA results. TCL2 0 () is the case P E (1=T ) with fitted T values. From top to bottom: (a) T =f13:68; 44:06; 104:50gmK and (b) T =f23:72; 24:22; 22:95gmK. Parameter values were chosen to be consistent with quantum annealing using flux qubits and the necessary condition t f g 2 . 102 is the average thermalization rate, and where P E () e E 0 =2 Z ; Z = 2 cosh(E 0 =2) (7.25) is the ground state probability in the adiabatic limit, given by the thermal equilibrium value asso- ciated with H S (1) [Eq. (7.2)]. As seen in Fig. 7.3, the agreement is excellent with both the RWA result when we use P E (0) = 1=2 (the infinite temperature limit), and with the TCL2 results when we use P E () and fit ; we find that the fitted is consistently slightly lower than the actual values used in our simulations. 7.4 Discussion and Conclusions We have proposed a double-slit approach to quantum annealing experiments, exhibiting “giant” interference patterns, motivated by the role of coherent diabatic evolution in enabling quantum speedups. Our analytical approach based on a simple time-dependent expansion in the adiabatic interaction picture accurately describes the associated dynamics. The experimental observation of such interference oscillations then becomes a clear and easily testable signature of coherence in the instantaneous energy eigenbasis. The test is simple in principle: it involves a quantum annealing protocol that employs the proposed schedules, with a measurement of only the ground state population as a function of the anneal time t f . When the relative phase between the upper and lower paths to the ground state is randomized, the interference effect is weakened. To explain these results we proposed an effective model that accurately explains the interference oscillations in terms of a few simple parameters. Namely, upon replacing P G (t f ) in Eq. (7.23) by p (1) 0 0 (t f ) as given in Eq. (7.14a), the three timescales t coh , t ad , and 1= d respectively characterize the oscillation period, Gaussian damping due to approach to the adiabatic limit, and exponential damping due to coupling to the thermal bath. We expressed all three timescales in terms of the input physical parameters of the problem [Eqs. (7.14b) and (7.24)], and together they completely characterize the oscillations and their damping. We expect that an experimental test of our “double-slit” proposal will reveal the predicted inter- ferenceoscillationsforqubitsthataresufficientlycoherent, suchasaluminum-basedfluxqubits[178– 180], Rydberg atoms [101, 224], or trapped ions [225, 226]. Such an experiment can be viewed as a necessary condition for quantum annealing implementations of algorithms exhibiting a quantum speedup, e.g., the glued trees problem [185], which rely on coherence between energy eigenstates. It appears relevant (if not essential) to use such coherence in order to bypass the common objec- tion that stoquastic quantum annealing or adiabatic quantum computing are subject to, which is that they can be efficiently simulated using the quantum Monte Carlo algorithm when restricted to ground-state evolution (with some known exceptions [227, 228]), due to the absence of a sign problem [229, 230]. Therefore an experimental observation of the quantum interference pattern predicted here will bolster our confidence in the abilities of coherent quantum annealers to one day deliver a quantum speedup. 103 7.5 Methods and Computational Detals 7.5.1 Dyson and Magnus Series We repeatedly use the following elementary identity for su(2) angular momentum operators: exp(i'J x )J z exp(i'J x ) =J z cos'J y sin' : (7.26) Note that the Pauli matrices are related via J i = i =2, i2fx;y;zg. Let us denote the solution of the adiabatic frame Hamiltonian given in Eq. (7.6) byU ad (). The adiabatic interaction picture propagator, U I () =U y 0 ()U ad () (7.27a) =T + e i R 0 d 0 ( 0 )X I ( 0 ) ; (7.27b) the solution of Eq. (7.7), can be computed using the Dyson series expansion: U I () =Ii Z 0 d 1 ( 1 )X I ( 1 ) + (i) 2 Z 0 d 1 Z 1 0 d 2 ( 1 )X I ( 1 )( 2 )X I ( 2 ) +::: (7.28) Note that each term in the Dyson series contributes to the ground state amplitude if and only if it is an even power, and likewise to the excitation amplitude if and only if it is an odd power. Consequently, the amplitudes calculated from the Dyson series may not be unitary to a desired precision until the terms are calculated to a high enough order. For this reason we prefer the Magnus expansion [219], for which U I () = lim N!1 exp h iK (N) () i ; K (N) () = N X n=1 K n () : (7.29) The first few terms are given by K 1 () = Z 0 dt 1 ( 1 )X I ( 1 ) (7.30a) K 2 () = i 2 Z 0 d 1 Z 1 0 d 2 ( 1 )( 2 )[X I ( 1 );X I ( 2 )] : (7.30b) 104 Using U (N) I () = exp iK (N) () and Eq. (7.8) we thus find U (1) I () = exp (i[S + +h.c.]) (7.31a) = cos(jj) i sin(jj)e i' i sin(jj)e i' cos(jj) ! (7.31b) =e i'Z=2 M jj e i'Z=2 (7.31c) M jj e ijjX = cos(jj)Ii sin(jj)X : (7.31d) where we wrote as a shorthand for (E 0 t f ), and where ' = arg(). This directly results in Eq. (7.12). TocomputethesecondorderMagnustermweuseX I () =e iE 0 t f S + +h.c.forthecommutation relation [X I (t 1 );X I (t 2 )] = 2i sin[E 0 t f ( 2 1 )]Z ; (7.32) so that K 2 () = Z 0 d 1 Z 1 0 d 2 ( 1 )( 2 ) sin[E 0 t f ( 1 2 )]Z : (7.33) 7.5.2 Error Analysis of Gaussian Angular Progression Schedules We discuss the general Gaussian angular progression d d = p e 2 () 2 (7.34) where is the Bloch sphere rotation angle. In the main text we assumed that we can perform a full Fourier transform (i.e., integration limits extended to1) to find = 2 e i! e (t f =t ad ) 2 (7.35) and thus arrive at the first order Magnus term K 1 = 2 e (t f =t ad ) 2 e i! S + +e i! S : (7.36) We now show that the assumption of a full Fourier transform results in an exponentially small error in where = minf; f g : (7.37) Let I be the finite time integral I = Z f 0 de i! r 2 e 2 () 2 (7.38) 105 with 0<< f , and let F be the full Fourier integral F = Z 1 1 de i! r 2 e 2 () 2 (7.39) and define the error =jFIj ; (7.40) where FI = r 2 ( Z 0 1 + Z 1 f ) de i! e 2 () 2 : (7.41) Thus, in terms of the standard normal cumulative density function G (x) = 1 2 1 + erf[x= p 2] , r 2 ( Z 0 1 + Z 1 f ) d e i! e 2 () 2 (7.42a) = r 2 ( Z 0 1 + Z 1 f ) de 2 () 2 (7.42b) = G p 2 + 1 G p 2( f ) (7.42c) = 1 2 [erfc(u 1 ) + erfc(u 2 )] ; (7.42d) where erfc(x) = 1 erf(x) is the complementary error function, and we have set u 1 , and u 2 ( f ). The complementary error function is known to have exponentially small bounds [231]. We can quickly derive an even tighter bound by writing the error in terms of the Faddeeva function w(z) 1 2 e u 2 1 w(iu 1 ) +e u 2 2 w(iu 2 ) (7.43) where w(z) =e z 2 erfc(iz) =e z 2 1 + 2i p Z z 0 e t 2 dt ; (7.44) which is real and positive for imaginary z. If Imz > 0, the Faddeeva function has the integral representation [232, Eq. 7.7.2] w(z) = i Z 1 1 e t 2 zt dt (7.45) from which we note jw(z)j 1 Z 1 1 e t 2 jztj dt; (7.46) and as 1=jztj 1= Imz, jw(z)j 1 p 1 Imz : (7.47) 106 With this bound on the Faddeeva function, it is straightforward to obtain the error bound 1 2 p e u 2 1 u 1 + e u 2 2 u 2 ! 1 p e ( ) 2 : (7.48) If > p 1:77, then 0:014. If > 2 p 3:5, then with e 4 =2, the percent error in assuming a full Fourier transform is under 0:6 parts per million. Extending the limits of integration will not result inan appreciable error if & 2. In other words, if 2 = 1=2 2 , where 2 is the variance of the normal distribution, the interval [0; f ] should contain the confidence interval of at least 2 p 2 2:8. 7.5.3 Second Order Term of the Magnus Expansion Evaluating the second order of the Magnus expansion [Eq. (7.33)] yields: K 2 = 2 4 2 Z f 0 d 2 Z 2 0 d 1 e 2 ( 1 ) 2 e 2 ( 2 ) 2 sin[!( 2 1 )]Z : (7.49) The integral is antisymmetric under the exchange of 1 and 2 due to the sine, so we can extend the time-ordered integrals into the whole square domain as K 2 = 2 8 2 Z f 0 d 2 Z f 0 d 1 e 2 ( 1 ) 2 e 2 ( 2 ) 2 sin(!j 2 1 j)Z : (7.50) For large we can let A extend over the entire plane. The error bound due to extending the integration limits can be found straightforwardly: the square [0; f ] [0; f ] contains the circle C centered at (;) with radius , so the region R 2 C contains a probability mass of e ( ) 2 (from a 2D Gaussian distribution). Sincejsin(!j 2 1 j)j< 1, the error in extending the region of integration is therefore bounded by 2 e ( ) 2 : (7.51) Thus, up to an error of 2 , we may write K 2 = 2 8 hsin(!jT 2 T 1 j)iZ; (7.52) whereT 2 andT 1 areindependentGaussianrandomvariableswithvariance 1=(2 2 ). Wecanperform a change of variables intoT + =T 2 +T 1 andT =T 2 T 1 , which are independent random variables with sum and difference means 2 and 0 respectively, and both with variance 1= 2 . Finally, the randomvariablejT jisknowntobedistributedaccordingtothefolded-normaldistributioncentered 107 at 0 (i.e. the half-normal distribution). Thus, the expectation value is precisely the imaginary part of the folded-normal characteristic function [233]: g t (!) =e ! 2 =2 2 (7.53) (1 G (i!= p 2)) + (1 G (i!= p 2)) ; where the parent normal distribution has a mean = 0. In this case, the characteristic function simplifies to the Faddeeva function g t =e 2r 2 erfc(i p 2r) =w p 2r ; (7.54) where again r =!=2. From Eq. (7.44), we see that Img t = 2 p D p 2r (7.55) where D(z) is the Dawson function D(z) =e z 2 Z z 0 e x 2 dx : (7.56) Thus, ashsin(!jT j)i = Im(g t (!)), the second order term in the Magnus propagator is K 2 = 2 4 p D p 2r Z : (7.57) 7.5.4 Magnus Expansion Convergence and Error Bounds Let SkK 1 k be a bound on the operator norm of the first order term in the Magnus expansion. A sufficient condition for the convergence of the Magnus expansion is that [234] S = 1:08686870 : (7.58) With a Gaussian angular progression as given in Eq. (7.36), and noting that e i! S + +e i! S 1, it is then sufficient that S = 2 e (t f =t ad ) 2 < (7.59) fortheMagnusexpansiontobeconvergent. Thismeansthat 2, andinparticularthephysically relevant range 2 [0;=2] (=2 represents a balanced beam-splitter, and >=2 is equivalent to ) is within the convergence radius. IfK (n) is thenth order truncation of the Magnus expansion, the error in the truncation is given by ME(n) KK (n) 1 X m=n+1 S m b m (7.60) 108 wherefb m g is a sequence defined in Ref. [234] via various recurrence relations. For ==2 and for !=(2) = 0; 0:5 and 1:0, the corresponding second order truncation errors are 0:25; 0:1, and 0:008. 7.5.5 Double-slit Interpretation Having derived the adiabatic frame Hamiltonian given in Eq. (7.6) H ad () = 1 2 d d XE 0 t f Z ; (7.61) we see that the angular progression d d of an annealing schedule is the perturbation that causes transitions between the two levels of the system. While this perturbation is steady and small in the case of a linear schedule, Gaussian schedules in which the perturbation is localized suggest an appealing physical picture similar to a double-slit or interferometer model. 7.5.5.1 Single Gaussian Step Letusfirstconsiderasingle Gaussianstep, whichEq.(7.13)reducestowhen = 0,c = p =2. Un- der the same assumptions as those leading to Eq. (7.14), we then find f (!) = 4 e i! f =2 e (t f =t ad ) 2 , with ! = E 0 t f . Thus, Eq. (7.31) gives us the first order Magnus expansion propagator in the interaction picture with jj = 4 e [E 0 t f =(2)] 2 = 4 e (t f =t ad ) 2 (7.62) and ' =E 0 t f f =2. The X-rotation matrix in Eq. (7.31c) thus becomes: M G = 0 @ cos 2 e (t f =t ad ) 2 i sin 2 e (t f =t ad ) 2 i sin 2 e (t f =t ad ) 2 cos 2 e (t f =t ad ) 2 1 A ; (7.63) with the superscript G serving as a reminder that this is the Gaussian step case. Now let us suppose that the Gaussian profile is narrow: E 0 t f , or equivalently t ad t f . The perturbation is then sudden relative to the adiabatic timescale, and acts like a beamsplitter in a Mach-Zehnder (MZ) interferometer [203]. In this limitjj=4 and Eq. (7.31c) gives U (1) I ( f ) =e i(E 0 t f f =2)Z M G =2 e i(E 0 t f f =2)Z (7.64) M G =2 = 1 p 2 1 i i 1 ! : Recall that in the adiabatic interaction picturej I (0)i =j0i. Thus, the first phase factor e i'Z has no effect, and we can picture a process by which the ground statej0i is instantly split into an equal superposition 1 p 2 (j0iij1i) by the “Mach-Zender” matrix M G =2 . These two states are then propagated freely by U y 0 ( f ) = e i(E 0 t f f =2)Z , so they accumulate a relative phase of ie iE 0 t f f . For 109 a single Gaussian, interference due to this phase difference is clearly not picked up via a Z basis measurement. 7.5.5.2 Two Gaussian Steps: Indirect Derivation of the Interferometer Model in the Narrow Gaussian Limit If instead we consider a two-step Gaussian schedule [Eq. (7.13)], then as we already found before Eq. (7.14), f (!) = 4 e i! f =2 e (t f =t ad ) 2 cos(!), with ! =E 0 t f . Eq. (7.31) now gives us the first order Magnus expansion propagator in the interaction picture withjj = 4 j cos(E 0 t f )je (t f =t ad ) 2 and again ' =E 0 t f f =2. ∗ Let us now derive an equivalent MZ interferometer model. On the one hand, we already know from Eq. (7.12) that p (1) 0 0 = cos 2 (jj), i.e. p (1) 0 0 = cos 2 ( 4 j cos(E 0 t f )je (t f =t ad ) 2 ) : (7.65) This function has a quasiperiod (the distance between consecutive maxima) of=(E 0 ), a minimum of cos 2 (=4) = 1=2 at t f = 0, and a maximum of 1. On the other hand, we may model the two-step narrow ( E 0 t f ) Gaussian schedule as two consecutive, localized (at f =2) and non-overlapping ( 1=) “beam-splitter” steps, separated by a dimensionless time interval of 2. Each beam-splitter is of the form given in Eq. (7.64), the only difference being that the first acts at f =2 (preceded by free evolution) and the second acts at f =2 + (followed by free evolution). In between the beam-splitter action there is free evolution of duration 2. Ignoring the initial and final free evolutions (since the initial and final state we are interested are bothj0i, which is invariant under U 0 ) we expect to be able to write the propagator as the following ansatz: ~ U (1) ( f ) =M G U 0 (2)M G (7.66) where we left the angle in the beam splitter matrix (7.63) unspecified in order to determine it by matching to the properties of p (1) 0 0 = cos 2 (jj). Carrying out the matrix multiplication and computing the expectation value, we find h0j ~ U (1) ( f )j0i 2 = cos 2 ( =2) sin 2 ( =2)e 2iE 0 t f 2 : (7.67) In order for this to match Eq. (7.65), we require a quasiperiod of =(E 0 ) (which is already the case), a minimum of 1=2 at t f = 0, and a maximum of 1. The latter two conditions force ==4. Therefore, considering Eq. (7.66), we have shown that the two-step Gaussian model is equivalent (in the large limit) to a MZ interferometer with two unbalanced beamsplitters, separated by free propagation of duration 2 (the separation between the two Gaussians). ∗ Notethatwithouttheexponentialdecayfactore (t f =t ad ) 2 =e (t f =t ad ) 2 theoscillationsarecompletelyundamped andtheadiabaticlimitisneverreached. Thusitisclearthatthe finite widthoftheGaussianstepsissolelyresponsible for the onset of adiabaticity. 110 The double-slit (or MZ interferometer model) is remarkably accurate in terms of predicting the ground state probability. This is shown in Fig. 7.1, where we compare the numerically exact result and the solution of the simple interferometer model given by Eq. (7.67). Namely, we use the interference model given in Eq. (7.67), with ==4. To calculate the interference fringe, the position of each of the two Gaussians is given bys = ( f =2)=. The phase factorE 0 t f , which only holds in the large limit, is replaced by E 0 t f [(s + )(s )] =E 0 t f R s + s ds 0 (s 0 ), where (s) is the cumulative dimensionless gap [Eq. (7.4)]. The reason for this replacement is given in the following, alternative and more direct derivation of the interferometer model. 7.5.5.3 Two Gaussian Steps: Direct Derivation of the Interferometer Model Given the two-step Gaussian schedule, Eq. (7.13), d d =c e [( + )] 2 +e [( )] 2 ; (7.68) where = f =2, we can split the unitary generated by the adiabatic frame Hamiltonian, Eq. (7.61), into two parts: U ad ( f ; 0) =U ad ( f ; f 2 )U ad ( f 2 ; 0) (7.69) We now wish to apply the Magnus expansion separately to each of the unitaries U ad ( f 2 ; 0) and U ad ( f ; f 2 ). ConsiderU ad ( f 2 ; 0). InvertingEq.(7.27a), thefirstorderMagnusexpansion[Eq.(7.31)] gives U ad ( f 2 ; 0) =U 0 ( f 2 ; 0)U (1) I ( f 2 ; 0) (7.70a) =U 0 ( f 2 ; 0)e i'Z=2 M jj e i'Z=2 ; (7.70b) where, using Eq. (7.9), now f =2;0 (E 0 t f ) = 1 2 Z f =2 0 d d 1 e iE 0 t f 1 d 1 : (7.71) For 1 we may extend the limits of integration over the interval [0; f =2] to1 without considering the second Gaussian step: c 2 Z 1 1 e [( 1 )] 2 e iE 0 t f 1 d 1 (7.72a) = 8 e iE 0 t f e (t f =t ad ) 2 ; (7.72b) 111 where we used c = p =4 as we found in the derivation of Eq. (7.14). We may thus write the explicit form of the interaction picture unitary as U (1) I ( f 2 ; 0) =e i(E 0 t f =2)Z M G =4 e i(E 0 t f =2)Z (7.73a) =U y 0 ( ; 0)M G =4 U 0 ( ; 0) ; (7.73b) and the adiabatic frame unitary becomes: U ad ( f 2 ; 0) =U 0 ( f 2 ; 0)U y 0 ( ; 0)M G =4 U 0 ( ; 0) (7.74a) =U 0 ( f 2 ; )M G =4 U 0 ( ; 0) : (7.74b) Repeating this calculation for the second adiabatic frame unitary U ad ( f ; f 2 ), we obtain U ad ( f ; f 2 ) =U 0 ( f ; + )M G =4 U 0 ( + ; f 2 ) : (7.75) Thus, Eq. (7.69) becomes U ad ( f ; 0) =U 0 ( f ; + )M G =4 U 0 ( + ; )M G =4 U 0 ( ; 0) ; (7.76) which describes an interferometer composed of two unbalanced (=4) double beam-splitters, inter- rupted by free propagation of duration + (ignoring the initial and final phases). The phase accumulated between j0i and j1i is solely determined by the free evolution in Eq. (7.76), U 0 ( + ; ) =e i[E 0 t f ( + )=2]Z (7.77) whose value is given by =E 0 t f ( + ) =E 0 t f Z s + s (s 0 ) ds 0 ; (7.78) where in the second equality we used Eq. (7.4). 7.5.6 Interference Oscillations in the Double-slit Experiment Imply Quantum Coherence in the Computational Basis Here we prove that coherence in the energy eigenbasis implies, in general, coherence in the compu- tational basis. Let H(t) denote an arbitrary, time-dependent TLS Hamiltonian, with instantaneous energy eigenbasisfj i (t)ig. The TLS density matrix can be written in this basis as (t) = X ij ~ ij (t)j i (t)ih j (t)j : (7.79) 112 Let us define “coherence” with respect to a given basis as the off-diagonal elements of the density matrix in the same basis. We can compute the coherence in the computational basisfj0i;j1ig via 01 =h0j(t)j1i = X ij h0j~ ij (t) ij (t)j1i; (7.80) where ij (t) =j i (t)ih j (t)j. The two bases are related via a unitary rotation: j 0 (t)i = cos(t)j0i +e i(t) sin(t)j1i (7.81a) j 1 (t)i = sin(t)j0ie i(t) cos(t)j1i ; (7.81b) so that Eq, (7.80) reduces to: h0j(t)j1i =e i n (~ 00 1 2 ) sin(2) Re(~ 10 ) cos(2) +i Im(~ 10 ) o : (7.82) where we used ~ 00 + ~ 11 = 1 and ~ 01 = ~ 10 . Equation (7.82) can be further simplified using (~ 00 1 2 ) sin(2) Re(~ 10 ) cos(2) =C(cos' sin(2) sin' cos(2)), where C = r (Re ~ 10 ) 2 + (~ 00 1 2 ) 2 (7.83a) tan' = Re(~ 10 ) ~ 00 1 2 : (7.83b) Additionally, by making use of the trigonometric identity sin(2') = sin 2 cos' sin' cos 2, Eq. (7.82) can be written as h0j(t)j1i =e i (C sin(2') +i Im ~ 10 ) : (7.84) Since C sin(2')2R, it follows that Im(~ 10 (t))6= 0 implies h0j(t)j1i6= 0. Therefore we next establish that indeed, Im(~ 10 (t))6= 0 in our double-slit proposal. Consider the the ground state just before the first beam-splitter, ( ") =j 0 ih 0 j (7.85) with"=( + ) 1. This state evolves through the double-beam-splitter region [recall Eq. (7.76)]: M jj U 0 ( + ; )M jj ; (7.86) where U 0 is given in Eq. (7.77) and M jj is given in Eq. (7.31d). 113 After passing through the first beam-splitter, the system density matrix in the energy eigenbasis becomes ( +") = cos 2 (jj) i sin(jj) cos(jj) i sin(jj) cos(jj) sin 2 (jj) ! : (7.87) It is useful to include a simple model of decoherence between energy eigenstates during the time interval [ ; + ], complementary to our master equation treatment. We can do so by introducing a continuous dephasing channel. This damps the phases by the factor e , where = + = 2, and > 0 is the dephasing rate. Right before the second beam-splitter, the system density matrix is then: ( + ") = cos 2 (jj) ie e it f E 0 sin(jj) cos(jj) ie e it f E 0 sin(jj) cos(jj) sin 2 (jj) ! : (7.88) After passing through the second beam-splitter, the state becomes ( + +") =M jj ( + ")M y jj . We find, after some algebra: P G = ~ 00 = sin 4 (jj) + cos 4 (jj) 2e sin 2 (jj) cos 2 (jj) cos(E 0 t f ) !1 ! 1 4 [cos(4jj) + 3] (7.89a) ~ 01 = 1 2 sin(2jj) e [ sin(E 0 t f ) +i cos(2jj) cos(E 0 t f )] +i cos(2jj) !1 ! i 1 4 sin(4jj): (7.89b) WenownotefromEq.(7.72b)thatjj = 8 e (t f =t ad ) 2 . Thereforewemayconcludethat Im(~ 10 (t f ))> 0, and Im(~ 10 )! 0 only in the adiabatic limit (t f t ad , which impliesjj! 0). Note that Eq. (7.89a) generalizes Eq. (7.67) by including the effect of dephasing in the energy eigenbasis. It is clear from Eq. (7.89) that oscillations in the ground state probability P G (t f ), which are presentforfinite , implyanon-vanishing Im(~ 10 (t f )). Thereforewemayconcludethattheobserva- tion of interference oscillations in our proposed double-slit experiment are also evidence of coherence in the computational basis at t f . For finite , such coherence vanishes only in the adiabatic limit. 7.5.7 Derivation of the Adiabatic-frame TCL2/Redfield Master Equation We start from the Hamiltonian given in Eq. (7.17), which we write as H tot (s) =H I (s) + ~ H SB (s) (7.90a) H I (s) = 1 2 _ (s)X I (s) (7.90b) ~ H SB (s) =~ (s)~ ~ B(s) ; (7.90c) where gt f . Our goal is to derive a master equation for the system evolution. It is con- venient to do so using the time-convolutionless (TCL) approach [220]. To do so we must first 114 perform yet another interaction picture transformation, defined by H I (s), with the associated uni- tary U I (s;s 0 ) = T + exp i R s s 0 H I (s 00 )ds 00 , where T + denotes forward time-ordering. In this frame the total Hamiltonian H tot (s) becomes ~ H tot (s) = ~ ~ (s)~ ~ B(s); ~ ~ (s) =U y I (s; 0)~ (s)U I (s; 0): (7.91) We can now calculate the TCL expansion generated by the superoperator L(s) =i h ~ H tot (s); i ; (7.92) whereupon _ ~ (s) = 1 X n=1 2n K 2n (s)~ (s): (7.93) The different orders are called TCL2, TCL4, etc. We give details on the convergence criteria of this expansion in Sec. 7.5.8. To second order the TCL generator is: K 2 (s)[~ S B ] (7.94) = Z s 0 ds 0 Tr B h ~ H tot (s); h ~ H tot s 0 ; ~ S (s) B ii ; where B is the initial state of the bath, and the joint initial state is assumed to be in the factorized form S B . Note that the TCL2 approximation coincides with the Redfield master equation [220]. Let C(s;s 0 ) = Tr[ ~ B(s) ~ B s 0 B ] =C (s 0 ;s) (7.95) denote the bath correlation function. By explicitly tracing out the bath,K 2 (s) can be written as K 2 (s)~ S = 2 h ~ ~ (s)~ ; ~ (s)~ S i +h.c. (7.96) where ~ (s) = Z s 0 ds 0 C(s;s 0 ) ~ ~ (s 0 )~ : (7.97) After transforming back to the Schrödinger frame with respect to H I (s) we obtain: _ S (s) =i[H I (s); S (s)] 2 [~ (s)~ ; (s) S (s)] +h.c.; (7.98) where (s) = Z s 0 ds 0 C(s;s 0 )U I (s;s 0 )~ (s 0 )U y I (s;s 0 )~ : (7.99) 115 7.5.8 Necessary Convergence Criterion 7.5.8.1 General Criterion Assuming [ B ;H B ] = 0 the correlation function becomes homogeneous in time, so we use the shorthand notation C(xy) =C(xy; 0) =C(x;y). We define the following quantities to bound the error of the expansion: (n) B =t n f Z 1 0 dss n1 jC(s)j; (7.100) and denote (1) B B , which has a natural interpretation as the bath correlation time [235]. Note that kK 2 (s)[~ S B ]kc 2 2 Z s 0 ds 0 C s 0 c 2 2 B =t f ; (7.101) where c 2 = O(1) is a constant arising from the number of terms in the TCL2 double commutator expression (7.94). We can similarly estimate the magnitude of the TCL4 terms: kK 4 (s)[~ S B ]k c 4 4 Z s 0 Z s 1 0 Z s 2 0 jC(ss 2 )jjC(s 1 s 3 )j ds 1 ds 2 ds 3 +c 0 4 4 Z s 0 Z s 1 0 Z s 2 0 jC(ss 3 )jjC(s 1 s 2 )j ds 1 ds 2 ds 3 (7.102) where c 4 ;c 0 4 = O(1) are constants arising from the number of terms in the TCL4 sum over mul- tiple commutators and triple integral. We can bound the two integrals in Eq. (7.102) as follows. Considering the first expression, we first make a change of variables as x 1 =ss 3 x 2 =s 1 s 3 x 3 =s 2 s 3 : (7.103) Because 1ss 1 s 2 s 3 , the new integration limits can be obtained as s 3 0 =) sx 1 ss 1 =) x 1 x 2 (7.104a) s 1 s 2 =) x 2 x 3 s 2 s 3 =) x 3 0 (7.104b) which issx 1 x 2 x 3 0. The JacobianjdetJj = 1 in this case and the new integral becomes Z s 0 dx 1 Z x 1 0 dx 2 Z x 2 0 dx 3 jC(x 1 x 3 )jjC(x 2 )j (7.105a) Z s 0 dx 1 Z x 1 0 dx 2 Z x 1 0 dx 3 jC(x 1 x 3 )jjC(x 2 )j (7.105b) Z s 0 dx 1 Z x 1 0 dx 3 jC(x 1 x 3 )j Z x 1 0 dx 2 jC(x 2 )j (7.105c) Z s 0 dx 1 Z x 1 0 dx 3 jC(x 1 x 3 )j B t f : (7.105d) 116 Now we make another change of variables, with v =x 1 x 3 u =x 1 +x 3 : (7.106) The new integration limits can be obtained by x 3 0 =) uv sx 1 =) 2svu (7.107a) x 1 x 3 0 =) 0vx 1 : (7.107b) The first line means that 2sv u v, while the second line gives 0 v s since 0 x 1 s. Thus: Z s 0 dx 1 Z x 1 0 dx 3 jC(x 1 x 3 )j (7.108a) = Z s 0 dvjC(v)j Z 2sv v dujdetJj (7.108b) = Z s 0 1 2 (2s 2v)jC(v)j dv (7.108c) Z 1 0 sjC(v)j dv B t f ; (7.108d) where in the last inequality we used s 1. The same can be done for the second integral in Eq. (7.102): Z s 0 ds 1 Z s 1 0 ds 2 Z s 2 0 ds 3 jC(ss 3 )jjC(s 1 s 2 )j (7.109a) = Z s 0 dx 1 Z x 1 0 dx 2 Z x 2 0 dx 3 jC(x 1 )jjC(x 2 x 3 )j (7.109b) Z s 0 dx 2 Z x 2 0 dx 3 jC(x 2 x 3 )j Z s 0 dx 1 jC(x 1 )j (7.109c) ( B =t f ) 2 : (7.109d) Combining these two bounds thus finally yields: kK 4 (s)[~ S B ]k c 4 +c 0 4 4 ( B =t f ) 2 (7.110) In particular, to ensure the validity of the TCL2 approximation it should be the case that the TCL4 term is much smaller than TCL2, i.e.: g 2 t f B < c 2 c 4 +c 0 4 or g 2 t f B 1: (7.111) 117 7.5.8.2 Ohmic Bath Case Let us assume a spin-boson noise model, for which H SB =gY X k k b y k + k b k (7.112a) H B = X k ! k b y k b k ; (7.112b) where b k is a bosonic annihilation operator for mode k with frequency ! k , and g k = g k is the associated system-bath coupling strength, where k is dimensionless and g has units of energy. A standard approach is to introduce a spectral density such thatjg k j 2 7!J(!)d!. For an Ohmic bath we have J(!) =!e !=!c ; (7.113) where is a parameter with dimensions of time squared. After transforming to the bath interaction picture and replacingt bys =t=t f to arrive at ~ H SB (s), the bath correlation function for the Ohmic spectral density is C(s) = Z 1 0 d!!e !=!c (7.114) coth ! 2 cos(!st f )i sin(!st f ) ; an integral which may be evaluated explicitly in terms of the Polygamma function [235]. In partic- ular, for large ! c and t f =, the correlation function can be expanded as C(s) = 2 4 2 e st f = B + 1 (st f = M ) +O e 2st f = B ; (st f ) 3 ! : (7.115) This form indicates a transition from a Markovian regime of purely exponential decay with a timescale of B !c!1 ! =(2), followed by a non-Markovian regime of power-law decay with a timescale of M = p 2=! c . The transition occurs at a time tr ln(! c ) [235]. In the Markovian limit ! c !1 we may thus replace Eq. (7.115) by jC(s)j = 2 2 e 2st f = ; (7.116) and hence the correlation function integral of Eq. (7.100) becomes Z 1 0 dsjC(s)j = 2t f ; (7.117) 118 whichreplaceseveryfactorof B =t f arisingfromthesameintegralintheboundsintheprevioussub- section. Inparticular,wenowhavethenecessaryconditionkK 4 (s)[~ S B ]k (c 4 +c 0 4 ) 4 (=(2t f )) 2 < kK 2 (s)[~ S B ]kc 2 2 =(2t f ). Eq. (7.111) can thus be rewritten in the Markovian Ohmic case as g 2 t f < 2c 2 (c 4 +c 0 4 ) or g 2 t f 1: (7.118) For finite ! c , one can refine this bound by replacing Eq. (7.100) with B =t f Z str 0 dsjC(s)j + Z 1 str dsjC(s)j ; (7.119) where s tr = tr =t f . For our purposes the bound (7.118) suffices and is satisfied in all the numerical results presented in the main text. Namely, we have g 2 t f 0:16. 7.5.9 Rotating Wave Approximation Let (!) = Z 1 0 dte i!t C(t) =t f s (!t f ) (7.120) be the one-sided Fourier transform of the bath correlation function, where s (!) Z 1 0 dse i!s C(s) = 1 2 s (!) +iS s (!); (7.121) and where s (!)=2 and S s (!) are the real and imaginary parts of s (!). Explicitly [220]: s (!) = Z 1 1 e i!s C(s)ds (7.122a) S s (!) = 1 2 Z 1 1 (! 0 )P 1 !! 0 d! 0 : (7.122b) HereP denotes the Cauchy principal value, and the s subscript is a reminder that t f has been factored out. Toperformtherotatingwaveapproximation,letusfirstdefinetheeigenspaceprojectionoperator of H I (s) as ((s)) =j(s)ih(s)j; (7.123) wherej(s)i is an eigenstate of H I (s) with instantaneous energy (s). We can then define the operator A(!(s)) X 0 (s)(s)=!(s) ((s)) ~ (s)~ 0 (s) ; (7.124) where !(s)2 n 0; _ (s) o (7.125) 119 is the dimensionless Bohr frequency, and the sum is over all pairs(s); 0 (s) subject to the constraint 0 (s)(s) = !(s). The interaction picture master equation (7.93) can then be written to second order, with the TCL2 generator (7.94) as _ ~ S = Z s 0 ds 0 Tr B h ~ H tot (s); h ~ H tot s 0 ; ~ S (s) B ii (7.126) = 2 X !;! 0 e i(! 0 !)s (!) A(!)~ S A y ! 0 (7.127) A ! 0 A(!)~ S +h.c. (7.128) To obtain this master equation, we apply the standard Markovian approximation: change the integration variables 0 7!ss 0 and replace the upper limit with1. The RWA consists of neglecting terms in Eq. (7.128) for which ! 0 6=!. A necessary condition for the validity of the RWA is [150]: 1= B < min !6=! 0 !! 0 ; (7.129) which, unfortunately, is not always satisfied for the two-step Gaussian schedule (7.13) because [recall Eq. (7.125)] min !6=! 0 !! 0 = _ (s) 0 (7.130) for s outside the Gaussian pulse region. Nevertheless, the RWA results in the interaction picture adiabatic Markovian master equation in Lindblad form [235]: _ ~ S =i[H LS ; ~ S ] +D(~ S ); (7.131) where H LS = 2 X ! S s (!)A y (!)A(!) (7.132) is the Lamb shift, and D(~ S ) = 2 X ! s (!) (A(!)~ S A y (!) 1 2 n A y (!)A(!); ~ S o (7.133) is the dissipator. We can explicitly calculate A(!(s)). First, recalling that H I () = 1 2 d d U y 0 ()XU 0 () [Eq. (7.7)], we realize that the eigenvalues and eigenvectors of H I (s) can be written as (s) = 1 2 _ (s) j (s)i =U y 0 (s)ji : (7.134) Also, from the sequence of transformations leading to Eq. (7.17), the interaction terms have the form ~ (s)~ =U y 0 (s) cos(s)Y + sin(s)Z U 0 (s) : (7.135) 120 Substituting these expressions back into Eq. (7.124), we obtain A(0) = 0 (7.136a) A _ (s) =ie i j (s)ih + (s)j (7.136b) A _ (s) =ie i j + (s)ih (s)j : (7.136c) After undoing the interaction picture transformation with respect to H I (s) and ignoring the phase factorsintheA(!)operators,weobtaintheSchrödingerpicturemasterequation,namelyEqs.(7.20)- (7.22) given in the main text. In deriving this result we made use of the Kubo-Martin-Schwinger (KMS) condition [220] () =e () ; (7.137) where is the dimensionless Bohr frequency in units of 1=t f : (s) =!(s)=t f : (7.138) 7.5.10 Derivation of the Semi-empirical Eq. (7.23) The semi-empirical formula (7.23) can be derived directly from Eq. (7.131). Let us first write Eq. (7.131) in terms of the quantities defined in Eq. (7.21b): _ ~ S =i[H LS ; ~ S ] t f d ~ + j + ih j + ~ + j ih + j (7.139) +t f t (~ ++ e ~ )(j ih jj + ih + j) : We now follow the steps in Ref. [236] to obtain the solution in this interaction picture. Eq. (7.139) can be split into two decoupled ordinary differential equations: d~ ds = d~ ++ ds = [F + (s)~ ++ F (s)~ ] (7.140a) d~ + ds = d~ + ds =[i (s) + (s)]~ + ; (7.140b) where F + (s) =g 2 t f t (s) (7.141a) F (s) =g 2 t f t (s)e (s) ; (7.141b) 121 and (s) =g 2 t f (S((s))S((s))) (7.142a) (s) =g 2 t f d (s) : (7.142b) Additionally, the KMS condition allows us to write d (s) in terms ofF + (s) F + (s)(1 +e (s) ) = 2g 2 t f d (s) : (7.143) The solution of Eqs. (7.140) is given by: (s) = exp 2t f g 2 Z s 0 ds 0 d (s 0 ) ( (0) (7.144a) + Z s 0 ds 0 F + (s 0 ) exp " 2t f g 2 Z s 0 0 ds 00 d (s 00 ) #) + (s) = exp Z s 0 ds 0 [i (s 0 ) +t f g 2 d (s 0 )] + (0) (7.144b) ++ (s) = 1 (s) (7.144c) + (s) = + (s) ; (7.144d) where the initial conditions are: ij (0) = 1 2 ; i;j2f+;g: (7.145) The next step is to move back to Schrödinger picture S (t) =U I (t)~ S (t)U y I (t) ; (7.146) and write the open system ground state probability in terms of ~ S : P 0 G (t f ) =h0j(t f )j0i (7.147a) =h0jU I (t f )~ (t f )U y I (t f )j0i (7.147b) = X i;j2f+;g ij h0j i ih j j 0i ; (7.147c) where j i (t f )i =U I (t f )j i (t f )i =U I (t f )U y 0 (t f )jii : (7.148) For simplicity, we further denote U a (t) = U I (t)U y 0 (t), whose elements can be related to those of U I (t) in thefj0i;j1ig basis: U a kl (t) =hkjU I (t)U y 0 (t)jli =e (1) l i(t) hkjU I (t)jli ; (7.149) 122 where k;l2f0; 1g and (t) =E 0 t=2. Then: h0j + ih + j 0i = 1 2 (jU a 00 j 2 +U a 00 U a 01 +U a 01 U a 00 +jU a 01 j 2 ) (7.150a) h0j ih j 0i = 1 2 (jU a 00 j 2 U a 00 U a 01 U a 01 U a 00 +jU a 01 j 2 ) (7.150b) h0j + ih j 0i = 1 2 (jU a 00 j 2 U a 00 U a 01 +U a 01 U a 00 jU a 01 j 2 ) (7.150c) h0j ih + j 0i = 1 2 (jU a 00 j 2 +U a 00 U a 01 U a 01 U a 00 jU a 01 j 2 ) : (7.150d) Because U I (t) is the closed system unitary, we have jU a 00 (t f )j 2 =jh0jU I (t f )j0ij 2 =P G (t f ) ; (7.151) and jU a 00 j 2 +jU a 01 j 2 = 1 : (7.152) Eq. (7.147) becomes: P 0 G (t f ) = 1 2 + ( + (t f ) + + (t f ))(P G (t f ) 1 2 ) (7.153a) + ( ++ (t f ) (t f )) Re(U a 00 U a 01 ) (7.153b) +i( + (t f ) + (t f )) Im(U a 00 U a 01 ) : (7.153c) This result is exact and corresponds to the numerical solution in the TCL2 case shown in Fig. 7.3. We now make two additional approximations in order to arrive at a simpler expression. First, we ignore the Lamb shift term (s) in Eqs. (7.144), which leads to: + (t f ) + + (t f ) exp g 2 t f Z 1 0 ds d (s) (7.154a) + (t f ) + (t f ) 0 : (7.154b) Second, we substitute the solution given in Eqs. (7.144) into line (7.153b): ( ++ (t f ) (t f )) Re(U a 00 U a 01 ) = Re(U a 00 U a 01 ) ( 1 2e 2t f g 2 R 1 0 ds 0 d (s 0 ) 1 2 +t f g 2 Z 1 0 ds 0 t (s 0 )e 2g 2 t f R s 0 0 ds 00 d (s 00 ) ) (7.155a) (1 2 1 2 ) Re(U a 00 U a 01 ) = 0 ; (7.155b) where in the last line we used the weak coupling assumption, g 2 t f 1. 123 With these two approximations, Eq. (7.153) becomes the semi-empirical formula (7.23) with P E (0) = 1=2. We note that it is well known that for time-independent Lindbladians the RWA master equation has the Gibbs state as its steady state [220]. We do not recover this result for the time-dependent case. Rather, we find that the time-dependent Redfield master equation (TCL2) converges to the Gibbs state P E () = e E 0 =2 Z , but with a temperature that differs from that of the bath state, as illustrated in Fig. 7.3. Acknowledgments The content in this chapter was originally published in Ref. [237]. The open system analysis and simulations in this publication were contributed by Huo Chen. We are grateful to L. Campos-Venuti, L. Fry-Bouriaux, M. Khezri, J. Mozgunov, and P. War- burton for insightful comments and discussions. We used the Julia programming language [238] and the DifferentialEquations.jl package [239] for some of the numerical calculations reported in this work. The research is based upon work (partially) supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office contract W911NF-17-C-0050. 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