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Tunneling, cascades, and semiclassical methods in analog quantum optimization
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Tunneling, cascades, and semiclassical methods in analog quantum optimization
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TUNNELING, CASCADES, AND SEMICLASSICAL METHODS IN ANALOG QUANTUM OPTIMIZATION by Siddharth Muthu Krishnan 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 AND ASTRONOMY) August 2018 Copyright 2018 Siddharth Muthu Krishnan To Amma and Appa, for life, love, knowledge, and wisdom. ii Acknowledgments Everyday, it astonishes me how much support, encouragement, help, well-wishes, and outright love I receive from all those around me. This dissertation wouldn’t exist without their generosity. I’m deeply grateful to my advisor Prof. Daniel Lidar for his mentorship during all these years. He taught me how to think, write, and talk about physics. He gave me freedom when I needed it and guidance when I needed it (and lots of free coffee as well!). I’m indebted to Dr. Tameem Albash, who was in many ways a second advisor. He taught me how to work through the messy details in physics, and the value of getting an intuitive argument in place before worrying about rigor. Collaborating with him on the research presented in Chap. 2 was one of the great pleasures of my PhD. I also thank Prof. Paolo Zanardi for teaching me some wonderful bits of physics and mathematics and for advice, guidance, and conversations. I thank Prof. Naresh Sharma, of the Tata Institute of Fundamental Research, who provided critical guidance at a crucial stage of my intellectual development. Thanks also to Prof. Ben Reichardt, Dr. William Kaminsky, and Prof. Hidetoshi Nishimori for pivotal insights at various stages of my PhD research. I also thank the many professors who taught me so many things over the years: especially, Profs. Itzhak Bars, Todd Brun, Clifford Johnson, Lorenzo Venuti, David Wallace, and Ben Williams. iii I thank Ms. Miriam Snyder and Ms. Jennifer Ramos. Their fantastic administrative support significantly improved my life and freed up my time and attention to focus on research. The wonderful folks I met at USC and outside are the ones who made this PhD special. EndlessconversationswithIgnacio, Josh, andMiladaboutphysics, mathematics, philosophy, andlife,were,inmanyways,amorevaluableexperiencetomethanmyresearch. Thanksalso to Ryan and Anurag for their friendship and for all the fun we had cranking out homework exercises in the beginning of the PhD. Thanks to Zhihui and Kristen for their friendship in the first few years of my PhD. Thanks to Richard for his companionship on campus and his strange sense of humor. Thanks to Scott for conversations and inviting me to fascinating events. Thanks to Christian for his infectious enthusiasm and energy and for discussing Lie groups with me on a rollercoaster. I thank Yukiko for her love, kindness, and support. Finally, Ithankmywonderfulparents, sister, brother-in-law, grandmother, andmyamaz- ing extended family in India and in the US. Their support over the years is what has kept me going. iv Contents Acknowledgments iii List of Figures viii Abstract x 1 Introduction 1 1.1 Quantum Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Quantum Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Tunneling and a very brief history of quantum annealing . . . . . . . . . . . 7 1.4 Oracle problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Semiclassical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Tunneling and Cascades in Perturbed Hamming Weight Problems 14 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Perturbed Hamming weight optimization problems and the examples studied 17 2.3 The semi-classical potential and tunneling . . . . . . . . . . . . . . . . . . . 21 2.4 Fixed Plateau: Performance of algorithms . . . . . . . . . . . . . . . . . . . 24 2.4.1 Adiabatic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.2 Simulated annealing using random spin selection . . . . . . . . . . . . 29 2.4.3 Optimal QA via Diabatic Transitions . . . . . . . . . . . . . . . . . . 35 2.4.4 Spin Vector Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.5 Simulated Quantum Annealing . . . . . . . . . . . . . . . . . . . . . 39 2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.6 Some open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3 Quasiadiabatic Grover Search via the Wentzel-Kramers-Brillouin Approx- imation 46 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Quasi-adiabatic WKB for interpolating Hamiltonians . . . . . . . . . . . . . 47 3.2.1 WKB as an asymptotic expansion . . . . . . . . . . . . . . . . . . . . 48 3.2.2 Interpolating Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . 49 v 3.3 The Grover problem via the quasi-adiabatic WKB approximation . . . . . . 50 3.4 Constructing the WKB solutions . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5.1 Single Qubit in a magnetic field . . . . . . . . . . . . . . . . . . . . . 58 3.5.2 The n-qubit Grover problem . . . . . . . . . . . . . . . . . . . . . . . 62 3.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4 Gaussian-noise-induced adiabaticity in quantum annealing for glued trees 70 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2 The Glued-Trees problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.1 The annealing algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3.1 Four noise models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.4 Noisy glued-trees: Results from numerical simulations . . . . . . . . . . . . . 83 4.5 Noise-induced adiabaticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5 Conclusion 94 5.1 Toy models, approximations, and explanations . . . . . . . . . . . . . . . . . 94 5.2 Open questions and directions for future work . . . . . . . . . . . . . . . . . 96 A Appendices to Chapter 2 100 A.1 Review of the Hamming weight problem and Reichardt’s bound for PHWO problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 A.1.1 The Hamming weight problem . . . . . . . . . . . . . . . . . . . . . . 100 A.1.2 Reichardt’s bound for PHWO problems . . . . . . . . . . . . . . . . . 105 A.2 (Non-)Locality of PHWO problems . . . . . . . . . . . . . . . . . . . . . . . 108 A.3 Derivation of Eq. (2.14) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 A.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 A.4.1 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 A.4.2 Quantum Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 A.4.3 Spin-Vector Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 114 A.4.4 Simulated Quantum Annealing . . . . . . . . . . . . . . . . . . . . . 119 A.5 Behavior of p GS vs. t f curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 B Appendices to Chapter 3 122 B.1 Comparison with the method of Hagedorn and Joye . . . . . . . . . . . . . . 122 C Appendices to Chapter 4 128 C.1 Writing in the column basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 C.2 Qubit versions of small glued trees instances . . . . . . . . . . . . . . . . . . 129 C.3 The Gaussian Orthogonal Ensemble . . . . . . . . . . . . . . . . . . . . . . . 132 vi C.4 Perturbative decay of overlap between the noisy and noiseless ground states . 133 Reference List 135 vii List of Figures 2.1 The plateau cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Quantum tunneling in the Fixed Plateau problem . . . . . . . . . . . . . . . 24 2.3 Difference between predicted and actual minimum gap locations and Ham- ming Weight drop locations . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Scaling of barrier heights and minimum gaps . . . . . . . . . . . . . . . . . . 26 2.5 Fixed plateau: Performance of different algorithms . . . . . . . . . . . . . . 29 2.6 Diabatic QA vs SA and SVD for the Fixed Plateau problem . . . . . . . . . 36 2.7 Spectrum implicated in the diabatic cascade . . . . . . . . . . . . . . . . . . 37 2.8 Optimal TTS and Population Dynamics for three PHWO problems . . . . . 38 2.9 Average Hamming weight dynamics for SQA, QA, and SA . . . . . . . . . . 41 2.10 Optimal TTS for a constant gap PHWO problem . . . . . . . . . . . . . . . 42 3.1 Single qubit population dynamics: WKB and Numerical . . . . . . . . . . . 57 3.2 Final ground state population of a single qubit (n = 1) in a magnetic field, under the g 0 (r) = 1 schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 Time-averaged WKB-Numerical trace-distance for a single qubit . . . . . . . 59 3.4 Numerical threshold-timescale scalings for four schedules . . . . . . . . . . . 61 3.5 Numerical threshold-timescale scaling for the g 3 schedule . . . . . . . . . . . 61 3.6 WKB threshold-timescale scalings for four schedules . . . . . . . . . . . . . . 62 3.7 Time-averaged WKB-Numerical trace-distance for four schedules . . . . . . . 63 viii 3.8 Final ground state probability as a function of anneal time for four schedules 63 3.9 Time-averaged trace-distance for the g 2 schedule for n = 1, 2, 3, 4, 5 . . . . . 64 3.10 Norm of the WKB approximation as a function of time . . . . . . . . . . . . 66 3.11 Difference between the time-averaged trace-norm distances for the renormal- ized and unnormalized WKB approximants for four schedules . . . . . . . . 67 3.12 Threshold time scalings for the renormalized WKB . . . . . . . . . . . . . . 68 4.1 Graph structure of the glued trees problem . . . . . . . . . . . . . . . . . . . 72 4.2 Low-energy spectrum of the glued-trees problem . . . . . . . . . . . . . . . . 76 4.3 Threshold time scaling for the noiseless glued-trees . . . . . . . . . . . . . . 77 4.4 p GS vs. t f for the noiseless glued-trees . . . . . . . . . . . . . . . . . . . . . . 78 4.5 Population dynamics of the noiseless glued-trees . . . . . . . . . . . . . . . . 79 4.6 Median p GS vs n for the four noise models . . . . . . . . . . . . . . . . . . . 86 4.7 Median p GS vs. log() for several n . . . . . . . . . . . . . . . . . . . . . . . 87 4.8 Exponential fits to median p GS vs. n for the four noise models . . . . . . . . 88 4.9 Scaling coefficients and speedup thresholds . . . . . . . . . . . . . . . . . . . 89 4.10 Optimal dynamics for the noiseless glued-trees at n = 20 . . . . . . . . . . . 91 4.11 Gap and population dynamics for = 10 −3 . . . . . . . . . . . . . . . . . . . 92 4.12 Gap and population dynamics for = 10 −2 . . . . . . . . . . . . . . . . . . . 93 A.1 1−p GS vs. t f for the plain Hamming weight problem . . . . . . . . . . . . . 103 A.2 Precision of t f needed to hit the diabatic cascade for the Fixed Plateau . . . 120 B.1 The performance of the Hagedorn-Joye asymptotic expansion . . . . . . . . . 126 C.1 Smallest instance of the glued-trees problem . . . . . . . . . . . . . . . . . . 130 ix Abstract This dissertation studies analog quantum optimization, in particular, quantum anneal- ing (QA). QA is a limited model of quantum computation that might offer speedups for optimization problems. We study several QA oracle algorithms using semiclassical methods. First, we evaluate the claim that QA derives its computational power from tunneling. We study permutation-symmetric problems and show that tunneling is neither necessary nor sufficient for a speedup. We also discover in many of these algorithms a novel mechanism which we call a diabatic cascade, arising due to a lining-up of avoided level-crossings in the spectrum of the Hamiltonian. Second, we analyze the adiabatic Grover search problem using a quasiadiabatic variation on the WKB method. We find that the WKB approximation is able to capture features of the dynamics that a naive adiabatic approximation is unable to capture. While the approximation is able to recover the Grover speedup in the case of the optimal schedule, the approximation breaks down for other schedules. Finally, we turn to the glued-trees problem: a search problem on two randomly glued trees for which QA is known to provide an exponential speedup via a diabatic pathway. We analyze additive random-matrix noise models for this problem. We find that models which induce interactions between distant nodes of the graph are able to retain a quantum speedup for a larger range of noise strengths when compared to models that only induce interactions between neighboring vertices of the graph. We also find that a certain symmetry essential to the working of the closed-sytem algorithm can be broken without significantly affecting performance. x Chapter 1 Introduction 1.1 Quantum Optimization Quantum optimization is a young field which hopes to capitalize on quantum effects to obtain speedups over classical optimization algorithms. Typically, these algorithms are heuristics, i.e., they’re not fine-tuned to the details of the problem at hand. This is distinct from “structured” quantum algorithms in which one identifies a relevant class of problems and comes up with an algorithm which solves the problem more efficiently than any known classical algorithm, or, more efficiently than a specific family of classical algorithms [1]. The most famous structured quantum algorithm is Shor’s algorithm to solve the problem of factoring integers, which it does in polynomial time on a quantum computer, exponentially faster than any known classical algorithm [2]. Optimization problems in general can be phrased as minimization or maximization prob- lems, in which we are given a cost function f(x), with the inputs to the function f being restricted to some domain: x∈D. The goal is to try and find x min = arg min x∈D f(x). (1.1) In this dissertation we mostly work with the domain being the set of length-n bit-strings, i.e.D ={0, 1} n . Optimization problems also include search problems. In a search problem over a domain D, the goal is to find a marked element x marked ∈D. Note that search problems can also be thought of as minimization and maximization problems and vice versa. Optimization can 1 be thought of search: simply set x marked =x min . Search can be thought of as optimization: for instance, define the function f to be f(x) = 0 x =x marked 1 x6=x marked , (1.2) which makes x marked =x min . The domainD might be structured or unstructured. The structure is usually isomorphic to a graphG = (V,E). Here, each element x∈D is identified with a vertex in the graph x∈ V and some elements are connected by an edge (x,x 0 )∈ E. The structure makes the problem more specific in that the marked element or the minimum might have some special properties, such as a specific degree in the graph, so that the class of algorithms to try and solve the problem can be chosen appropriately. Note that there is a distinction between structured/unstructured optimization domains (i.e. structured/unstructured problems) and structured/heuristic algorithms. Structured domains invite the use of structured algorithms that exploit regularities in the domain. But we could also use heuristic algorithms to solve structured problems. Unstructured problems on the other hand, are typically only amenable to heuristic algorithms, since there is no pattern or information for the algorithms to latch on to. In this dissertation, we consider three kinds of problems. The first, perturbed Hamming weight problems, is minimization of a certain class of functions on the hypercube of lengthn bit-strings[Chap.2]. Thesecond, Groversearch, issearchoveranexponentiallylargedomain with no graph structure [Chap. 3]. Third, and finally, the glued-trees problem is search over a large domain, in which a subset of the domains has a graph structure isomorphic to two binary trees randomly glued together [Chap. 4]. There are two main kinds of algorithms to solve optimization problems: discrete and ana- log. Discrete quantum optimization is based on the circuit model of quantum computation. 2 It uses a sequence of quantum gates, i.e. unitary operators, which prepares a state that, upon measurement, returns a solution to the problem we want to solve. Until very recently, there have not been unstructured discrete quantum optimization algorithms [3, 4, 5]. On the other hand, analog, or continuous, quantum optimization was invented to handle unstruc- tured problems. Analog quantum optimization involves the evolution of a quantum system under a continuously changing Hamiltonian. The focus of this dissertation is analog quantum optimization. We focus on a particular family of analog quantum optimization algorithms: quantum annealing. We turn to this next. 1.2 Quantum Annealing The term “quantum annealing (QA)” is often used in differing ways. Usually, it refers to an analog quantum optimization technique that involves interpolating from a Hamiltonian whose ground state is easy to prepare to a Hamiltonian whose ground state is the answer to the problem we want to solve. Generically it is operated in the adiabatic regime: that is, the interpolation timescale is long compared to the inverse of the energy gap between the lowest two energy levels of the system. This is because the adiabatic theorem provides us a guarantee that for a sufficiently long interpolation timescale, if the evolution started in the ground state of the initial Hamiltonian, it would end in the ground state of the final Hamiltonian. While operating QA adiabatically is common, it isn’t the only way to operate QA. “QA” can also refer to cases where the system undergoes diabatic transitions to excited states and returns to the ground state. This might also include cases where the system starts in an excited state [6]. “QA” is sometimes used to refer to both the closed and the open system settings, however, it is increasingly being reserved for the open-system setting. In this dissertation, we consider QA in all its guises. In Chap 2, we study closed-system QA in both the adiabatic and diabatic settings. In Chap 3, we study closed-system QA 3 in a quasiadiabatic setting, i.e., in the regime where QA is nearly adiabatic, though some non-adiabaticities occur. In Chap 4, we study some open system models for QA in both the adiabatic and diabatic settings. We now describe the adiabatic version of QA. One begins in an easy-to-prepare ground state (GS) of a Hamiltonian, and slowly changes the Hamiltonian until a final Hamilto- nian is reached whose ground state encodes the solution to the problem we want to solve. Canonically, this is written as H(s) =A(s)H init +B(s)H fin . (1.3) s∈ [0, 1] is the anneal parameter which changes with time ass =t/t f , wheret is the physical time andt f is the anneal timescale. The functionsA(s),B(s) are real-valued functions on the unit interval, i.e., A(s),B(s) : [0, 1]→R. They represent parameters or fields that we can experimentally control. Typically we choose these functions such that A(0) = 1,B(0) = 0 and A(1) = 0,B(1) = 1. This means that in the beginning, i.e. at s = 0, only the initial HamiltonianH init is on, while at the end, i.e. ats = 0, only the final HamiltonianH fin is on. Sometimes, an intermediate term is turned on and off during the evolution H(s) =A(s)H init +C(s)H middle +B(s)H fin , (1.4) with C(0) =C(1) = 0. The evolution, in the closed system case, is simply the Schrödinger evolution i d|ψ(t)i dt =H(t)|ψ(t)i, (1.5) which becomes, after using s =t/t f , i d|ψ(s)i ds =t f H(s)|ψ(s)i. (1.6) 4 A standard set of choices for H init ,H fin ,A(s), and B(s) is H init =− X i σ x i (1.7a) H fin = X i,j J ij σ z i σ z j (1.7b) A(s) = 1−s B(s) =s, (1.7c) where the last line is called the linear interpolation. Given the large number of possible choices of these various parameters, there is a signif- icant body of literature devoted to understanding the effects of different interpolations and theeffectsofdifferentinitial, final, andintermediateHamiltonians(see, e.g., Refs.[7,8,6,9]). In this dissertation we consider many variations on this theme. In Chap. 2 we study interpolations of the form H init = 1 2 X i (I i −σ x i ) (1.8a) H fin = X x∈{0,1} n f(x)|xihx| (1.8b) A(s) = 1−s B(s) =s C(s) = 0, (1.8c) where the function f(x) depends only on the Hamming weight of x. In Chap. 3 we study interpolations of the form H init =I−|uihu| (1.9a) H fin =I−|mihm| (1.9b) A(s) = 1−r(s) B(s) =r(s), (1.9c) 5 where|ui is the uniform superposition over the set of all length-n bit-strings and|mi is a bit-string we want to find and we consider many choices for the r(s) functions. In Chap. 4 we study interpolations of the form H init =−|ENTRANCEihENTRANCE| (1.10a) H fin =−|EXITihEXIT| (1.10b) H middle = ADJ (1.10c) A(s) = 1−s B(s) =s C(s) =s(1−s), (1.10d) where|ENTRANCEi and|EXITi are certain special vertices in a particular graph and ADJ represents the adjacency matrix of this graph. We now briefly describe the adiabatic theorem of quantum mechanics. Roughly speaking, the theorem states that if a quantum system starts in an eigenstate of a Hamiltonian, and the Hamiltonian is changed slowly, then system will continue to be in the same eigenstate. To state it a little more precisely, we need a few definitions. Let|φ 0 (s)i be the ground state of H(s). Let the system start in the ground state of H(0), i.e.,|ψ(0)i =|φ 0 (0)i. Let|ψ t f (s)i be the time-evolved quantum state, which depends on the anneal time t f . Let Δ(s) =E 1 (s)−E 0 (s) be the spectral gap between the lowest two energy levels ofH(s). The adiabatic theorem states that if t f max s∈[0,1] || ˙ H(s)|| Δ(s) 2 (1.11) then |ψ t f (1)i≈|φ 0 (1)i. (1.12) (More precise versions are available in Ref. [10]. See also Ref. [11] for an overview.) Note that the adiabatic theorem is a sufficient condition, not a necessary condition. While there are some necessary conditions for adiabaticity that have been proposed in the literature [12, 13], they are often difficult to compute. 6 1.3 Tunneling and a very brief history of quantum annealing In the previous section, we described quantum annealing. In this section we briefly describe the development of an idea in quantum annealing that is often considered to under- write its computational power: tunneling, which is the ability of quantum wavefunctions to render non-zero probability to classically forbidden regions. Tunneling is an important focus of this dissertation. We will also give a brief overview of the history of quantum annealing. A remark on the word “tunneling”: it is used ambiguously in the literature. It looks as if it is a verb, which seems to suggest a dynamical process. Usually though, it refers to a kinematic property, i.e. the property of a wavefunction at a particular instant of time, or the property of a certain eigenstate of the Hamiltonian: the property of having a non-zero amplitude in classically forbidden regions. Sometimes though it is also used to refer to a dynamical process, e.g., the dynamics of a wavepacket localized in one well of a double-well moving to the other well. Quantum annealing is directly inspired from simulated annealing (SA). In SA, first pro- posed in Ref. [14], we slowly decrease thermal fluctuations in a system so that at the end of the evolution the system is in a low temperature state. From this, we can read off the low-temperature physics of the system, and most importantly, the ground state. Thus, it can also serve as an optimization algorithm. Quantum annealing started with the idea that, just as in SA we slowly decrease ther- mal fluctuations, similarly, we can also slowly decrease quantum fluctuations in a quantum Hamiltonian. It was first proposed by Appolloni et. al. in Ref. [15] and then studied further inthechemicalphysicsliteratureasaclassical heuristic[16,17,18,19]. There, insimulations of the imaginary-time Schrödinger equation −~ ∂ψ(x,τ) ∂τ = " − ~ 2 2m ∂ 2 ∂x 2 +V (x) # ψ(x,τ), (1.13) 7 they slowly sent~→ 0, or, equivalently, the mass of the particle m→∞, while sending the imaginary timeτ→∞. At the end of the simulation, the system will end in the ground state of the potential energy functionV (x). In this setting, tunneling is essentially identified with the ability of our representation of the quantum wavefunction to live in classically forbidden regions. The hope was that this might increase the probability of exploring the landscape faster than SA. Similar ideas of introducing “quantum” fluctuations in classical simulations of spin- systems were considered in [20, 21]. Here too, claims of tunneling were made, but are difficult to justify, because it unclear what the potential energy function is which respect to which the tunneling occurs, since there are no clear kinetic-energy and potential-energy terms. These methods, which are really classical optimization algorithms, suffer from the prob- lem of exponential growth of dimension of the quantum Hilbert space with the number of particles under consideration. We may get around this problem by several approximations, as and when applicable. But the other approach to overcome this problem, is to follow Feynman [22], and have an actual physical quantum system performing these dynamics. For quantum annealing, this was done in Ref. [23], which can be considered the first truly quantum annealing proposal, and also its first implementation. There, a quantum anneal along the lines described in the previous section (1.2) was experimentally performed: The Ising spin-glass arising from the LiHo x Y 1−x F 4 crystal was kept at a low-temperature, while a transverse magnetic field was decreased. It was experimentally demonstrated that this method was superior to a classical thermal anneal in finding the ground-state of the spin- glass. (Here, too, claims of tunneling were made, though tunneling was left undefined.) Soon after that, in [24], a clean mathematical framework for these kinds of problems was provided in terms of the adiabatic theorem of quantum mechanics. As long as the quantum evolution was slow enough to keep the system in the instantaneous ground state, we should expect to see a significant probability of finding the ground state at the end of the 8 evolution. In [25], this framework was applied to random instances of an NP-hard problem, and for small problem sizes, a polynomial scaling in the complexity was observed. Later it was realized that this kind of polynomial scaling could not continue to larger problem sizes [26, 27, 28]. Since then there has been rapid experimental progress in the implementation of quantum annealers, most notably by the Canadian company D-Wave Systems. They have built large- scale annealers (≈ 10 3 qubits). Unfortunately, there has been, as yet, no clear sign of a quantum speedup in these machines. It is thought that this is because the annealers are relatively noisy. It could also be that there are theoretical reasons that preclude quantum annealers from offering speedups [29]. For an overview of the experimental testing and benchmarking of their annealers, see Ref. [30]. 1.4 Oracle problems In theoretical computer science, there is a rich tradition of studying oracle problems. An oracle is a mathematical construct. It is imagined as a black box that returns the answer to some question. To illustrate, in classical computation, a typical oracle takes as input a length n bit-string, x∈{0, 1} n , and returns as output a certain a certain real-valued function of that bit-string f(x)∈R. In the quantum case, the space of possible oracles is larger. The laws of quantum mechanics allow for us have states that are superpositions of bit-strings. Thus, we can consider oracles that take as input a certain quantum state|ψi return as output the quantum state obtained by applying some general linear operator on that quantum state|ψi7→O|ψi. Oracles can be considered in both the circuit model and the Hamiltonian model of quan- tum computing. In the circuit model, the oracle is to be thought of as simply a unitary operator that we get to apply at any stage in the quantum circuit. In the Hamiltonian model, they are Hamiltonians that are available to be turned on or off at various stages of 9 the evolution (see Ref. [31] for more on Hamiltonian oracles). In QA, the oracle is usually the final Hamiltonian, and it encodes the problem that we wish to solve. In the circuit model, since an oracle call is a discrete event—one use of the oracle unitary is an oracle call—there is a straigtforward way to develop the theory of query complexity, which asks what is the number of oracle calls needed in order to solve a given computational problem. In the Hamiltonian model, there is no easy way to separate query complexity and time complexity, since the oracle is a Hamiltonian, which may very well be always on. Nevertheless, this question is developed in an interesting way in Ref. [32]. Oracles are not meant to be physically realizable. Rather, they are to be thought of as models that are useful in illustrating the capabilities of various computational models. E.g., the fact that quantum oracles can take superposition states as inputs and give superposition states as outputs, allows them to perform computational tasks that a similar classical oracle cannot perform. Thus by comparing algorithms that use different kinds of oracles, we can identify the strengths and weaknesses of different computational models. Another reason to study oracle problems is that they are often stepping stones to non- oracle algorithms. E.g., Shor’s algorithm can be thought as obtained from derelativizing, i.e. making it oracle-free, the quantum algorithm for solving the Hidden Subgroup Problem corresponding to period-finding. In this dissertation, we consider different kinds of oracle problems. First, we study the so-called perturbed Hamming weight oracle (PHWO) problems. These were first proposed in [33]. Next we turn to the Grover problem. This is an oracle problem where the oracle tells us whether a particular element is marked or unmarked. Finally we study the glued-trees problem, where the oracle encodes the adjacency matrix of a graph. Given a particular vertex of the graph, the oracle tells you which vertices are adjacent to the given vertex. 10 1.5 Semiclassical methods A physical system is considered semiclassical if it is not amenable to a fully quantum mechanical description and it is not amenable to a fully classical description. Semiclassical methods have been front and center since the beginning of quantum mechanics because quantum systems are usually much too complicated to deal with rigorously. Semiclassical methodsinvolvemakingapproximationstothedynamicalequationsofthesystemthatallows one to describe the behavior of the quantum system as classical dynamics with quantum mechanical corrections added in. In this dissertation, we analyze the annealing algorithms for the three oracle problems mentioned above—perturbed Hamming weight, Grover search, and the glued trees—using threedifferentsemiclassicalmethods. TheperturbedHammingweightproblemsareanalyzed using the spin-coherent path-integral method. This involves writing the Schrödinger dynam- ics of the system as a path-integral in the spin-coherent basis, and taking the semiclassical limit of this system using the stationary phase approximation. The Grover search problem is analyzed using a novel application of the Wentzel-Kramers-Brillouin (WKB) method. The WKB method was originally developed in order to construct series expansions for wave- functions in powers of ~. In this dissertation, we use the WKB method to obtain series expansion in powers of the anneal timescale t f . Thus, in this case, the physics is not semi- classical, though the mathematical method is. Finally, the glued-trees problem is analyzed by using random matrices to approximate an open system evolution. This is also a semiclas- sical method because we are introducing classicality into the system by approximating the effects of a quantum bath by a phenomenological model. 11 1.6 Summary of results We now summarize our results. Tunneling is often claimed to be the key mechanism underlying possible speedups in quantum optimization via quantum annealing (QA), espe- cially for problems featuring a cost function with tall and thin barriers. In Chap. 2, we present and analyze several counterexamples from the class of perturbed Hamming-weight optimization problems with qubit permutation symmetry. We first show that, for these problems, the adiabatic dynamics that make tunneling possible should be understood not in terms of the cost function but rather the semi-classical potential arising from the spin- coherentpathintegralformalism. Wethenprovideanexamplewheretheshapeofthebarrier in the final cost function is short and wide, which might suggest no quantum advantage for QA, yet where tunneling renders QA superior to simulated annealing in the adiabatic regime. However, the adiabatic dynamics turn out not be optimal. Instead, an evolution involving a sequence of diabatic transitions through many avoided level-crossings, involving no tunnel- ing, is optimal and outperforms adiabatic QA. We show that this phenomenon of speedup by diabatic transitions is not unique to this example, and we provide an example where it provides an exponential speedup over adiabatic QA. In yet another twist, we show that a classical algorithm, spin vector dynamics, is at least as efficient as diabatic QA. Finally, in a different example with a convex cost function, the diabatic transitions result in a speedup rel- ative to both adiabatic QA with tunneling and classical spin vector dynamics. This chapter is based on Ref. [34]. In various applications one is interested in quantum dynamics at intermediate evolution times, for which the adiabatic approximation is inadequate. In Chap. 3 we develop a quasi- adiabatic approximation based on the WKB method, designed to work for such intermediate evolution times. We apply it to the problem of a single qubit in a time-varying magnetic field, and to the Hamiltonian Grover search problem, and show that already at first order, the quasi-adiabatic WKB captures subtle features of the dynamics that are missed by the 12 adiabatic approximation. However, we also find that the method is sensitive to the type of interpolation schedule used in the Grover problem, and can give rise to nonsensical results for the wrong schedule. Conversely, it reproduces the quadratic Grover speedup when the well-known optimal schedule is used. This chapter is based on Ref. [35]. Finally, in Chap. 4, we analyze the quantum annealing algorithm for the glued-trees problem under different noise models. We consider two dichotomies of noise models. One dichotomyisbetweennoisemodelswhichinducelong-rangeinteractionsamongdistantnodes in the graph and noise models which induce interactions between nearest neighbor nodes. The other dichotomy is between noise models which break the reflection symmetry of the spectrum and noise models which retain the reflection symmetry of the spectrum. We show that, for a certain range of values of the noise-strength, the long-range noise models, both symmetry-preserving and symmetry-breaking, are able to preserve the quantum speedup while the short-range models are not. We also see that for the long-range noise models, for a certain range of the noise-strength, there is an increase in success with increasing noise. We explain these phenomena by arguing that, in the relevant range of noise strengths, the long-range models are able to lift the spectral gap of the problem so that the evolution changes from being diabatic to being adiabatic. We also find, somewhat surprisingly, that the symmetry of the spectrum does not make a significant difference to the performance of the noisy algorithms. The work in this chapter has not yet been published. We conclude in Chap. 5. The Appendix 5.2 contains technical details and other relevant information. 13 Chapter 2 Tunneling and Cascades in Perturbed Hamming Weight Problems 2.1 Introduction The possibility of a quantum speedup for finding the solution of classical optimization problems is tantalizing, as a quantum advantage for this class of problems would provide a wealth of new applications for quantum computing. The goal of many optimization problems can be formulated as finding an n-bit string x opt that minimizes a given cost function f(x), which can be interpreted as the energy of a classical Ising spin system whose ground state isx opt . Finding the ground state of such systems can be hard if, e.g., the system is strongly frustrated, resulting in a complex energy landscape that cannot be efficiently explored with any known algorithm due to the presence of many local minima [36]. This can occur, e.g., in classical simulated annealing (SA) [37], when the system’s state is trapped in a local minimum. Thermal hopping and quantum tunneling provide two starkly different mechanisms for solving optimization problems, and finding optimization problems that favor the latter con- tinues to be an open theoretical question [38, 39]. It is often stated that quantum annealing (QA) [40, 41, 42, 43, 44] uses tunneling instead of thermal excitations to escape from local minima, which can be advantageous in systems with tall but thin barriers that are easier to tunnel through than to thermally climb over [39, 44, 45]. It is with this potential tunneling- induced advantage over classical annealing that QA and the quantum adiabatic algorithm [46] were proposed. Our goal in this work is to address the question of the role played by 14 tunneling in providing a quantum speedup, and to elucidate it by studying a number of illustrative examples. We shall demonstrate that the role of tunneling is significantly more subtle than what might be expected on the basis of the “tall and thin barrier” picture. In order to make progress on this question, the potential with respect to which tunnel- ing occurs must be clearly specified. Tunneling is defined with respect to a semi-classical potential which delineates classically allowed and forbidden regions. In QA, one typically initializes the system in the known ground state of a simple Hamiltonian and evolves the system towards a Hamiltonian representing the final cost function. We shall argue that when one takes a natural semi-classical limit, the semi-classical potential does not become the final cost-function. Instead one obtains a potential appearing in the action of the spin-coherent path-integral representation of the quantum dynamics. This potential, which here we call the spin-coherent potential, has been used profitably before [47, 48, 49, 50]. We provide comprehensive evidence that multi-spin tunneling can be understood with respect to this spin-coherent potential. We analyze the spin-coherent potential for several examples from a well-known class of problems known as perturbed Hamming weight oracle (PHWO) problems. These are prob- lems for which instances can be generated where QA either has an advantage over classical random search algorithms with local updates, such as SA [47, 51], or has no advantage [33, 51]. Moreover, because PHWO problems exhibit qubit permutation symmetry, their quantum evolutions are easily classically simulatable, and furthermore, their spin-coherent potential is one-dimensional. Tunneling becomes clear and explicit for these problems when using the spin-coherent potential. We focus on a particular PHWO problem that has a plateau in the final cost function (henceforth,“the Fixed Plateau”). This problem offers a counter-example to two commonly held views: (1) QA has an advantage, due to tunneling, over SA only on problems where the barrier in the final cost function is tall and thin; (2) tunneling is necessary for a quantum speedup in QA. We refute the first statement by showing that for the Fixed Plateau, which is 15 a short and wide cost function, QA significantly outperforms SA by using tunneling. Indeed, we find numerically that adiabatic QA (AQA) needs a time ofO(n 0.5 ) to find the ground state, wheren is the number of spins or qubits. Moreover, using the spin-coherent potential, we observe the presence of tunneling during the quantum anneal. On the other hand, we prove that single-spin update SA takes a time ofO(n plateau width ). Thus, we have essentially an arbitrary polynomial tunneling speedup of QA over SA on a cost-function that is not tall and thin. We remark that the result about SA’s performance is also a rigorous proof of a result due to Reichardt [51] that classical local search algorithms will fail on a certain class of PHWO problems and is of independent interest. We refute the second statement by showing that, for the Fixed Plateau, it is actually optimal to run QA diabatically (henceforth, DQA for diabatic quantum annealing). The system leaves the ground state, only to return through a sequence of diabatic transitions associated with avoided-level crossings. In this regime, the runtime for QA isO(1). More- over, in this regime, we do not observe any of the standard signatures of tunneling. We show that this feature — that the optimal evolution time t f for QA is far from being adiabatic — is present in a few other PHWO problems and that this optimal evolution involves no multi-qubit tunneling. Given that the optimal evolution involves no tunneling, we are inspired to investigate a classical algorithm, spin vector dynamics (SVD), which can be interpreted as a semi-classical limit of the quantum evolution with a product-state approximation. We observe that SVD evolves in an almost identical manner to DQA, and is able to recover the speedup seen by DQA. Thus, in these problems, we show that what may be suspected to be a highly quantum-coherent process—diabatic transitions—can be mimicked by a quantum-inspired classical algorithm. The structure of this chapter is as follows. In Sec. 2.2, we list the PHWO problems we study. In Sec. 2.3, we use these problems to present evidence that tunneling can be understood with respect to the spin-coherent potential. In Sec. 2.4, we focus on the Fixed 16 Plateau PHWO problem, and exhaustively analyze the performance of various algorithms for this problem. In particular we numerically characterize AQA (Sec. 2.4.1), provide a rigorous proof of SA’s performance (Sec. 2.4.2), and numerically analyze DQA (Sec. 2.4.3), SVD (Sec. 2.4.4), and a quantum Monte Carlo algorithm (Sec. 2.4.5). We conclude in Sec. 2.5 by discussing the implications of our work and possible directions for future work. Additional background information and technical details can be found in the Appendix. 2.2 Perturbed Hamming weight optimization prob- lems and the examples studied The cost function of a PHWO problem is defined as, f(x) = |x| +p(|x|) l<|x|<u, |x| elsewhere , (2.1) where|x| denotes the Hamming weight of the bit string x∈{0, 1} n . For SA, this is the cost-function. For QA, this will be the final Hamiltonian. More precisely, we define QA as the closed-system quantum evolution governed by the time-dependent Hamiltonian, H(s) = 1 2 (1−s) X i (I−σ x i ) +s X x f(x)|xihx| , (2.2) where we have chosen the standard transverse field “driver” HamiltonianH(0) that assumes no prior knowledge of the form of f(x), and a linear interpolating schedule, with s≡ t/t f being the dimensionless time parameter. The initial state is the ground state of H(0). Below, we list several of PHWO examples that we study in greater detail. We refer to the case with p = 0 as the Plain Hamming Weight problem. 17 |x| 0 2 4 6 8 10 f(x) 0 2 4 6 8 10 Figure 2.1: l = 3,u = 8 1. Fixed Plateau: f(x) = u− 1, l<|x|<u, |x|, otherwise . (2.3) Clearly, this forms a plateau in Hamming weight space. We takeu,l =O(1). Since the location of the plateau does not change with n, we refer to it as “fixed.” An instance of this cost function with l = 3 and u = 8 is illustrated in Fig. 2.1. By numerical diagonalization we find that QA has a constant gap for this cost-function. 2. Reichardt: f(x) = |x| +h(n), l(n)<|x|<u(n) |x| otherwise , (2.4) with h u−l √ l =o(1). For this case, Reichardt [51] proved a constant lower bound on the minimum spectral gap during the quantum anneal. In Appendix A.1 we provide a 18 pedagogical review of this proof and fill in some details not explicitly provided in the original proof. 3. Moving Plateau: f(x) = u− 1, l(n)<|x|<u(n) |x| otherwise , (2.5) with l(n) =n/4, and u(n) =O(1). This is termed “moving” since the location of the plateau changes with n. Note that this is a special case from the Reichardt class. 4. Grover: f(x) = n, |x|≥ 1, 0, |x| = 0. (2.6) This is a minor modification of the standard Grover problem: the marked state is the all-zeros string with energy 0, and the energy of all the other states is n. Scaling the energy by n keeps the maximum energy of all the PHWO problems we consider comparable. 5. Spike: f(x) = n, |x| =n/4, |x|, otherwise . (2.7) ThiswasstudiedbyFarhiet al. in[47], whereitwasarguedthatthequantumminimum gap scales asO(n −1/2 ) and that SA will take exponential time to find the ground state. However, weshowbelow(Fig.2.8)thatSVDismoreefficientthanQAforthisproblem. 6. Precipice: f(x) = −1, |x| =n, |x|, otherwise . (2.8) 19 This was studied by van Dam et al. in [33], where it was proved that the quantum minimum gap for this problem scales asO(2 −n/2 ). 7. α-Rectangle: f(x) = |x| +n α , n 4 − 1 2 cn α <|x|< n 4 + 1 2 cn α , |x|, otherwise . (2.9) We call this anα-Rectangle because the width of the perturbation (cn α ) isc times the height. This was studied in [52], where evidence for the following conjecture for the scaling of the quantum minimum gap g min was presented, g min = constant, α< 1 4 , 1/poly(n), 1 4 <α< 1 3 , 1/ exp(n), α> 1 3 . (2.10) Note that α < 1/4 is a member of the Reichardt class, and thus the constant lower- bound on the minimum gap is a theorem, and not a conjecture. We restrict ourselves to the case of c = 1. Weremarkthatalltheproblemslistedabovearerepresentativemembersofalargefamily of problems: if the input bit-string to any of the above problems is transformed by an XOR mask, then all of our analysis below will hold. For QA, the XOR mask can be represented as a unitary transformation: N n i=1 (σ x i ) a i , with a∈{0, 1} n being the mask string. As this unitary commutes with the QA Hamiltonian at all times, none of our subsequent analysis is affected. Similar arguments go through for SA and all the other algorithms that we consider. WenotethatPHWOproblemsarestrictlytoyproblemssincetheseproblemsaretypically represented by highly non-local Hamiltonians (see Appendix A.2) and thus are not physically implementable, in the same sense that the adiabatic Grover search problem is unphysical [53, 20 54]. Nevertheless, these problems provide us with important insights into the mechanisms behind a quantum speed-up, or lack thereof. 2.3 The semi-classical potential and tunneling In order to study tunneling, we need a potential arising from a semi-classical limit, which defines classically allowed and forbidden regions. One approach to writing a semi-classical potential for quantum Hamiltonians is to use the spin-coherent path-integral formalism [55]. This semi-classical potential has been used profitably in various QA studies, e.g., Refs. [47, 49, 48, 50], and we extend its applications here. For the quantum evolution, since the initial state [the ground state of H(0)] is symmetric under permutations of qubits and the unitary dynamics preserves this symmetry (it is a symmetry of H(s) for all s), we can consistently restrict ourselves to spin-1/2 symmetric coherent states|θ,φi: |θ,φi = n O i=1 " cos θ 2 ! |0i i + sin θ 2 ! e iϕ |1i i # . (2.11) The spin-coherent potential is then given by: V SC (θ,φ,s) =hθ,φ|H(s)|θ,φi . (2.12) We show that for all the examples defined above except the Reichardt class (we address this below), this potential captures important features of the quantum Hamiltonian [Eq. (2.2)] and reveals the presence of tunneling. Specifically: 1. The spin-coherent potential displays a degenerate double-well almost exactly at the point of the minimum gap. In Fig. 2.2(a) we plot, for the Fixed Plateau the potential near the minimum gap. The potential transitions from having a single minimum on the right to a single minimum on the left. In between, it becomes degenerate and displays 21 a degenerate double well. Since the instantaneous ground state corresponds to the position of the global minimum, which exhibits a discontinuity, the degeneracy point is where tunneling should be most helpful. In Fig. 2.3(a), we show that the location of the minimum gap of the quantum evolution is very close to the location of the degenerate double-well in the spin-coherent potential. 2. The ground state predicted by the spin-coherent potential is a good approximation to the quantum ground state except near the degeneracy point. As expected from a potential that arises in a semi-classical limit, the ground state predicted by the spin-coherent potential (i.e., the spin-coherent state corresponding to the instantaneous global min- imum in V SC ) agrees well with the quantum ground state, except where tunneling is important. In particular, delocalization when the spin-coherent potential is a degener- ate double-well (or is close to being one) should imply that approximating the ground state with a wavefunction localized in one of the wells fails. Indeed, we find this to be the case. We illustrate this for the Fixed Plateau in Fig. 2.2(b); similar results hold for the other examples we have considered. 3. There is a sharp change in the ground state of the adiabatic quantum evolution at the degeneracy point. Tunnelingshouldbeaccompaniedbyasharpchangeintheproperties of the ground state at the degeneracy point as the state state shifts from being localized in one well to the other. We quantify this change by calculating the expectation value of the Hamming weight operator, defined as HW = 1 2 P n i=1 (I−σ z i ). We expect a discontinuity in the spin-coherent ground state expectation valuehHWi, because the spin-coherent ground state changes discontinuously at the degeneracy point. We find that there is a nearly identical change in the quantum ground state expectation value hHWi, for all of the examples listed above. This is illustrated explicitly for the Fixed Plateau in Fig. 2.2(c). In Fig. 2.3(b), we show that there is close and increasing 22 agreement (as a function of n) between the position of the sudden drop inhHWi and the position of the degeneracy point, for all of the problems considered. 4. The scaling of the barrier height in the spin-coherent potential is positively correlated with the scaling of the minimum gap of the quantum Hamiltonian. In Fig. 2.4, we see that as the barrier height increases, the inverse of the quantum minimum gap also increases. Note that the Reichardt class is absent from the discussion above. The reason is that for these problems, the barrier in the spin-coherent potential is very small, which makes its numerical detection difficult. Fortunately, we can make some analytical claims about this class of problems. By adapting Reichardt’s proof (reviewed in Appendix A.1) that these problems have a constant minimum gap, we are able to prove that the barrier height in the spin-coherent potential for this class vanishes as n→∞. Therefore, for these easy-for-AQA problems, there is a vanishing barrier in the spin-coherent potential. More precisely, we can show, for any perturbed Hamming weight problem, V pert SC −V unpert SC =s X l<k<u f(k) n k ! p(θ) k (1−p(θ)) n−k =O h u−l √ l ! (2.13) where the unperturbed case refers to h(n) = 0 in Eq. (2.4). Recall that h u−l √ l = o(1) for the Reichardt class. Thus asymptotically, the spin-coherent potential for this class approaches the spin-coherent potential of the unperturbed Hamming weight problem. It is easy to check that the latter has a single minimum throughout the evolution, and hence no barriers. Taken together, these observations indicate that the spin-coherent potential (not the cost function alone) is the appropriate potential with respect to which tunneling is to be understood for these problems. 23 0 0.05 0.1 0.15 0.2 0.25 θ 0 0.5 1 1.5 V SC (θ,s)− V SC (θ min ,s) s=0.88 s≈0.89 s=0.90 (a) 0.6 0.7 0.8 0.9 1 t/t f 0 0.2 0.4 0.6 0.8 1 Distance u = 6 u = 16 u = 26 (b) 0.6 0.7 0.8 0.9 1 t/t f 0 10 20 30 40 50 Average Hamming weight GS, u=6 GS, u=16 GS, u=26 SC GS, u=6 SC GS, u=16 SC GS, u=26 (c) Figure 2.2: Results for the Fixed Plateau problem with l = 0 and n = 512. (a) The semi- classical potential with u = 6 exhibits a double-well degeneracy at the position s≈ 0.89 (solid), but is non-degenerate before and after this point (dotted and dashed), thus leading to a discontinuity in the position of its global minimum. The same is observed for other the PHWO problems we studied (not shown). (b) The trace-norm distance between the quantum ground state (obtained by numerical diagonalization) and the spin-coherent state corresponding to the instantaneous global minimum in V SC , as a function of t/t f . The peak corresponds to the location of the tunneling event, at which point the semi-classical approximationbreaksdown. (c)hHWiintheinstantaneousquantumgroundstatestate(GS) and the instantaneous quantum ground state as predicted by the semi-classical potential (SC GS), as a function oft/t f . The sharp drop in the GS and SC GS curves is due to a tunneling event wherein∼u qubits are flipped which occurs at the degeneracy point observed in the spin-coherent potential. 2.4 Fixed Plateau: Performance of algorithms Having motivated the spin-coherent potential for understanding tunneling, we now exhaustively analyze the Fixed Plateau. We choose this problem because it forces us to confront some intuitions about the performance of certain algorithms. Considering the final cost function, the Fixed Plateau has neither local minima nor a barrier going from large to small|x|: it just has a long, flat section before the ground state at|x| = 0. This might suggest that it is easy for an algorithm such as SA, and is not a candidate for a quantum speedup. Moreover, given the absence of a barrier, one might suspect that the quantum evolution would not even involve multi-qubit tunneling. 24 log 10 n 0 1 2 3 4 log 10 (Δs drop ) -7 -6 -5 -4 -3 -2 -1 Fixed Plateau Spike 0.5-Rectangle 0.33-Rectangle Precipice Grover (a) log 10 n 0 1 2 3 4 log 10 (Δs gap ) -7 -6 -5 -4 -3 -2 -1 Fixed Plateau Spike 0.5-Rectangle 0.33-Rectangle Precipice Grover (b) Figure 2.3: (a) The difference in the position of the minimum gap from exact diagonalization and the position of the double-well degeneracy (as seen in Fig. 2.2(a) for the Fixed Plateau) from the semi-classical potential, as a function ofn for the Fixed Plateau, the Spike, the 0.5- Rectangle, the 0.33-Rectangle, the Precipice, and Grover (log-log scale). (b) The difference in the position of the sudden drop in Hamming weight [as seen in Fig. 2.2(c)] and the position of the double-well degeneracy from the semi-classical potential [as seen in Fig. 2.2(a)], as a function of the number of qubits n for the Fixed Plateau, the Spike, the 0.5-Rectangle, the 0.33-Rectangle, the Precipice, and Grover (log-log scale). We dispel both of these intuitions and summarize our findings first. In the previous section, we already provided evidence that tunneling is unambiguously present for this prob- lem. The spin-coherent potential involves energy barriers, despite their absence in the final cost function, and the adiabatic quantum evolution is forced to tunnel in order to follow the ground state. By a simulation of the Schrödinger equation, we find that AQA needs a time ofO(n 0.5 ) in order to reach a given success probability (see Sec. 2.4.1). Therefore, the adiabatic algorithm, via tunneling, is able to solve this problem efficiently. Turning to SA, an algorithm which performs a local stochastic search on the final cost function, we prove that simulated annealing with single spin-updates will take time O(n u−l−1 ) =O(n plateau width ) to find the ground state (see Sec. 2.4.2). This result is due to the fact that a random walker on the plateau has no preferred direction and becomes 25 log 10 n 0 1 2 3 4 log 10 (Barrier Height) -2 -1 0 1 2 3 4 Fixed Plateau Spike 0.5-Rectangle 0.33-Rectangle Precipice Grover (a) log 10 n 0 1 2 3 4 log 10 (1/Min Gap) 0 2 4 6 8 10 Fixed Plateau Spike 0.5-Rectangle 0.33-Rectangle Precipice Grover (b) Figure 2.4: (a) The height of the barrier between the two wells at the degeneracy point of the spin-coherent potential [as seen in Fig. 2.2(a)], as a function of n for the Fixed Plateau, the Spike, the 0.5-Rectangle, the 0.33-Rectangle, the Precipice, and Grover (log-log scale). (b) The inverse of the minimum gap as a function ofn for the Fixed Plateau, the Spike, the 0.5-Rectangle, the 0.33-Rectangle, the Precipice, and Grover (log-log scale). The important thing to note is that the order of scaling is preserved in both plots. That is, steeper the scaling of the barrier height, steeper the scaling of the inverse minimum gap. trapped there. More precisely, the probability of a leftward transition while on the plateau is proportional to the probability of flipping one of a constant number of bits (given by the Hamming weight) out ofn, which scales as∼ 1/n ifl,u =O(1). And since the walker needs to make as many consecutive leftward transitions as the width of the plateau in order to fall off the plateau, the time taken for this to happen scales asO(n plateau width ). Consequently, we obtain a polynomial speedup of AQA over SA that can be made as large as desired. Therefore, using the Fixed Plateau, we are able to demonstrate that a quantum speedup over SA is possible via tunneling in the adiabatic regime. However, is the adiabatic evolution optimal? In order to find the optimal evolution time, we employ the optimal time to solution (TTS opt ), a metric that is commonly used in 26 benchmarking studies [56] (also see Appendix A.3). It is defined as the minimum total time such that the ground state is observed at least once with desired probability p d : TTS opt = min t f >0 ( t f ln(1−p d ) ln (1−p GS (t f )) ) , (2.14) wheret f is the duration (in QA) or the number of single spin updates (in SA) of a single run of the algorithm, and p GS (t f ) is the probability of finding the ground state in a single such run. The use of TTS opt allows for the possibility that multiple short runs of the evolution, each lasting an optimal annealing time (t f ) opt , result in a better scaling than a single long (adiabatic) run with an unoptimized t f . The quantum evolution that gives the optimal annealing time relative to this cost function is actually DQA, with an asymptotic scaling of O(1). Importantly, thisdiabaticevolutiondoesnotcontainanyofthesignaturesoftunneling discussed in the previous section. Therefore, for the Fixed Plateau, tunneling does not give rise to the optimal quantum performance. Motivated by the fact that the optimal quantum evolution involves no multi-qubit tun- neling, we consider spin-vector dynamics [57] (see, also Refs. [58, 59]), a model that evolves according to the spin-coherent potential in Eq. (2.12). SVD can be derived as the saddle- point approximation to the path integral formulation of QA in the spin-coherent basis [59]. The SVD equations are equivalent to the Ehrenfest equations for the magnetization under the assumption that the density matrix is a product state, i.e., ρ =⊗ n i=1 ρ i , whereρ i denotes the state of the ith qubit. This algorithm is useful since it is derived under the assumption of continuity of the angles (θ,φ), so tunneling, which here would amount to a discrete jump in the angles, is absent. We also consider a quantum Monte Carlo based algorithm, often called simulated quan- tum annealing (SQA) [60, 61]. We show that SQA has a scaling that is better than SA’s. Indeed, this is consistent with the fact that SQA thermalizes not just relative to the final cost function, but also during the evolution. 27 WeprovidefurtherdetailsofourimplementationsofeachofthesealgorithmsinAppendix A.4. We now turn to each of the algorithms individually and detail their performance for the Fixed Plateau problem. 2.4.1 Adiabatic dynamics In order to study the scaling of adiabatic dynamics, we consider the minimum time τ 0 required to reach the ground state with some probability p 0 , where we choose p 0 to ensure that we are exploring a regime close to adiabaticity for QA. We call this benchmark metric the “threshold criterion,” and setp 0 = 0.9. As seen in Fig. 2.5, we observe a scaling for AQA that is approximately∼ n 0.5 . As is to be expected given that the tunneling for the Fixed Plateau problem is controlled by the width of the plateau, which is constant (does not scale withn), we find thatτ 0 scales in the same way for the Fixed Plateau and the Plain Hamming Weight problems (see Appendix A.1). This suggests that the dominant contribution to the scaling at large n is not the time associated with tunneling but rather the time associated with the Plain Hamming Weight problem. As also seen in Fig. 2.5, we find that the textbook adiabatic criterion [62] given by t f & max s∈[0,1] |hε 0 (s)|∂ s H(s)|ε 1 (s)i| Gap(s) 2 , (2.15) serves as an excellent proxy for the scaling of AQA 1 . The scaling of AQA is matched by the scaling of the numerator of the adiabatic condition, which is explained by the fact that we find a constant minimum gap for the case l,u =O(1). This numerator turns out to be well approximated in our case by the matrix element of H(s) between the ground and first excited states, leading to t f ∼n 0.5 in the adiabatic limit. Note that calculating this matrix 1 We note that while this adiabatic criterion matches the numerical scaling we observe for the quantum evolution, it is well known to be neither exact nor general; see, e.g., Refs. [10, 63, 64]. 28 10 2 10 3 n 10 2 10 4 10 6 10 8 10 10 τ 0 QA SQA Adiabatic Cond. SA Figure 2.5: Performance of different algorithms for the Fixed Plateau problem with l = 0 and u = 6. Shown is a log-log plot of the scaling of the time to reach a threshold success probability of 0.9, as a function of system size n for AQA, SQA (β = 30, N τ = 64) and SA (β f = 20). The time for SQA and SA is measured in single-spin updates (for SQA this isN τ times the number of sweeps times the number of spins, whereas for SA this is the number of sweeps times the number of spins), where both are operated in ‘solver’ mode as described in Appendix A.4. Also shown is the scaling of the numerator of the adiabatic condition as defined in Eq. (2.15). The scaling for AQA and the adiabatic condition extracted by a fit using n& 10 2 is approximately n 0.44 . However, the true asymptotic scaling is likely to be ∼n 0.5 since the scaling for the Fixed Plateau problem is clearly lower-bounded by the Plain Hamming Weight problem, for which we have verified τ 0 ∼ n 0.5 (see Appendix A.1), and we expect the effect of the plateau to become negligible in the large n limit. SQA scales more favorably (∼ n 1.5 ) than SA (∼ n 5 ). We have checked that the scaling of SQA does not change even if we double the number of Trotter slicesN τ and keep the temperature 1/β fixed. element can easily be done for arbitrarily large systems, and is hence much easier to check directly than the scaling of AQA. 2.4.2 Simulated annealing using random spin selection We consider a version of SA with random spin-selection as the rule that generates can- didates for Metropolis updates. Our main motivation is to understand the behavior of a 29 local, stochastic search algorithm which has access only to the final cost function. We note that our analysis below is general for any Plateau problem, and is not limited to the Fixed Plateau or the Moving Plateau. If we pick a bit-string at random, then for largen we will start with very high probability at a bit-string with Hamming weight close to n/2. The plateau may be to the left or to the right of n/2; if the plateau is to the right, then the random walker is unlikely to encounter it and can quickly descend to the ground state. Thus, the more interesting case is when the random walker arrives at the plateau from the right. We proceed to analyze these two cases separately. Walker starts to the right of the plateau In this case, how much time would it take, typically, for the walker to fall off the left edge? It is intuitively clear that traversing the plateau will be the dominant contribution to the time taken to reach the ground state, as after that the random walker can easily walk down the potential. We show below (for the walker that starts to the left of the plateau) that this time can be at mostO(n 2 ) if β = Ω(logn). To evaluate the time to fall off the plateau, note that the perturbation is applied on strings of Hamming weightl + 1,l + 2,...,u− 1, so the width of the plateau isw≡u−l− 1. Consider a random walk on a line ofw+1 nodes labelled 0, 1,...w. Nodei represents the set of bit strings with Hamming weightl +i, with 0≤i≤w. We may assume that the random walker starts at node w. Only nearest-neighbor moves are allowed and the walk terminates if the walker reaches node 0. Ouranalysiswillprovidealowerboundontheactualtimetofallofftheleftedge, because in the actual PHWO problem one can also go back up the slope on the right, and in addition we disallow transitions from strings of Hamming weight l to l + 1. This is justified because the Metropolis rule exponentially (in β) suppresses these transitions. 30 The transition probabilities p i→j for this problem can be written as a (w + 1)× (w + 1) row-stochastic matrixp ij =p i→j . Herep is a tridiagonal matrix with zeroes on the diagonal, except at p 00 and p ww . First consider 1≤ i≤ w− 1. If the walker is at node i, then the transition to nodei + 1 (which has Hamming weightl +i + 1) occurs with probability n−(l+i) n (the chance that the bit picked had the value 0). Similarly, for 1≤ i≤ w, the Hamming weight will decrease tol +i− 1 with probability l+i n (the chance that the bit picked had the value 1). Combining this with the fact that a walker at node 0 stays put, we can write: b i ≡p i→i = 1 if i = 0 0 if 1≤i≤ (w− 1) 1− l+w n if i =w , (2.16a) c i ≡p i−1→i = 0 if i = 1 1− l+i−1 n if i = 2,...,w , (2.16b) a i ≡p i→i−1 = l +i n if i = 1, 2,...,w. (2.16c) LetX(t)bethepositionoftherandomwalkerattime-stept. Therandomvariablemeasuring the number of steps the random walker starting from node r would need to take to reach node s for the first time is τ r,s ≡ min{t> 0 :X(t) =s,X(t− 1)6=s|X(0) =r} . (2.17) The quantity we are after isEτ w,0 , the expectation value of the random variableτ w,0 , i.e., the mean time taken by the random walker to fall off the plateau. Since only nearest neighbor moves are allowed we have Eτ w,0 = w X r=1 Eτ r,r−1 . (2.18) 31 Stefanov [65] (see also Ref. [66]) has shown that Eτ r,r−1 = 1 a r 1 + w X s=r+1 s Y t=r+1 c t a t ! , (2.19) where c w+1 ≡ 0. Evaluating the sum term by term, we obtain: Eτ w,w−1 = n l +w , (2.20a) . . . Eτ w−k,w−k−1 = n l +w−k " 1 + n− (l +w−k) l +w− (k− 1) +... + n− (l +w−k) l +w− (k− 1) ×··· × n− (l +w− 2) l +w− 1 × n− (l +w− 1) l +w # . (2.20b) Now consider the following cases: 1. Fixed Plateau, l,u =O(1): Here, using the fact that k =O(w) =O(1), we conclude thatEτ w−k,w−k−1 =O(n k+1 ). Since the leading order term isEτ w−(w−1),w−w =Eτ 1,0 , the time to fall off the plateau isO(n w ) =O(n u−l−1 ). This result about SA’s perfor- mance is confirmed numerically in Fig. 2.5. 2. In order for Reichardt’s bound (see Appendix A.1) to give a constant lower-bound to the quantum problem, we need u = l +o(l 1/4 ). Since at most we can have l =O(n), we can conclude Eτ w−k,w−k−1 =O n l k+1 . Therefore, the time to fall-off becomes Eτ w,0 =O w( n l ) w . • Moving Plateau: Ifl = Θ(n) andw =O(1), we can see thatEτ w,0 =O(1), which is a constant time scaling. • Moving Plateau with changing width: If l = Θ(n) and w =O(n a ), where 0<a< 1/4, thenEτ w,0 =O(n a O(1) n a ), which is super-polynomial. 32 • Most general plateau in the Reichardt class: More generally, if l =O(n b ), with b≤ 1 and w =O(n a ), where 0≤ a < b/4, then we get the scaling Eτ w,0 = O(n a O(n 1−b ) n a ) Walker starts to the left of the plateau Note that this case is equivalent to the unperturbed Hamming weight problem, which is a straightforward gradient descent problem. We may therefore consider a simple fixed temperature version of SA (i.e., the standard Metropolis algorithm). We will show that the performance of SA on this problem provides an upper bound ofO(n 2 ) on the time for a random walker to arrive at the plateau, and on the time for a random-walker to reach the ground state after descending from the plateau. Moreover, our analysis provides a lower bound ofO(n logn) on the efficiency of such algorithms. For this problem, the transition probabilities are: c i ≡p i−1→i = n−i + 1 n e −β , (2.21a) a i ≡p i→i−1 = i n , (2.21b) withi = 1, 2,...,n denoting strings of Hamming weighti, andβ is the inverse temperature. Using the Stefanov formula (2.19), we can write (after much simplification): Eτ n−k,n−k−1 = n n−k n k ! −1 k X l=0 e −lβ n k−l ! . (2.22) We will bound Eτ n,0 = n−1 X k=0 n n−k n k ! −1 k X l=0 e −lβ n k−l ! , (2.23) the expected time to reach the all-zeros string starting from the all-ones string. This is the worst-case scenario as we are assuming that we are starting from the string farthest from the all-zeros string. Note again that if we start from a random spin configuration, then with 33 overwhelming probability we will pick a string with Hamming weight close to n/2. Thus, most probably,Eτ n/2,0 will be the time to hit the ground state. We first show that β =O(1) will lead to an exponential time to hit the ground state, irrespective of the walker’s starting string. Toward that end, Eτ 1,0 =Eτ n−(n−1),n−n (2.24a) = n−1 X l=0 e −lβ n n− 1−l ! (2.24b) =e β h (e −β + 1) n − 1 i , (2.24c) which is clearly exponential in n if β =O(1). Next, let β(n) = logn, i.e., we decrease the temperature logarithmically in system size. In this case, Eτ 1,0 =n 1 + 1 n n − 1 ≤n(e− 1) =O(n) . (2.25) NowitisintuitivelyclearthatEτ 1,0 >Eτ r,r−1 forallr> 1, whichimpliesthatnEτ 1,0 ≥Eτ n,0 . Thus, if β = logn, thenEτ n,0 =O(n 2 ) at worst. To obtain a lower-bound on the performance of the algorithm, we take β→∞. Thus, for each k in Eq. (2.23), only the l = 0 term will survive. Hence, lim β→∞ Eτ n,0 = n−1 X k=0 n n−k =n n X i=1 1 i ≈n(logn +γ) , (2.26) for largen, withγ being the Euler-Mascheroni constant. The scaling here isO(n logn). This is the best possible performance for single-spin update SA with random spin-selection on the plain Hamming weight problem. Therefore, if β = Ω(logn), the scaling will be between O(n logn) andO(n 2 ). Of course, this cost needs to be added to the time taken for the walker starting to the right of the plateau. 34 Two clarifications are in order regarding the comparison between our theoretical bound on SA’s performance and the associated numerical simulations we have presented. First, while Fig. 2.5 displays the time to cross a threshold probability, our theoretical bound of O(n u−l−1 ) is on the expected time for the random walker to hit the ground state [Eq. (2.18)]. However, we found that both metrics show identical scaling. Second, while the SA data in Fig. 2.5 was generated using sequential spin updates, the theoretical bound assumes random spin updates (see Appendix A.4.1 for more details on the update schemes). However, we found that the asymptotic scaling for both cases is nearly identical in the long-time regime, and thus have plotted only the former. 2.4.3 Optimal QA via Diabatic Transitions Having established that for the Fixed Plateau AQA enjoys a quantum speedup over local search algorithms such as SA via tunneling, we are motivated to ask: Is tunneling necessary to achieve a quantum speedup on these problems? In order to answer this question, we demonstrateusingtheoptimalTTScriteriondefinedinEq.(2.14)thattheoptimalannealing time for QA is far from adiabatic. Instead, as shown in Fig. 2.6(a), the optimal TTS for QA is such that the system leaves the instantaneous ground state for most of the evolution and only returns to the ground state towards the end. The cascade down to the ground state is mediated by a sequence of avoided energy level-crossings as seen in Fig. 2.7. We consider this a diabatic form of QA (DQA) and call this mechanism through which DQA achieves a speedup a diabatic cascade. As n increases for fixed u, repopulation of the ground state improves for fixed (t f ) opt , hencecausingTTS opt todecreasewithn, asseenFig.2.6(b),untilitsaturatestoaconstantat the lowest possible value, corresponding to a single run at (t f ) opt . At this point the problem is solved in constant time (t f ) opt , compared to the∼O(n 0.5 ) scaling of the adiabatic regime. Moreover, as shown in Fig. 2.6(c), there are no sharp changes inhHWi, suggesting that the non-adiabatic dynamics do not entail multi-qubit tunneling events, unlike the adiabatic 35 0 0.2 0.4 0.6 0.8 1 t/t f 0 0.2 0.4 0.6 0.8 1 P i 0 1 2 3 4 5 6 7 8 (a) 10 2 10 3 n 10 1 10 2 10 3 10 4 10 5 TTS opt SVD DQA SA 0 20 40 u 10 0 10 5 TTS opt (b) 0.75 0.8 0.85 0.9 0.95 1 t/t f 0 2 4 6 8 10 Average Hamming weight SVD DQA 0 0.5 1 t/tf 0 0.5 1 Trace norm distance (c) Figure 2.6: Diabatic QA vs SA and SVD for the Fixed Plateau problem with l = 0. (a) Population P i in the ith energy eigenstate along the diabatic QA evolution at the optimal TTS for n = 512 and u = 6. Excited states are quickly populated at the expense of the ground state. By t/t f = 0.5 the entire population is outside the lowest 9 eigenstates. In the second half of the evolution the energy eigenstates are repopulated in order. This kind of dynamics occurs due to a lining-up of avoided level crossings as seen in Fig. 2.7. (b) Scaling of the optimal TTS with n for u = 6, with an optimized number of single-spin updates for SA, and equal (t f ) opt for DQA and SVD. SA scales asO(n), a consequence of performing sequential single-spin updates. DQA and SVD both approachO(1) scaling as n increases. Here we setp d = 0.7 in Eq. (2.14), in order to be able to observe the saturation of SVD’s TTS to the point where a single run suffices, i.e., TTS opt = (t f ) opt . The conclusion is unchanged if we increase p d : this moves the saturation point to larger n for both SVD and DQA, and we have checked that SVD always saturates before DQA. Inset: scaling as a function ofu for n = 1008. SVD is again seen to exhibit the best scaling, while for this value of n the scaling of DQA and SA is similar (DQA’s scaling with n improves faster than SA’s as a function of n, at constant u). (c)hHWi of the QA wavefunction and the SVD state (defined as the product of identical spin-coherent states) for n = 512 andu = 6. The behavior of the two is identical up to t/t f ≈ 0.8, when they begin to differ significantly, but neither displays any of the sharp changes observed in Fig. 2.2(c) for the instantaneous ground state. Inset: the trace-norm distance between the DQA and SVD states, showing that they remain almost indistinguishable until t/t f ≈ 0.8. case. Thus, thisestablishesthatwemayhavespeedupsinQAthatdonotinvolvemulti-qubit tunneling. One may worry that for this diabatic evolution to be successful, the optimal annealing time may need to be very finely tuned. We address this concern in Appendix A.5, where we show that if is the precision desired in p GS , we need only have a precision of polylog(1/) in setting t f , which means that the diabatic speedup is robust. 36 0.6 0.7 0.8 0.9 1 t/t f 0 20 40 60 80 Eigenenergy Figure 2.7: The eigenenergy spectrum along the evolution for the Fixed Plateau with n = 512, l = 0, and u = 6. Note the sequence of avoided level crossings that unmistakably line up in the spectrum to reach the ground state. This is the pathway through which DQA is able to achieve a speedup over AQA. Figure 2.8 shows that the speedup of DQA and SVD over AQA exists for three other PHWO problems: the Moving Plateau, the Spike, and the 0.5-Rectangle problems. Impor- tantly, DQAandSVDhaveanexponentialspeedupoverAQAforthe0.5-Rectangleproblem. We do not observe a diabatic speedup for the Precipice or Grover problems. 2.4.4 Spin Vector Dynamics Given the absence of tunneling in the time-optimal quantum evolution, we are motivated to consider the behavior of Spin-Vector Dynamics (SVD), which arise in a semi-classical limit (see Appendix A.4.3 for an overview of this algorithm). As we show in Fig. 2.6(b), the scaling of SVD’s optimal TTS also saturates to a constant time, i.e., (t f ) opt . Moreover, it reaches this value earlier (as a function of problem size n) than DQA, thus outperforming DQA for small problem sizes, while for large enough n both achieveO(1) scaling. As seen in the inset, SVD’s advantage persists as a function of u at constant n. 37 200 400 600 800 1000 n 0 100 200 300 400 500 600 700 TTSopt DQA SVD 0 50 100 0 1 2 ×10 5 (a) Spike 200 400 600 800 1000 n 0 10 20 30 40 50 60 70 TTSopt DQA SVD (b) Moving Plateau 400 600 800 1000 n 0 100 200 300 400 500 TTSopt DQA SVD 0 100 200 0 5 ×10 5 (c) 0.5-Rectangle t/t f 0 0.2 0.4 0.6 0.8 1 P i 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 (d) Spike t/t f 0 0.2 0.4 0.6 0.8 1 P i 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 (e) Moving Plateau 0 0.2 0.4 0.6 0.8 1 t/t f 0 0.2 0.4 0.6 0.8 1 P i 0 1 2 3 4 5 6 7 8 (f) 0.5-Rectangle Figure 2.8: (a-c) The optimal TTS for the Spike, Moving Plateau, and 0.5-Rectangle prob- lems respectively. Inset for (a) and (c): the optimal TTS for small problem sizes, where we observe SVD at first scaling poorly. However, as n grows, this difficulty vanishes and it quickly outperforms DQA. (d-f) Population P i in the i-th energy eigenstate along the diabatic QA evolution at the optimal TTS. We observe similar diabatic transitions for these problem (shown are the cases withn = 512 andt f = 9.85 for the Spike,n = 512 andt f = 10 for the Moving Plateau, and n = 529 andt f = 9.8 for the 0.5-Rectangle) as we observed for the Fixed Plateau [Fig. 2.6(a)]. The dynamics of DQA are well approximated by SVD until close to the end of the evolution, as shown in Fig. 2.6(c): the trace-norm distance between the instantaneous states of DQA and SVD is almost zero until t/t f ≈ 0.8, after which the states start to diverge. This suggests that SVD is able to replicate the DQA dynamics up to this point, and only deviates because it is more successful at repopulating the ground state than DQA. In Fig. 2.8, we show that SVD’s speedup over AQA is replicated for the Spike, Moving Plateau, and 0.5-Rectangle problems as well. Remarkably, while the 0.5-Rectangle problem has an exponentially small gap [see Eq. (2.10) and Fig. 2.4(b)], SVD and DQA both achieve 38 O(1) scaling, and hence the diabatic cascades provides an exponential speedup relative to AQA. It is important to note that SVD is ineffective if one desires to simulate the adiabatic evolution. In the absence of unitary dynamics (which allow for tunneling) or thermal acti- vation (to thermally hop over the barrier), SVD gets trapped behind the barrier that forms in the semi-classical potential separating the two degenerate minima [see Fig. 2.2(a)] and is unable to reach the new global minimum. In this sense, SVD does not enjoy the guarantee provided by the quantum adiabatic theorem for the unitary evolution [10, 63, 64], that for sufficiently longt f dictated by the adiabatic condition, the ground state can be reached with any desired probability. Likewise, it is important to keep in mind the distinction between a classical algorithm being able to match, or sometimes outperform, a quantum algorithm (as SVD does here), and the classical algorithm approximating the evolution or instantiating the physics of the quantum algorithm (as SVD fails to do here). Indeed, in both the diabatic and adiabatic regimes, SVD provides a poor approximation to the instantaneous quantum state. For example, in the diabatic regime, it is clear from Fig. 2.6(c) that the trace-norm distance between the instantaneous SVD state and the instantaneous quantum state starts to increase significantly fors& 0.8. In the same spirit, consider the instantaneous semi-classical ground state, i.e., the spin-coherent state evaluated at the minimum of the spin-coherent potential, which may be suspected to provide a good approximation to the instantaneous quantum ground state, but does not as shown in Fig. 2.2(b). Thus the unentangled semi-classical ground state also fails to provide a good approximation to the quantum ground state. 2.4.5 Simulated Quantum Annealing Simulated Quantum Annealing (SQA) is a quantum Monte Carlo algorithm performed along the annealing schedule (see Appendix A.4.4 for further details). It is often used as a benchmark against which QA is compared (though see Ref. [39] for caveats). SQA scales 39 better than SA for the Fixed Plateau problem using the threshold criterion (see Fig. 2.5). In order to understand why SQA enjoys an advantage over SA using this benchmark metric, it isusefultostudythebehaviorofthestateofSQAalongtheannealingschedule. Weshowthe behavior ofhHWi for SQA in Fig. 2.9, where we observe that SQA at the optimal number of sweeps (the case of 1500 sweeps shown in Fig. 2.9) does not follow the instantaneous ground state. Instead it reaches the threshold success probability by thermally relaxing to the ground state after the minimum gap point (and tunneling event) of the quantum Hamiltonian. Therefore, SQA’s advantage over SA stems from the fact that it thermalizes in a different energy landscape than SA. WealsocontrastthebehaviorofSQAandAQAusingthethresholdcriterion. WhileSQA is able to follow the instantaneous ground state for a sufficiently large number of sweeps and thus mimic the tunneling of AQA (see Fig. 2.9), this is not the optimal way for it to reach the threshold criterion. For a fixed threshold success probability, the process of thermal relaxation after the minimum gap point uses fewer sweeps (and hence is more efficient) than following the instantaneous ground state closely throughout the anneal 2 . This is in contrast to AQA, where tunneling is the only means for it to reach a high success probability and nevertheless is more efficient than SQA, as seen in Fig. 2.5. We note that SQA’s threshold criterion advantage over SA does not carry over to the optimal TTS criterion. In fact, we find that using the optimal TTS criterion, SQA scales as O(n 1.5 ), while SA scales as O(n), as seen in Fig. 2.6(b). The reason for the latter scaling is that the optimal number of sweeps for SA is 1, simply because there is a small but non-zero probability that in the first sweep all the 1s are flipped to 0s. 2 ThismaybeanartifactofourimplementationofSQA,wherebyweonlyincludeclusterupdatesalongthe imaginary-time direction and not along the spatial (problem) direction. An implementation with space-like cluster updates may allow SQA via its thermal relaxation to mimic the tunneling of AQA more efficiently. Whether this is the case will be addressed in future work. 40 0.8 0.85 0.9 0.95 1 t/t f 0 2 4 6 8 10 Average Hamming weight GS QA, t f =4931.16 SQA, N sw =1.5k SQA, N sw =10k SQA, N sw =100k SQA, N sw =1000k Figure 2.9: The expectation value of the Hamming weight operator for the quantum ground state, SQA, and AQA for the Fixed Plateau problem with n = 512, l = 0, and u = 6 and annealing time chosen so as to reach a success probability of 0.9. The expectation value for SQA(β = 30,N τ = 64)atagivent/t f iscalculatedbyaveragingovertheHammingweightof theN τ imaginary time states at that time and over 10 5 independent trials. The case of 1500 sweeps is the minimum number of sweeps required for SQA (in ‘annealer’ mode) to reach the threshold ground state probability of p 0 = 0.9, and similarly for the annealing time value of t f = 4931.16 for AQA. While AQA is able to approximately follow the quantum ground state (i.e., the evolution is very close to being adiabatic), the optimal SQA evolution (i.e., that requires the fewest sweeps) for achieving the threshold criterion involves not following the ground state at the minimum gap point and instead thermally relaxing towards the ground stateafter thispoint. Asshown usingthehigherN sw values, only afterincreasingthe number of sweeps by more than two orders of magnitude does SQA follow the instantaneous ground state closely. 2.5 Discussion It is often assumed that the shape of the final cost-function determines how hard it is for QA to solve the problem (in fact, this was partly the motivation for the Spike problem in Ref. [47]), and that potentials with tall and thin barriers should be advantageous for AQA, since this is where tunneling dominates over thermal hopping (e.g., [39, p.215], [44, p.1062], [45, p.226]). It is then assumed that problems where the final potential has this feature are 41 200 400 600 800 1000 n 20 40 60 80 100 TTS opt QA SVD 5 10 15 20 t f 0 0.5 1 p GS Figure 2.10: The optimal TTS for the potential given in Eq. (2.27). QA outperforms SVD over the range of problem sizes we were able to check. The reason can be seen in the inset, which displays the ground state probability for SVD and QA for different annealing times t f , withn = 512. The optimal annealing time for SVD occurs at the first peak in its ground state probability (t f ≈ 8.98), whereas the optimal annealing time for QA occurs at the much larger second peak in its ground state probability (t f ≈ 10.91). those for which there should be a quantum speedup. We have given several counterexamples to such claims, and shown that tunneling is not necessary to achieve the optimal TTS. Instead, the optimal trajectory may use diabatic transitions to first scatter completely out of the ground state and return via a sequence of avoided level crossings. That diabatic transitions can help speed up quantum algorithms has also been noted and advantageously exploited in Refs. [67, 6, 68, 69]. Moreover, we have shown that the instantaneous semi- classical potential provides important insight into the role of tunneling, while the final cost function can be rather misleading in this regard. While both adiabatic and diabatic QA outperform SA for the Fixed Plateau problem, the faster quantum diabatic algorithm is not better than the classical SVD algorithm for this problem. The PHWO problems due to Reichardt [51], which includes problems very similar to the Fixed Plateau, have widely been considered an example where tunneling provides a 42 quantum advantage; we have shown that this holds if one limits the comparison to SA, but that there is in fact no quantum speedup in the problem when one compares the quantum diabatic evolution (which outperforms adiabatic quantum annealing) to SVD. These results of the diabatic optimal evolution extend beyond the plateau problems: even the Spike problem studied in Ref. [47]—which is in some sense the antithesis of the plateau problem since it features a sharp spike at a single Hamming weight—also exhibits the diabatic-beats-adiabatic phenomenon, indicating that tunneling is not required to effi- ciently solve the problem. Thus diabatic evolution, especially via diabatic cascades, is an important and relatively unexplored mechanism in quantum optimization that is different from tunneling. The fact that we observe a speedup relative to AQA for several problems, especially an exponential speedup for the 0.5-Rectangle, motivates the search for algorithms exploiting this mechanism and may yield fruitful results. However, we also already know that diabatic cascades are not generic. E.g., we have checked that this mechanism is absent in the Grover and Precipice problems, even though the Grover problem is equivalent to a ‘giant’ plateau problem. In summary, our work provides a counterargument to the widely made claims that tun- neling should be understood with respect to the final cost function, that speedups due to tunneling require tall and thin barriers; and that tunneling is needed for a quantum speedup in optimization problems. Which features of Hamiltonians of optimization problems favor diabatic or adiabatic algorithms remains an open question, as is the understanding of tun- neling for non-permutation-symmetric problems. We finish on a positive note for QA. We have given several examples where SVD outper- forms QA, e.g., the Spike problem [47]. However, we make no claim that SVD will always have an advantage over QA. A simple and instructive example comes from the class of cost 43 functions that are convex in Hamming weight space, which have a constant minimum gap [70]: f(x) = 2, |x| = 0 |x|, otherwise . (2.27) We have observed similar diabatic transitions for this problem as for the Fixed Plateau (not shown), but find that DQA outperforms SVD, as shown in Fig. 2.10. This results because the optimal TTS for QA occurs at a slightly higher optimal annealing time, i.e., there is an advantage to evolving somewhat more slowly, though still far from adiabatically. Thus, this provides an example of a “limited” quantum speedup [56]. 2.6 Some open questions In this section, we list some open questions that are raised by the work presented in this chapter. 1. Thedescriptionoftunnelingusingthespin-coherentpathintegralformalismisusefulto understand permutation symmetric problems, but can be it also be used to understand systems that don’t have this symmetry? 2. Can we define “tunneling time” in these systems? Can the spin-coherent path-integral formalism yield rigorous estimates for such a quantity? Can we quantify the connection between tunneling and speedup? 3. Can we develop a resource theory of quantum tunneling, along the lines of the resource theory of entanglement? What is the connection between tunneling and entanglement? 4. Diabatic cascades were discovered to be generic in the class of perturbed Hamming weight problems. Why are they so common? When do they occur? Are they stable 44 to noise? Do they occur in more physical systems? Some preliminary work has been done in Refs. [71, 72], but much more work is needed. 45 Chapter 3 Quasiadiabatic Grover Search via the Wentzel-Kramers-Brillouin Approximation 3.1 Introduction There exist only a handful of Hamiltonian-based quantum algorithms [73, 74], designed to run on analog quantum computers [75, 76, 77], that exhibit a provable quantum speedup [78]. The adiabatic version of the Grover search problem is one such example [53]. The existence of this speedup is proven using the adiabatic theorem [10], i.e., it is based on an asymptotic analysis in the total evolution time. This is in contrast to the circuit model version of the Grover problem [79], for which a closed-form analytical solution is known for arbitrary evolution times and arbitrary initial amplitude distributions [80, 81]. No such closed form analytical solution of the Hamiltonian version of Grover’s algorithm is known as of yet. The Wentzel-Kramers-Brillouin (WKB) method is a famous technique for approximating differential equations which has found applications in many domains of physics and math- ematics, including optics, acoustics, astrophysics, elasticity, and quantum mechanics (see Ref. [82] for a mathematical history of the WKB method). In this work, we adopt the WKB method to provide approximate analytical solutions to the Hamiltonian Grover problem. The WKB method we use is quasi-adiabatic (as opposed to semiclassical [62]): the small parameter is the inverse of the total evolution time (not~, which we set to 1). We choose to 46 focuson theGroverproblemsincethis problem iswellstudiedand understood, buttheWKB method is widely applicable and easily generalizable to other Hamiltonian-based quantum algorithms. We thus expect it to be a useful tool in analyzing such algorithms beyond the adiabatic approximation. We compare the results of the WKB approximation with a numerically exact solution. Strikingly, we find that the quality of WKB results depends strongly on the interpolation schedule from the initial to the final Hamiltonian. The WKB approximation is reliable already at low order for the schedule that generates a quantum speedup for the Grover prob- lem [53], but fails for the other schedules we tested. These other schedules are characterized by a different dependence on the power of the inverse spectral gap. The structure of the chapter is as follows. We briefly review the quasi-adiabatic WKB method in Sec. 3.2. The method is applied to the Grover problem in Sec. 3.3, and the WKB solutions are derived in Sec. 3.4. The results are discussed and analyzed in Sec. 3.5, where we perform a comparison with the numerically exact solution. We conclude in Sec. 3.6. We remark that there are other tools available to study quasi-adiabatic dynamics: Adia- batic perturbation theory is a popular method [83]. In Appendix B.1 we study a particular variant of adiabatic perturbation theory from Ref. [84] and compare it to our method. Our method is not a variant of adiabatic perturbation theory because at the lowest order we do not recover the adiabatic evolution. 3.2 Quasi-adiabatic WKB for interpolating Hamiltoni- ans We start by briefly reviewing the asymptotic WKB expansion technique (for background see, e.g., Ref. [85]), and connect it to interpolating Hamiltonians of the type used in adiabatic quantum computing. 47 3.2.1 WKB as an asymptotic expansion The WKB expansion y(r)∼e θ(r)/ [y 0 (r) +y 1 (r) + 2 y 2 (r) +... ], (3.1) isanansatzusedforthesolutionofordinarydifferentialequationsiny(r)thatcontainasmall parameter, , multiplying the highest derivative. This ansatz is an asymptotic expansion in , i.e., there is no guarantee that it will provide a unique or even a convergent solution. In fact, the asymptotic series for y(r) is usually divergent; the general term n y n (r) starts to increase after a certain value n =n max , which can be estimated for second order differential equations of the form 2 y 00 (r) =Q(r)y(r), if Q(r) is analytic [86]. The number n max can be interpreted as the number of oscillations between r 0 [the point at which y(r) needs to be evaluated] and the turning pointr ? [i.e., whereQ(r ? ) = 0] closest tor 0 . In this work we will only be concerned with the expansion up to order for a second order differential equation. For later convenience, we list the expressions for the derivatives of the ansatz: y∼e θ/ ∞ X j=0 j y j (3.2a) y 0 ∼e θ/ ∞ X j=0 j−1 (θ 0 y j +y 0 j−1 ) | {z } ≡z (1) j (3.2b) y 00 ∼e θ/ ∞ X j=0 j−2 [(θ 0 ) 2 y j +θ 00 y j−1 + 2θ 0 y 0 j−1 +y 00 j−2 ] | {z } ≡z (2) j (3.2c) with y k ≡ 0 if k< 0, and where the number of primes denotes the order of the derivative. 48 3.2.2 Interpolating Hamiltonians We consider interpolating Hamiltonians of the form H[r(s)] = [1−r(s)]H initial +r(s)H final . (3.3) which depend on time only via the dimensionless time s≡ t/t f . Here t f denotes the total evolution time and is the only timescale in the problem. The “interpolation schedule” r(s) is strictly increasing, differentiable, and satisfies the boundary conditions r(0) = 0 and r(1) = 1. The derivative of the inverse of r(s), viz. s 0 (r), is therefore also strictly positive. This allows us to divide by s 0 when we need to. Consider now the Schrödinger equation for this evolution i d dt |χ(t)i =μH(r[s(t)])|χ(t)i, (3.4) where μ is an energy scale, and H(·) is dimensionless, e.g., a linear combination of Pauli matrices. Writing everything in terms of s, we get i d ds |χ(s)i =μt f H[r(s)]|χ(s)i. (3.5) One can also write the problem in terms of r. This yields: i d dr |χ(r)i =g(r)H(r)|χ(r)i, (3.6) where g(r)≡s 0 (r), s(r) : [0, 1]7→ [0, 1], and where ≡ 1 μt f , (3.7) 49 is the dimensionless small parameter for our WKB expansion. Since is small for large t f , we call our method “quasi-adiabatic WKB”. 1 3.3 The Grover problem via the quasi-adiabatic WKB approximation Recall that the Grover problem can be formulated as finding an item in an unsorted list of N = 2 n items, in the smallest number of queries [87]. This admits a quadratic quantum speedup, as was first shown by Grover in the circuit model [79]. It is also one of the few instances where an adiabatic algorithm was discovered which recovers the quantum speedup. The crucial insight, which eluded the first attempt [74], was that the speedup obtained in the circuit model could be recovered in the adiabatic model provided the right interpolation scheduler(s) is chosen, namely, a schedule that drives the system more slowly when the gap is smaller [53] (see also Refs. [10, 54]). In the Hamiltonian Grover algorithm one uses the n-qubit interpolating Hamiltonian H Grover [r(s)] = [1−r(s)](I−|uihu|) +r(s)(I−|mihm|), (3.8) 1 We remark that the quasi-adiabatic WKB approximation should not be confused with the traditional WKB approximation associated with the~→ 0 limit. The latter is typically used as a semiclassical approx- imation in one-dimensional position-momentum quantum mechanics, involving a potential barrier (see, e.g., Ref. [62]). The quasi-adiabatic and semiclassical WKB approximations are not interchangeable. This can be seen from the Schrödinger equation for a particle in a one-dimensional potential: i ~ t f d ds |χi = − ~ 2 2m ∂ 2 x +V(x,st f ) |χi, where again s = t/t f and V(x,t) is a space- and time-dependent potential energy function. It is evident that there is no way to trade both~ and 1/t f for a single small parameter, since they appear together as the product~t f . 50 where m∈{0, 1} n is the marked state and |ui≡ 1 √ 2 n X x∈{0,1} n |xi, (3.9) is the uniform superposition state. The system is initialized in the state|ui. It can be easily checkedthatthedynamicsdescribedbythisHamiltonianisrestrictedtoS≡ Span{|ui,|mi}. Let K≡ 2 n − 1 and define |m ⊥ i≡ 1 √ K X x∈{0,1} n x6=m |xi, (3.10) so that|ui = (|mi + √ K|m ⊥ i)/ √ K + 1. Note that{|mi,|m ⊥ i} is an orthonormal basis for S. In this basis, the Hamiltonian is H(s) = [1−r(s)] K K+1 −[1−r(s)] √ K K+1 −[1−r(s)] √ K K+1 1− [1−r(s)] K K+1 . (3.11) Henceforth we restrict our analysis to this two-dimensional Hamiltonian and will not return to the high-dimensional Hamiltonian that gave rise to it. Let |χ(s)i =ψ(s)|mi +φ(s)|m ⊥ i , (3.12) i.e., henceforthψ(s) is the amplitude of the marked state (ground state of the final Hamilto- nian), and φ(s) is the amplitude of the unmarked component (the excited state of the final Hamiltonian). From Eq. (3.6), the Schrödinger equation for a general interpolation becomes iψ 0 = g(r) K + 1 h K(1−r)ψ− √ K(1−r)φ i , (3.13a) iφ 0 = g(r) K + 1 h − √ K(1−r)ψ + (1 +rK)φ i . (3.13b) 51 The boundary conditions are ψ(0) = 1 √ K + 1 , φ(0) = s K K + 1 , (3.14) so it follows from Eqs. (3.13) that ψ 0 (0) =φ 0 (0) = 0. We now turn the above coupled first order system into two decoupled second order differential equations: 2 (1−r)ψ 00 +(a 1,1 +a 1,2 )ψ 0 +a 0 ψ = 0 (3.15a) 2 (1−r)φ 00 +(a 1,1 +a 1,2 )φ 0 + (a 0 +ig)φ = 0 , (3.15b) where a 0 =− g 2 K(1−r) 2 r K + 1 (3.16a) a 1,1 = 1− g 0 g (1−r), a 1,2 =i(1−r)g . (3.16b) The function g(r) =s 0 (r) uniquely determines the schedule r(s). We shall consider four different schedules corresponding to choices α∈{0, 1, 2, 3} in r 0 (s) =c α Δ(r) α , (3.17) wherec α is a constant that depends onK (see Refs. [10, 53]) and Δ(r) is the eigenvalue gap of the Hamiltonian in Eq. (3.11), given by: Δ(r) = s 1− 4Kr(1−r) K + 1 . (3.18) Equation(3.17)forcesthescheduletobecomeslower(faster)whenthegapissmaller(larger). 52 The linear schedule [r(s) =s] corresponds to the choice α = 0, and the schedule discov- ered by Roland and Cerf [53] corresponds toα = 2. We also analyze schedules corresponding to α = 1, 3. To find the constant c α , we integrate Eq. (3.17) and use the boundary condi- tion s(1) = 1. The expressions for the schedules thus obtained, expressed in terms of the corresponding g α (r) functions [recall that g(r)≡s 0 (r)], are as follows: g 0 (r) =g lin (r) = 1 (3.19a) g 1 (r) = 2 q K K+1 log √ K+1+ √ K √ K+1− √ K × 1 Δ(r) (3.19b) g 2 (r) =g RC (r) = √ K (K + 1) tan −1 ( √ K) × 1 Δ(r) 2 (3.19c) g 3 (r) = 1 K + 1 × 1 Δ(r) 3 (3.19d) We now turn to the construction of the WKB solutions for both amplitudes ψ andφ for each of the schedules. 3.4 Constructing the WKB solutions ToderivetheWKBsolutions, wesubstitutetheWKBansatz[Eqs.(3.2)]intoEqs.(3.15a) and (3.15b). 2 Then, we set the terms multiplying different orders j to zero, which yields the following recursive set of equations for j≥ 1: (1−r)z (2) j +a 1,1 z (1) j−1 +a 1,2 z (1) j +a 0 y j = 0 (3.20a) (1−r)z (2) j +a 1,1 z (1) j−1 +a 1,2 z (1) j +a 0 y j +igy j−1 = 0 , (3.20b) 2 A Mathematica R notebook containing code for obtaining the WKB expressions used in our analysis is provided at https://tinyurl.com/WKB-notebook. 53 whereψ [Eq. (3.15a)] is reconstructed from Eq. (3.20a), andφ [Eq. (3.15b)] is reconstructed from Eq. (3.20b). We consider only the lowest three orders in below. First, isolating the 0 term [i.e., settingj = 0 in both Eqs. (3.20a) and (3.20b)], we obtain the eikonal equation: (θ 0 ) 2 +igθ 0 −g 2 Kr(1−r) K + 1 = 0 , (3.21) which is a quadratic equation in θ 0 , so that: θ 0 ± =− ig 2 [1± Δ(r)] . (3.22) Turning to the 1 term, we obtain the transport equations: ψ 0 0 ψ 0 =− (1−r)θ 00 + h 1− g 0 g (1−r) i θ 0 (1−r)(2θ 0 +ig) , (3.23a) φ 0 0 φ 0 =− (1−r)θ 00 + h 1− g 0 g (1−r) i θ 0 +ig (1−r)(2θ 0 +ig) . (3.23b) Hereψ 0 ,φ 0 arethepartsoftheWKBapproximantthatcorrespondtoy 0 , whichwasageneral placeholder for the lowest order term. Further, Eq. (3.23a) is obtained from Eq. (3.20a), and Eq. (3.23b) is obtained from Eq. (3.20b), both after setting j = 1 and using the eikonal equation (3.21) to eliminate the y 1 term. Let Θ ± ≡ θ 0 ± /g(r). It is easy to check that the transport equations then become: ψ 0 0 ψ 0 =− (1−r)Θ 0 + Θ (1−r)(2Θ +i) , (3.24a) φ 0 0 φ 0 =− (1−r)Θ 0 + Θ +i (1−r)(2Θ +i) . (3.24b) Since, by Eq. (3.22), Θ ± =− i 2 [1± Δ(r)] is independent of g, it follows that ψ 0 and φ 0 do not depend on the interpolation g. 54 Further, using Θ + =−(i + Θ − ) and Θ 0 ± =∓ i 2 Δ 0 , it is straightforward to show that the r.h.s. of Eq. (3.24a) for ψ ± 0 is identical to the r.h.s. of Eq. (3.24b) for φ ∓ 0 . Thus, after integration we have ψ ± 0 (r) =c ± 0 φ ∓ 0 (r), where c ± 0 is (the exponential of) an integration constant. Moreover, the r.h.s. of Eq. (3.24a) corresponding to Θ ± is easily seen to be equal to − 1 2 h Δ 0 (r) Δ(r) + 1 1−r ± 1 (1−r)Δ(r) i . Hence, integrating Eqs. (3.24) yields: logψ ± 0 (r) = logφ ∓ 0 (r) + ˜ c ± 0 (3.25) = 1 2 log 1−r Δ(r) ∓ 1 2 Z 1 (1−r)Δ(r) dr + ˜ d ± 0 , where ˜ c ± 0 and ˜ d ± 0 are integration constants. Or, using the explicit form for the gap given in Eq. (3.18): ψ + 0 (r) =c + 0 φ − 0 (r) (3.26a) =d + 0 1−r q √ K + 1Δ(r)[K(2r− 1) + (K + 1)Δ(r) + 1] ψ − 0 (r) =c − 0 φ + 0 (r) (3.26b) =d − 0 v u u t K(2r− 1) + (K + 1)Δ(r) + 1 √ K + 1Δ(r) , where c ± 0 =e ˜ c ± 0 and d ± 0 =e ˜ d ± 0 . Finally, turning to the 2 term [i.e., setting j = 2 in Eqs. (3.20a) and (3.20b)] yields: w 0 =− y 00 0 (1−r) + h 1− g 0 g (1−r) i y 0 0 (1−r)(2θ 0 +ig)y 0 , (3.27) where w≡ y 1 y 0 . Here y represents both ψ and φ. We used the eikonal equation to eliminate they 2 term, and the transport equations to obtain y 0 0 /y 0 inw 0 . Solving Eq. (3.27) yields y 1 . Note that here we cannot remove the dependence of y 1 on the interpolation g. 55 We can now assemble the different functions into a solution. Given the interpolation g, we can integrate Eq. (3.22) to find θ ± , resulting in two solutions ψ ± and φ ± . This means that we have to consider linear combinations of these two solutions. Thus ψ∼A ψ e θ + / (ψ + 0 +ψ + 1 ) +B ψ e θ − / (ψ − 0 +ψ − 1 ) (3.28a) φ∼A φ e θ + / (φ + 0 +φ + 1 ) +B φ e θ − / (φ − 0 +φ − 1 ) , (3.28b) where the constantsA ψ,φ ,B ψ,φ are determined using the boundary conditions ψ(0) = 1 √ K+1 , φ(0) = q K K+1 , and ψ 0 (0) = φ 0 (0) = 0. Note that despite the fact that ψ 0 and φ 0 do not depend on g, the parameter θ does, via θ ± = R gΘ ± dr. Therefore even at the lowest order, the approximate solution retains a dependence on the interpolation g. The only constraints our solutions must satisfy are the differential equations (3.20) and the boundary conditions. Thus we are free to choose the integration constants (c ± 0 ,d ± 0 , and others that would arise at higher ordersj≥ 2), and henceforth we choose them to be equal at all orders, such that onlyA,B,C,D are undetermined until we use the boundary conditions. It is important to remember that the WKB approximation method does not enforce normalization. Hence, generically, the WKB approximation to a quantum state is unnor- malized, resulting in approximations to probabilities that may be greater than 1. 3 Thus, care must be taken when applying this approximation technique to estimate physical quantities, and in particular one must check that normalization holds. For some of the examples we study here, such nonsensical probabilities indeed arise. In Sec. 3.5.2, we study whether the norm of the WKB approximation is an indicator of approximation quality and also whether renormalization can be improve the WKB approximation. One final general comment is in order. It turns out that the differential equations (3.15a) and (3.15b) have the following unfortunate property: substituting the WKB approximation 3 For convenience, we will abuse terminology somewhat and refer to the WKB approximations to physical probabilities as “probabilities” even though they may not be normalized. It should be clear from the context whether we are referring to approximated probabilities or to actual probabilities. 56 to ψ into Eq. (3.13a) and solving for φ does not yield a good approximation to φ. On other hand, the WKB approximation to φ does yield a good approximation to φ. This is why we need to perform the WKB approximation separately for each of the amplitudes. 3.5 Results In this section we analyze the quality of the approximate solutions by comparing them with the solutions obtained via numerical integration of the Schrödinger equation. The results obtained by numerical integration are sufficiently accurate that we can take the numerical solution to be a good proxy to the exact solution. We denote the numerically obtained solution by|χ Num i and the solution obtained from the WKB approximation by |χ WKB i. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.6 0.7 0.8 0.9 1.0 (a) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 ×10 -3 (b) Figure 3.1: Evolution of a single qubit (n = 1) in a magnetic field, under the g 0 (r) = 1 schedule. (a) Population in the ground state|mi as a function of r = s = t/t f for t f = 50 according to the numerical solution (|χ Num i), the two lowest orders of the WKB approximation (|χ (0) WKB i and|χ (1) WKB i), and the naive adiabatic evolution (|χ GS i). The WKB predictionsandthenumericalsolutionexhibitoscillationsandareindistinguishablefromeach other on the scale shown. The adiabatic approximation does not exhibit oscillations. (b) The ground state population difference between the WKB approximation and the numerical simulation fort f = 50. The higher order WKB approximation provides a significantly better approximation. 57 20 40 60 80 100 0. 0.2 0.4 0.6 0.8 1. 1.2 1.4 x10 -3 (a) 20 40 60 80 100 -6 -4 -2 0 2 4 6 8 ×10 -4 (b) Figure 3.2: Final ground state population of a single qubit (n = 1) in a magnetic field, under the g 0 (r) = 1 schedule. (a) Depopulation of the ground state (i.e., population in the excited state|m ⊥ i) at r = 1 as a function of t f , comparing the numerical solution (|χ Num i) and the two lowest orders of the WKB approximation (|χ (0) WKB i and|χ (1) WKB i). The lowest order|χ (0) WKB i captures the asymptotic behavior of the exact solution, while|χ (1) WKB i becomes indistinguishable from the exact solution for t f & 50. (b) The difference between the true population in the state|m ⊥ i at time r = 1 and the asymptotic prediction of 1 4t 2 f obtained from the 1/t f expansion of hm ⊥ |χ (0) WKB (1)i 2 . The asymptotic approximation becomes more accurate as t f grows. 3.5.1 Single Qubit in a magnetic field Asasimpletest, wefirstapplytheformalismdevelopedinSec.3.2tothecaseK =n = 1, which models a qubit in a time-varying magnetic field that changes from the x-direction to the z-direction, with a linear interpolation r(s) =s: H(r) =−(1−r)σ x −rσ z , (3.29) where σ x ≡|mihm ⊥ | +|m ⊥ ihm| and σ z ≡|m ⊥ ihm ⊥ |−|mihm|. Thus, the eikonal equa- tion (3.22) becomes θ 0 ± =− i 2 [1± Δ(r)], (3.30) 58 0 10 20 30 40 50 60 0.00 0.02 0.04 0.06 0.08 0.10 Figure 3.3: The time-averaged trace-norm distance [see Eq. (3.33)] vs. t f for a single qubit (n = 1) in a magnetic field under theg 0 (r) = 1 schedule. The distances plotted are between the numerical solution and the adiabatic approximation, and the two lowest order WKB approximations. The adiabatic and the WKB distances decrease monotonically with t f , with the former being a prediction of the adiabatic theorem in the large t f limit. The WKB approximations at both orders are consistently better than the adiabatic approximation and the first-order WKB approximation improves upon the zeroth-order WKB approximation. where Δ(r)≡ q 1− 2r(1−r). Therefore the two energy levels of this problems areE ± (s) = −iθ 0 ± . Similarly, the solutions [Eqs. (3.26)] of the transport equations yield, after setting K = 1, ψ + 0 =φ − 0 = 1−r q Δ(r + Δ) (3.31a) ψ − 0 =φ + 0 = s r + Δ Δ , (3.31b) where we have chosen the integration constants to remove overall numerical factors. Next, we may use these solutions to obtain the first-order correction. For this we obtain from Eq. (3.27): ψ ± 1 (r) =∓iψ ± 0 (r) 16r 4 − 40r 3 + 42r 2 − 17r + 5± 6Δ(r) 12(1−r)Δ(r) 3 , (3.32a) φ ± 1 (r) =±iφ ± 0 (r) 16r 4 − 40r 3 + 42r 2 − 17r + 5∓ 6Δ(r) 12(1−r)Δ(r) 3 . (3.32b) 59 From these expressions and the boundary conditions we construct two solutions:|χ (0) WKB i (using ψ 0 ,φ 0 ) and|χ (1) WKB i (using ψ 0 ,φ 0 and ψ 1 ,φ 1 ). We expect|χ (1) WKB i to be a better approximation to the exact solution than|χ (0) WKB i and we expect the quality of approximation to improve with increasing t f , i.e., with decreasing . We also consider the naive adiabatic approximation, which we define as the instantaneous ground state of H(r). Figure 3.1 shows that the WKB approximation is able to capture the correct population dynamics. 4 In more detail, Fig. 3.1(a) shows that the approximation captures oscillations not present in a naive adiabatic approximation, and Fig. 3.1(b) shows that the quality of the approximation improves from the lowest order to the next order of the WKB approximation. Next, consider the final ground state probability, p GS (t f ). In Fig. 3.2(a), we see that |χ (0) WKB i is already sufficient to capture the asymptotic scaling of p GS with t f . Further, |χ (1) WKB i captures the oscillations in p GS (t f ), with an accuracy that grows with increasing t f . Performing a series expansion of h1|χ (0) WKB (1)i 2 in powers of 1 t f , we obtain the leading order term to be 1 4t 2 f . As we see in Fig. 3.2(b), this asymptotic prediction is close to the asymptotic scaling of the numerical solution. Finally, consider the time-averaged trace-norm distance between two time-evolving states |χ 1 (t)i and|χ 2 (t)i: D (|χ 1 i,|χ 2 i) = 1 t f Z t f 0 D (|χ 1 i,|χ 2 i)dt (3.33a) D (|χ 1 i,|χ 2 i)≡ 1 2 k|χ 1 ihχ 1 |−|χ 2 ihχ 2 |k 1 . (3.33b) The results comparing the numerical solution to the naive adiabatic approximation and the two lowest WKB approximation orders are shown in Fig. 3.3. As expected, the naive adiabatic approximation becomes better as t f increases, and the same is true for the WKB approximations, which are both more accurate than the adiabatic approximation. Moreover, 4 Figures were made with the help of the MaTeX package for Mathematica R by Szabolcs Horvát (see url: http://szhorvat.net/pelican/latex-typesetting-in-mathematica.html). 60 the first-order WKB approximation is better than the adiabatic approximation according to the time-averaged trace-norm distance metric. Taken together, the results for the n = 1 case show that both the zeroth-order WKB approximation and the first-order WKB approximation consistently improve upon the naive adiabatic approximation, and the first-order WKB approximation can be used to pick out more subtle features of the quantum evolution. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ■ ◆ ▲ 0 2 4 6 8 10 12 14 0 5 10 15 Figure 3.4: The time required to achieve a final ground state probability of 0.95 for the schedules defined in Eqs. (3.19) (log scale). The straight lines represent exponential scaling fits ofO(2 1.01n ),O(2 0.667n ),O(2 0.508n ), andO(2 0.463n ) respectively. ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 16 18 20 22 24 9 10 11 12 13 14 Figure 3.5: The scaling ofp GS (t f ) from the numerically exact solution under theg 3 schedule, for larger problem sizes than in Fig. 3.4. The straight line represents an exponential scaling fit ofO(2 0.499n ). Thus, the scaling converges to the expected scaling ofO(2 n/2 ) predicted by the query complexity bound [88]. 61 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ■ ◆ ▲ 0 5 10 15 20 25 30 5 10 15 20 25 30 Figure 3.6: The time required to achieve a final ground state probability of 0.95 for the interpolations defined in Eqs. (3.19)] (log scale), using the WKB approximation at the lowest order. The straight lines represent fits ofO(2 n ),O(2 0.51n ),O(2 3.5n 0.2 ), andO(1) respectively. Thus, the lowest-order WKB approximation predicts the right scaling only for the optimized schedule g 2 (r). 3.5.2 The n-qubit Grover problem We next turn to a study of the Grover problem as a function of problem size n, with n > 1. The quantity of interest to us is how long we need to run the adiabatic algorithm before a certain threshold probability of success p Th is exceeded. The associated threshold timescale is defined as: t Th f ≡ min{t f :p GS (t)>p Th ∀t>t f }. (3.34) Here, p GS (t) represents the probability of finding the ground state at the end an adiabatic evolution of time t. We choose p Th = 0.95 (we have checked that the results are insensitive to changing p Th ). First, in Fig. 3.4 we show how t Th f (n) scales for the numerical solution, under the four different schedules defined in Eqs. (3.19). It appears as though the scaling for the g 3 (r) schedule is better than the theoretically optimal scaling of 2 n/2 [88], but this is a small n effect as shown in Fig. 3.5. 62 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 (a) 0 10 20 30 40 50 60 0.00 0.05 0.10 0.15 0.20 0.25 0.30 (b) Figure 3.7: (a) The trace-norm distance between the lowest order WKB approximation and the numerically exact solution for the four different schedules [Eq. (3.19)] as a function of the evolution parameter r. Here n = 6 and t f = 60. (b) The time-averaged trace-norm distance [Eq. (3.33)] between the lowest order WKB approximation and the numerically exact solution for the four different schedules [Eqs. (3.19)]. Here n = 6. Both panels are consistent with Fig. 3.6 where g 2 recovers the correct scaling. Recall that g 2 represents the optimal schedule found in Ref. [53], which provides the best approximation to the numerical evolution. 0 100 200 300 400 0.98 0.99 1.00 1.01 1.02 0 2000 4000 6000 8000 10000 0.0 0.2 0.4 0.6 0.8 1.0 Figure 3.8: Final ground state probability p GS as function of total evolution time t f for the Grover problem with n = 4 for the four different schedules, g α with α∈{0, 1, 2, 3} as predicted by the WKB approximation at first order. (a) The g 0 and g 1 schedules. The rise in p GS as a function of t f is very steep, and quickly exceeds 1 for both schedules (the g 0 curve goes very slightly above 1). The g 0 curve rises faster than the g 1 curve for p GS ≤ 1. (b) The g 2 and g 3 schedules.The rise in the p GS curve for g 2 is much steeper than the rise for the g 3 curve. In general, the smaller is α, the larger the steepness in the p GS (t f ) curve. This is consistent with the t Th f (n) scalings obtained in Fig. 3.6. 63 0 10 20 30 40 50 60 0.0 0.1 0.2 0.3 0.4 Figure 3.9: The Grover problem with the g 2 (r) schedule for n∈{1,..., 5}. (a) The time- averaged trace-norm distance [Eq. (3.33)] between the numerical solution and the lowest order WKB approximation (solid), i.e, D(|χ (0) WKB i,|χ Num i); and the time-averaged trace- distance between the numerical solution and the adiabatic approximation (dashed), i.e, D(|χ GS i,|χ Num i). The lowest order WKB approximation is always better than the adia- batic approximation for all n values we have tested. (For both the solid and dashed lines, the lower the line on the plot, the lower the value of n.) Next, we examine how well the WKB approximation does in predicting these scalings. In Fig. 3.6 we plot the scaling of t Th f (n) for the same four schedules, under the lowest order WKB approximation. Only the g 2 (r) schedule (which slows as the inverse-square of the gap) yields the correct scaling of t Th f (n). This is also the schedule which yields the smallest instantaneous and time-averaged trace-norm distance, as shown in Figs. 3.7(a) and 3.7(b), respectively. For the other schedules, Fig. 3.6 shows that the WKB approximation gives answers that are dramatically different from the exact solution. Furthermore, for the g 0 and g 1 schedules, the scaling with n of t Th f (n) violates the query complexity bound [88]. Why do the approximations for the g 0 , g 1 , and g 3 schedules give us the wrong scalings, while the approximation for the g 2 schedule gives us the correct scaling? A possible answer lies in the steepness of the final-time approximate success probability curves for the different schedules. In Fig. 3.8 we show thep GS (t f ) curves for all four schedules forn = 4 (K = 15) as 64 predicted by the first-order WKB approximation (the highest order at which we are able to obtain analytic expressions). For the g 0 andg 1 schedules, we see that the final ground state probability rises very sharply and exceeds unity (very slightly for g 0 ), and thereby becomes nonsensical [see Fig. 3.8(a)], while for the g 2 and g 3 schedules, p GS (t f )≤ 1 [see Fig. 3.8(b)]. Further, we observe that the curves are ordered from steepest to shallowest rise as g 0 , g 1 , g 2 ,g 3 . We conjecture that this rise inp GS witht f continues to slow down with increasingα. Thus theg 2 schedule captures the right scaling [in Fig. 3.6] by capturing the right steepness: forα< 2 the rise is too steep, and forα> 2 the rise is too shallow. A full explanation of this phenomenon is left to future work, but we speculate that theg 0 andg 1 schedules correspond to effective Hamiltonians that no longer represent the Grover problem. Given that the WKB approximation gives consistent results only for the g 2 (r) schedule, we focus on this schedule and examine where the WKB approximation performs better than the naive adiabatic approximation. As can be seen in Fig. 3.9, for the g 2 (r) schedule, the WKB approximation always has an advantage over the naive adiabatic approximation. Further, it is clear that the advantage is bigger for smaller evolution times t f and for larger problem sizes n. As we have indicated above, the WKB approximants are generically not normalized: they can be sub-normalized or super-normalized. Two questions arise: (1) Is the degree of non-normalization a good indicator of approximation quality of the WKB aproximation? (2) Does renormalization by fiat improve the quality of the approximation? First, in Fig. 3.10 we plot the norm of the WKB approximation at the lowest order for the case of n = 6 and t f = 60 as a function of the anneal parameter r for all four schedules. In this regime, the WKB approximation is sub-normalized for all schedules. Arranging the schedules from farthest from normalization to closest to normalization, we have: g 3 ,g 2 ,g 1 ,g 0 , with g 0 and g 1 closest to being normalized. On the other hand, we have seen [Fig. 3.7(a) and Figs. 3.6,3.7(b)] that the best approximation was obtained for the g 2 schedule. Thus, 65 we conclude that the degree of non-normalization is not a good indicator of the quality of approximation. 0.0 0.2 0.4 0.6 0.8 1.0 0.96 0.97 0.98 0.99 1.00 Figure 3.10: The norm of the WKB approximation at the lowest order as a function of the evolution parameter r for different schedules. g 2 represents the optimal schedule, but does not maintain normalization. Note thatg 3 is significantly more sub-normalized that than the rest, and dips down to about 0.7 (not shown). Here n = 6 and t f = 60. Second, we consider renormalization of the WKB approximation, by which we mean: |χ rWKB i≡ |χ WKB i q hχ WKB |χ WKB i , (3.35) where we have denoted the renormalized WKB approximants as|χ rWKB i. Note that it is somewhat ad hoc to normalize the approximant: a renormalization step is not part of the standard WKB approximation procedure. With this caveat, we now analyze the behavior of the WKB approximants after renormalization. First we consider the time-averaged trace-norm distance,D [Eq. (3.33)]. As shown in Fig. 3.11, renormalizing the WKB approximation does improve the approximation. However, the situation changes when we consider the threshold timescale t Th f (n) [Eq. (3.34)], shown in Fig. 3.12. We see that renormalizing the WKB approximation yields highly unphysical results for the g 0 ,g 1 , and g 2 schedules. In particular, for the g 0 and g 1 66 0 10 20 30 40 50 60 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 Figure 3.11: The difference between the time-averaged trace-norm distances for the renor- malized and unnormalized WKB approximants, for the four different schedules [Eqs. (3.19)]. Here n = 6 and t f = 60. A negative value means that the unnormalized WKB approx- imation deviates more from the numerically exact solution than the renormalized WKB approximation. schedules, we see that the renormalized WKB approximation predicts a scaling for t Th f (n) that decreases with problem size n. For the g 2 schedule we see a scaling ofO(1). So, while the unnormalized WKB predicted the correct scaling for the g 2 schedule, the renormalized WKB does not retain that feature. On the other hand, for the g 3 schedule, we see that the renormalized WKB predicts the correct scaling ofO(2 n/2 ), fixing the incorrect scaling of the unnormalized WKB approximation for that schedule. 3.6 Summary and Conclusions We have presented a straightforward technique to obtain an analytic asymptotic approxi- mation to slowly evolving 2-level systems by adapting the WKB method. We have applied it to a problem that is motivated by adiabatic quantum computation: the Hamiltonian Grover search problem. This problem has a physical Hilbert space of dimension 2 n , but is effec- tively constrained to a 2-dimensional subspace. We have seen that in this case when n = 1, 67 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ■ ◆ ▲ 0 5 10 15 20 25 30 -15 -10 -5 0 5 10 15 20 Figure 3.12: The time required to achieve a final ground state probability of 0.95 for the interpolationsdefinedinEqs.(3.19)](logscale),usingtherenormalizedversionWKBapprox- imation at the lowest order. The straight lines represent fits ofO(2 n/2 ),O(1),O(2 −n/2 ), and O(2 −n/2 ) respectively. the WKB method provides good approximations, especially to the population dynamics. We saw that the WKB approximation can capture fluctuations in the population that are absent in the purely adiabatic (ground state) evolution. Thus, the WKB is quasi-adiabatic. For completeness, in Appendix B.1, we compare our WKB approximation to the asymptotic expansion method of Hagedorn and Joye [84], and show that the latter misses the oscillations that are captured by the quasi-adiabatic WKB expansion. Turning to the Grover problem with n > 1 and with different interpolation schedules, we observed that the WKB approximation yields meaningful results only for the schedule which slows quadratically with the ground state gap. For this g 2 (r) schedule, the WKB approximation is able to capture the scaling with n of t Th f , and hence recovers the quantum speedup, even at the lowest approximation order. On the other hand, for the schedules that slow down more slowly than quadratically with the gap, the WKB approximation violates normalization and predicts an impossible faster-than-quadratic quantum speedup for the Grover problem. 68 We also saw that, using the time-averaged trace-norm distance, for the g 2 (r) schedule, the WKB approximation always does better than a naive adiabatic approximation, and the advantage becomes more pronounced for larger system sizes and shorter evolution times. Turning to the question of the whether the norm of the WKB approximation is a good signal of approximation quality, we saw that this is not the case. Further, we saw that enforc- ing renormalization by fiat gives mixed results. On the one hand, it lead to an improvement in the time-averaged trace-distance and also gave the right predictions for scaling of the threshold timescale for the g 3 schedule. On the other hand, it gave incorrect predictions for the scaling of the threshold timescales for the g 0 ,g 1 , and g 2 schedules, especially degrading the prediction for the g 2 schedule compared to its unnormalized counterpart. An interesting problem for future work is to provide more rigorous justifications and explanations for when and where the WKB approximation provides good approximations. With a better understanding of the regions where the WKB approximation performs well, and if the approximation errors are better controlled, the method could be used in the design of quantum control protocols to implement quantum gates [89, 90, 91, 92, 93]. 69 Chapter 4 Gaussian-noise-induced adiabaticity in quantum annealing for glued trees 4.1 Introduction In this work, we consider the quantum annealing (QA) algorithm for solving the glued- trees problem. The algorithm was proposed in the closed-system setting. Here, we consider some open-system models. The glued-trees problem is an oracle problem. This problem was first introduced in Ref. [94], where it was shown that any classical algorithm must necessarily take exponential time, while a quantum walk algorithm was demonstrated that solved the problem in polynomial time. Hence, it showed an exponential quantum (oracle) speedup. Later, in Ref. [95], developed the QA algorithm and demonstrated that it also solved the problem in polynomial time. This is the only known QA algorithm for which an exponential speedup is known. It is not an adiabatic algorithm: the system goes to first-excited state and return back to the ground state. We consider the behavior of this algorithm under some noise models. These noise models add a time-independent random matrix with Gaussian entries to the Hamiltonian evolution. We consider two dichotomies of noise models. One dichotomy is between noise models which induce long-range interactions among distant nodes in the graph and noise models which only induce interactions between nearest-neighbor nodes. The other dichotomy is between noise models which break a certain reflection symmetry in the spectrum and noise models which retain the reflection symmetry. 70 These noise models are motivated by three concerns. First, they offer ways to perturb features of the problem that are considered explanatorily relevant to the performance of the algorithm. This will become clearer later, but the main idea can be illustrated as follows. The QA algorithm described in Ref. [95] works reliably because the spectrum is symmetric upon reflection about the middle of the evolution. This symmetry guarantees that if the system goes to the first-excited state in the first half of the evolution, it will then return back to the ground state in the second half. Therefore, a perturbation that breaks this reflection symmetry offers a control knob to explore the importance of this symmetry. The second motivation: Given that this is an oracle problem, in order to obtain physical noise models, one needs to consider physical realizations of the oracle. But oracles are generically unrealizable as local Hamiltonians. Thus, in the absence of physical implementations, we guess that the noise is Gaussian at the oracle level. The third and final motivation: We choose these noise models because they are numerically and analytically tractable. This chapter is organized as follows. In Sec. 4.2, we describe the glued-trees and the QA algorithm that solves it. In Sec. 4.3, we describe the noise that we study. In Sec. 4.4, we present numerical results on how the performance of the algorithm changes under the different noise models. In Sec. 4.5, we provide an explanation for these results and we conclude in Sec. 4.6. 4.2 The Glued-Trees problem Consider two identical perfect binary trees, of depth n, glued together as depicted in Fig. 4.1. The gluing is done such that each leaf on one tree is joined to two leaves on the other tree, and vice versa. This ensures that every vertex in the graph, except the two root vertices, have degree 3. One root node is called the ENTRANCE vertex and the other root 71 node is called the EXIT vertex. The goal of the problem is to start from the ENTRANCE vertex and find the EXIT vertex. 1 j=0 j=1 ... ... j = n j=2n+1 ... ... j = n+1 Figure4.1: Thegraphstructureofthegluedtreesproblem. j = 0, 1, 2,...,n,n+1,..., 2n+1 indexes the different columns of the graph, with j = 0 being the ENTRANCE vertex and j = 2n + 1 being the EXIT vertex. TheoracleistheadjacencymatrixAofthisgraph. Intheclassicalcase, givenaparticular vertex, the oracle returns the vertices connected to the given vertex. In the quantum case, given a particular vertex, the oracle returns a uniform superposition over all the vertices connected to it. This implies that the vertices must have names. Since n is the depth of one of the binary trees, the total number of vertices in the glued trees is 2 n+2 − 2 =O(2 n ), which is the minimum number of distinct names we need. Therefore, the entire graph can be named using (n + 2)-length bit-strings. But this naming system is insufficient to make the problem hard for classical algorithms. Instead, to prove classical hardness, we need 1 That all the vertices, except the ENTRANCE and the EXIT vertices, have equal degree is crucial to avoid the easy solution of this problem by a “backtracking” classical random walk. See Ref. [94]. 72 there to be exponentially more names than vertices, so we will choose a naming system with 2n-length bit-strings that are assigned randomly to the different vertices. Under this naming system, in Ref. [94], it was shown that any classical algorithm that makes fewer than 2 n/6 queries to the oracle, will not be able to find the EXIT vertex with probability greater than 4× 2 −n/6 . This means that a classical algorithm will at least take time Ω(2 n/3 ), because to get a high success probability it would have to repeat a procedure that takes time 2 n/6 , 2 n/6 times. On the other hand, in Ref. [94], it was also shown that a quantum walk algorithm which starts from the ENTRANCE vertex, and evolves under the Hamiltonian equal to the adjacency matrix of the graph, can find the EXIT vertex with probabilityO( 1 n ) if the algorithm is run for times chosen uniformly at random from the interval [0,O(n 4 )]. This means we can can get a probability of success arbitrarily close to 1 by simply repeating the algorithmO(n) times, and therefore the algorithm will take at mostO(n 5 ) time. This yields an exponential speedup over the classical algorithm. 2 4.2.1 The annealing algorithm Let us now turn to the QA version of the above algorithm presented in Ref. [95]. In this case, the initial Hamiltonian is a projector onto the ENTRANCE vertex: H 0 = −|ENTRANCEihENTRANCE|. The final Hamiltonian is a projector on to the EXIT ver- tex: H 1 =−|EXITihEXIT|. We then interpolate between these projectors while turning on and off the adjacency matrix A: H(s) = (1−s)αH 0 −s(1−s)A +sαH 1 , (4.1) with s = t t f , where t is the physical time and t f is the total evolution time. Therefore, s∈ [0, 1]. Also, 0<α< 1 2 is a constant. 2 The naming system is irrelevant to the quantum algorithm. 73 In Ref. [95], it was shown that the above interpolation ends with sufficiently high- probability in the the ground-state of H 1 , the EXIT vertex, if t f =O(n 6 ). The quantum dynamics are confined to the subspace spanned by the column basis whose elements are |col j i = 1 q N j X i∈column j |a(i)i, (4.2) where N j = 2 j 0≤j≤n 2 2n+1−j n + 1≤j≤ 2n + 1 (4.3) is the number of vertices in columnj (there are 2n+2 columns in total). It is straightforward to show (see Appendix C.1) that in the column basis, the matrix elements of the Hamiltonian [Eq. (4.1)] are H 0,0 =−α(1−s) (4.4a) H j,j+1 =H j+1,j =−s(1−s) for j6=n (4.4b) H n,n+1 =H n+1,n =− √ 2s(1−s) (4.4c) H 2n+1,2n+1 =−αs. (4.4d) Reflection symmetry This Hamiltonian is invariant under the composition of two transformations, which together we will call the reflection symmetry. The first transformation is the reflection of the graph around the central glue. In the column basis, this is represented by the permutation matrix P which has 1’s on the anti-diagonal and 0’s everywhere else: P ij =δ i,2n+1−j , i,j∈{0, 1, 2,..., (2n + 1)}. (4.5) 74 The second transformation iss7→ (1−s): the reflection of the evolution parameters around s = 0.5. The reflection symmetry is the invariance of the Hamiltonian [Eq. (4.1)] under the com- position of these two transformations: H(s) =PH(1−s)P. (4.6) One consequence of the reflection symmetry is that the spectrum of the Hamiltonian is symmetric under the second transformation s7→ (1−s) alone. This is because Eq. (4.6) implies that s7→ (1−s) corresponds to effectively conjugating the Hamiltonian by P, and since P is unitary, the spectrum is unchanged. Therefore, E k (s) =E k (1−s) for k∈{0, 1, 2,..., (2n + 1)}. (4.7) Moreover, another consequence of the symmetry is that if|φ k (s)i is the k-th eigenstate of H(s), then H(s)|φ k (s)i =E k (s)|φ k (s)i (4.8a) =⇒ PH(s)P † P|φ k (s)i =E k (s)P|φ k (s)i (4.8b) =⇒ H(1−s)(P|φ k (s)i) =E k (s)(P|φ k (s)i) (4.8c) =⇒ H(1−s)(P|φ k (s)i) =E k (1−s)(P|φ k (s)i) (4.8d) =⇒ |φ k (1−s)i =P|φ k (s)i. (4.8e) Together, Eqs. (4.7) and (4.8e) imply that H(1−s) has the same eigenvalues as H(s) and that the eigenvectors ofH(1−s) are the reversed-in-column-basis eigenvectors ofH(s). 75 4.2.2 Dynamics As shown in Ref. [95], the key feature of the Hamiltonian that results in polynomial time performance is the spectrum, depicted in Fig. 4.2. The evolution that solves the problem in polynomial time is as follows. The system starts in the ground state of the Hamiltonian at s = 0 (i.e., the ENTRANCE vertex). In the optimal evolution, the system then transitions to the first excited state at the first exponentially small gap (between s 1 and s 2 ). Then, it stays in the first excited state and does not transition to the second excited state because of the polynomially large gap between the first and second excited states. Finally, it returns back down to ground state through the second exponentially small gap (between s 3 ands 4 ). It was shown that at the end of annealing evolution described above, we get the EXIT vertex with high probability, as long as the evolution time t f is chosen to scale asO(n 6 ). 0 0.2 0.4 0.6 0.8 1 s -0.5 -0.4 -0.3 -0.2 -0.1 0 Eigenvalues s 1 s 2 s 3 s 4 10 / 2 n/2 21 c 0 /n 3 10 c/n 3 10 c/n 3 E 2 (s) <latexit sha1_base64="0iswiHiJIy2rulLXl8SfmQOpzls=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXspuKeixIILHCvYD2qVk02wbm02WJCuUpf/BiwdFvPp/vPlvTLd70NYHA4/3ZpiZF8ScaeO6305hY3Nre6e4W9rbPzg8Kh+fdLRMFKFtIrlUvQBrypmgbcMMp71YURwFnHaD6c3C7z5RpZkUD2YWUz/CY8FCRrCxUud2WK/qy2G54tbcDGideDmpQI7WsPw1GEmSRFQYwrHWfc+NjZ9iZRjhdF4aJJrGmEzxmPYtFTii2k+za+fowiojFEplSxiUqb8nUhxpPYsC2xlhM9Gr3kL8z+snJrz2UybixFBBlovChCMj0eJ1NGKKEsNnlmCimL0VkQlWmBgbUMmG4K2+vE469Zrn1rz7RqXZyOMowhmcQxU8uIIm3EEL2kDgEZ7hFd4c6bw4787HsrXg5DOn8AfO5w9Wbo5C</latexit> <latexit 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We choose a small n so that the exponentially small gaps are visible. 76 SinceO(n 6 ) is an analytically derived upper-bound, we expect the scaling obtained via numerical simulations to be better. Indeed this is the case. To see this, let’s look at the threshold timescale t Th f (n)≡ min{t f :p GS (t f )≥p Th }, (4.9) where p Th is a threshold probability we choose (in the rest of this chapter, we choose p Th = 0.95). In Fig. 4.3, we plot the scaling of t Th f (n) for the QA algorithm for the glued trees problem. We can see that we get a scaling ofO(n 2.86 ), which is significantly faster that O(n 6 ). 0 1 2 3 4 5 0 5 10 15 20 Figure 4.3: This is a plot of the threshold time-scale t Th f (smallest anneal time at which we get a probability of 0.95 for reaching the EXIT vertex) vs. n for the (noiseless) quantum annealing glued trees algorithm. We see that the threshold timescale scales asO(n 2.8613 ). It is also instructive to examine what the p GS (t f ) function looks like. This is exhibited for the casen = 10 in Fig. 4.4. Forn = 10, the threshold timescale ist Th f (10) = 1690. 3 This 3 A note on units. We work here in units where~ = 1 and the timescales should be compared to the scales of the gaps of the problem, which are determined by the scales of the matrix elements of the Hamiltonians, which areO(1). 77 corresponds to the second peak in the oscillations. So, in general, the QA algorithm works by being in a region of the p GS vs. t f curve before adiabaticity is achieved. It is also instructive to examine what the dynamics look like at different evolution timescales. We examine the populations in the instantaneous ground state, first excited state, and the second excited state as a function of the evolution parameter s for the prob- lem size of n = 4. [Fig. 4.5]. We see that at small evolution times, the evolution is close to optimal: The population starts off in the ground state, enters the first excited state at the first exponentially small gap, and returns to the ground state at the second exponentially small gap. At longer evolution times, we see that the dynamics is closer to adiabatic, with some interesting fluctuations that arise around the exponentially small gaps. In Fig. 4.5, we have chosen a small problem size,n = 4, so that we can simulate problem sizes large enough to reach adiabaticity. For larger problem sizes, the timescale at which adiabaticity is reached gets very large and numerical simulations start to fail. 0 1000 2000 3000 4000 5000 0 0.2 0.4 0.6 0.8 1 Figure 4.4: Probability of finding the ground state at the end of evolution as a function of t f for the noiseless glued-trees quantum anneal at problem size n = 10. 78 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 (a) 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 (b) 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 (c) Figure 4.5: Populations in the instantaneous ground state, first excited state, and second excited state as a function of the anneal parameter s for the glued-trees problem at size n = 4, without any added noise. (a) t f = 268; (b) t f = 1000; and (c) t f = 4000. 4.3 Noise Broadly speaking, one can consider two kinds of noise models when it comes to quantum algorithms. One kind is a noise model that applies to a particular physical realization, or family of possible physical realizations, of the quantum algorithm. A good example of this is the widely studied bosonic-bath noise models for Hamiltonian systems. The other kind of noise model is phenomenological. It ignores the details of how the algorithm might be implemented and instead posits some general abstract noise model. This method is especially well-suited to oracle algorithms. This is because oracles are typically very difficult to realize physically. Indeed, for the glued-trees problem we can show that the termsH 0 ,H 1 , and A in Eq. (4.1) would all need to be highly non-local and require interactions which are experimentally difficult to engineer (see Appendix C.2). Such oracle-level noise models are studied, e.g., in Refs. [96, 97, 98], for circuit algorithms, and for some quantum walk algorithms (see [99] for a review), including the quantum walk version of the glued-trees algorithm [100]. In our work, we will consider an oracle-level noise model for the QA glued-trees problem. Our noise model is inspired by a noise model due to Roland and Cerf [101]. Their noise model is a time-dependent random-matrix added to the Hamiltonian, with the entries of the 79 random matrix being distributed as white noise with a cut-off. They show that this noise doesn’t significantly affect the performance of the adiabatic algorithm as long as the cut-off frequency of the white noise is either much smaller or much greater than the energy scale of the noiseless Hamiltonian. They further explore this noise model in detail for the adiabatic Grover algorithm [7]. The noise models we study are as follows. We add a random matrix to our Hamiltonian, but our Hamiltonian is time-independent. Mathematically, this is ˜ H(t) =H(t) +h, (4.10) where H(t) is the noiseless Hamiltonian, while ˜ H(t) is the noisy Hamiltonian, and h is the added perturbation which is time-independent. (In the rest of this chapter, variables with a tilde denote quantities under the noisy evolution.) We restrict the noise matrix h to be inside the subspace spanned by the column basis. Therefore h has the same dimensions as H(t) when written in the column basis. The four noise models correspond to different ways of choosing the random matrix h, which we now specify. 4.3.1 Four noise models We will construct four noise models by selecting one branch in each of two dichomoties. The first dichotomy is between long-range and short-range noise models. The second dichotomy is between reflection-symmetric and reflection-asymmetric noise models. We now describe these four noise models in turn. 80 First consider the noise model in whichh is chosen from Gaussian Orthogonal Ensemble (GOE). This means that in any given basis, and in particular the column basis, the matrix elements of h are distributed as 4 h ij =h ji = N (0, 1), i6=j N (0, 2), i =j. (4.11) A standard way of generating such a matrix is to start with a matrix M all of whose entries are independentN (0, 1) random variables (and therefore non-Hermitian) and then set h = M +M T √ 2 . (4.12) That Eq. (4.11) is obtained from Eq. (4.12) can be seen from the fact that the addition of two independent Gaussian random variables obeys N (μ 1 ,σ 2 1 ) +N (μ 2 ,σ 2 2 ) =N (μ 1 +μ 2 ,σ 2 1 +σ 2 2 ), (4.13) combined with aN (μ,σ 2 ) = N (μ, (aσ) 2 ) (see Appendix C.3 for more details about the GOE). We will call this noise model—i.e., the model generated by simply adding a time- independent random matrix chosen from the GOE—the long-range asymmetric (LA) noise model. Long-range because h contains matrix elements which connect all columns to all columns, and asymmetric becauseh breaks the reflection symmetry of the Hamiltonian. To see that h breaks the reflection symmetry, notice that h is not invariant under conjugation with the permutation matrix P, which, together with the fact that h is time-independent, yields that ˜ H is not reflection symmetric. Let us call h in this case h LA . 4 The Gaussian random variables are denoted asN(μ,σ 2 ), with μ being the mean of the Gaussian and σ the standard deviation. 81 Next, we will consider what we call the long-range symmetric (LS) noise model. This noise model preserves the reflection symmetry of the problem. To generate this noise model, we first pick a matrix from the GOE; call it h. Then we reflection-symmetrize it: h LS ≡ h +PhP √ 2 . (4.14) Nowh LS is manifestly reflection symmetric and therefore so is ˜ H =H +h LS . Note thath LS still contains terms connecting distant columns, and hence is deemed long-range. From the above definition, we can check that the matrix elements of h LS as distributed as h ij =h ji (4.15) =h (2n+1)−i,(2n+1)−j =h (2n+1)−j,(2n+1)−i = N (0, 1), i6=j N (0, 2), i =j. The reflection symmetry of h LS implies that the spectrum of ˜ H is reflection symmetric as well in this case. Those were the long-range noise models. We next turn to the short-range noise models. These noise-models only connect neighboring columns in the glued-trees graph. Again, we examine both short-range asymmetric (SA) and short-range symmetric (SS) noise models. In the SA model, the perturbation is (h SA ) ij = N (0, 1) |i−j| = 1, 0 otherwise. (4.16) And in the SS model, we have h SS = h SA +Ph SA P √ 2 , (4.17) 82 which preserves the reflection symmetry of the Hamiltonian. We remark that the parameter controlling the strength of the noise , can be absorbed into the standard deviations of the Gaussian random variables: e.g., N (0, 1) =N (0, 2 ). Therefore, the larger the noise, the greater the spread of the Gaussians according to which the matrix elements are drawn. 4.4 Noisy glued-trees: Results from numerical simula- tions We will simulate the Schrödinger evolution i~ d dt | ˜ ψ(t)i = ˜ H(t)| ˜ ψ(t)i (4.18) with the added noise for the different noise models and then study the success probability. E.g., consider the LA model. We will simulate the Schrödinger evolution according ˜ H(t) = H(t) +h LA for many different instances of the random matrix h LA which is distributed according to Eq. (4.11). We will then study p GS [t Th f (n)]≡ hφ 0 [t Th f (n)]| ˜ ψ[t Th f (n)]i 2 , (4.19) which is probability of finding the EXIT vertex at the end of the anneal, or the success probability. Further, we have chosen the annealing timescale t f to be equal to threshold timescale of the noiseless algorithm, i.e. the timescales depicted in Fig. 4.3. We do this because it is natural to imagine that one operates the algorithm in the regime in which the noiseless algorithm succeeds. Note that this probability is a random variable whose value will depend on the specific noise realization; we will, in particular, focus on the median of 83 this random variable. The median p GS will depend on the noise model, the strength of the noise , and the problem size n. We will study its variation with respect to all three. In Fig. 4.6, we plot the median success probability for the four different noise models specified in Sec. 4.3 as a function of problem sizen. (Recall thatn is the depth of one of the binary trees in the glued trees.) First, notice that there is no significant advantage seen in the symmetric noise model when compared to the asymmetric noise model, for both the long-range and short-range models. This is surprising, because, prima facie, one might think that an important part of the reason why the QA algorithm succeeds for the noiseless case is that the symmetry of spectrum allows a transition to the first-excited in the first half of the spectrum to be followed by a transition back down to the ground state in the second half of the spectrum. ButFig.4.6suggeststhatthatmightnotbethewholestory. Notethoughthatthesymmetry is not entirely unhelpful: the symmetric noise does slightly outperform the asymmetric case. Next, one remarkable feature of Fig. 4.6 is that for, large enough n, i.e. for n& 13, we see that the success probability is non-monotonic with respect to the strength of the noise . This can be seen more clearly in Fig. 4.7. We see that, as expected, there is a fall in the success probability from the = 0 case to the = 10 −3 case, but, surprisingly, there is a rise from the = 10 −3 to = 10 −2 . For higher values of , the probability falls off, again as expected. We explain the counterintuitive behavior—i.e. the rise from the 10 −3 to 10 −2 —in the next section. Wealsoexamine, foragivennoisemodelandagivennoisestrength, whetherthequantum speedupseeninthenoiselessalgorithmisretained, orwhetheritislost. Supposethep GS (t Th f ) vs. n curve declines faster than 2 − n 3 . This would mean that one has to repeat the algorithm at least 2 n 3 times in order to recover a probability of success comparable to the noiseless algorithm. But 2 n/3 is the best possible time that the classical random walk can achieve (see Sec. 4.2 and Ref. [94]). Therefore, in this case, the quantum speedup over the classical algorithmwouldbelost. Ontheotherhand,ifthedeclineinsuccessprobabilityisslowerthan 84 2 − n 3 , then we can say that a speedup over the classical algorithm is retained. But it might not be an exponential speedup: if the success probability for the noisy quantum algorithm declines as an exponential function that decreases slower that 2 − n 3 , then the speedup over the classical algorithm will be a polynomial speedup and not an exponential speedup. Thus, to examine the speedups (or lack thereof) we perform exponential fits to the p GS (t Th f ) vs. n curves. These are displayed in Fig. 4.8. For the long-range models, the fits are performed in a domain that excludes small values of n because the asymptotic behavior has not started to show yet. For the short-range models, we only perform the fits on small values of n because the exponential decay causes the values of the success probability to be very small for larger values of n. The exponential fits are of the form c× 2 αn . In Fig. 4.9, we plot the scaling coefficient α as a function of the noise strength for the four different noise models. As discussed in the previous paragraph, if the α <−1/3, then we lose the quantum speedup. As we can see from Fig. 4.9, the quantum speedup is retained for the long-range models for all the values of that we study, while the quantum speedup is lost for the short-range models for & 10 −1.75 . 4.5 Noise-induced adiabaticity Two things stand out in the results presented in the previous section. The first is that for the long-range models, there is a rise in success probability from ≈ 10 −3 to ≈ 10 −2 . The second is that the long-range models retain their speedup for a large range of noise strengths, when compared to the short-range models. In this section, we provide explanations for this behavior. One phenomenon helps explain both these behaviors: In the long-range models, pertur- bation theory implies that the noise typically increases the spectral gap of the Hamiltonian. First, we argue for this claim. Then we show how it explains the behaviors. 85 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 (a) 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 (b) 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 (c) 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 (d) Figure 4.6: The median success probability, p GS , at the end of an evolution of duration t Th f (n) as a function of n for = 0, 10 −3 , 10 −2 , 5× 10 −2 , 10 −1 , 5× 10 −1 . = 0 (thin blue line in all the plots) is the noiseless evolution, witht Th f (n) chosen so that the success probability for the noiseless evolution is just above 0.95. The error bars are obtained by bootstrap sampling over 300 realizations of the noise. (a) The long-range symmetric noise model. (b) The long-range asymmetric noise model. (c) The short-range symmetric noise model. (d) The short-range asymmetric noise model. Consider what happens to the spectral gap at first-order perturbation theory when we add long-range noise. We look at the asymmetric case. Let E (1) 0 (s) and E (1) 1 (s) be the first-order corrections to the ground and first excited states respectively. We know that E (1) 0 (s) =hφ 0 (s)|h LA (s)|φ 0 (s)i (4.20) ∼N (0, 2), (4.21) 86 -4 -3.5 -3 -2.5 -2 -1.5 -1 0 0.2 0.4 0.6 0.8 1 Figure 4.7: Median success probability vs. the log (base 10) of the strength of the noise for the LS noise model for several, larger values of the problem size n. As can be seen there is a fall, then a rise, and then a fall again in this success probability as a function of . where|φ 0 (s)i represents the ground state of the unperturbed (noiseless) problem. We have used the fact thath LA (s) is drawn from the GOE, and that the diagonal elements of a GOE matrix are distributed with variance 2 [see Eq. (4.11)]. A parallel calculation will show that E (1) 1 (s)∼ N (0, 2). Putting these together, we can approximate the spectral gap of the perturbed Hamiltonian using first-order perturbation theory as follows. ˜ Δ LA (s) = ˜ E 1 (s)− ˜ E 0 (s) (4.22a) ≈ [E 1 (s) +E (1) 1 (s)]− [E 0 (s) +E (1) 0 (s)] (4.22b) = Δ(s) +N (0, 4), (4.22c) where Δ(s) is the spectral gap of the noiseless problem. To obtain the last line, we used the fact that the difference of two normal random variables obeys N (μ 1 ,σ 2 1 )−N (μ 2 ,σ 2 2 ) =N (μ 1 −μ 2 ,σ 2 1 +σ 2 2 ). (4.23) 87 5 10 15 20 -4 -3 -2 -1 0 (a) 5 10 15 20 -5 -4 -3 -2 -1 0 (b) 1 2 3 4 5 -8 -6 -4 -2 0 (c) 1 2 3 4 5 -8 -6 -4 -2 0 (d) Figure 4.8: Asterisks represent the base-2 logarithm of the median ground-state success probability. Straight lines are exponential fits, of the formO(2 αn ), to the median of the success probability, p GS (t Th f ). This is done for ∈{10 −3 , 10 −2 , 5× 10 −2 , 10 −1 , 5× 10 −1 } for the short-range models. For the long-range models, the set of values of is the same, except that we omit the = 10 −2 case since it shows anomalous behavior (i.e., a rise and a fall) which doesn’t fit an exponential decay. We display the behavior of the scaling coefficient α as a function of the noise in Fig. 4.9. (a) The long-range symmetric noise model. (b) The long-range asymmetric noise model. (c) The short-range symmetric noise model. (d) The short-range asymmetric noise model. Using the fact that Δ(s) scales either as inverse polynomially or inverse exponentially (shown in Ref. [95]; see also Fig. 4.2), and the fact that the random variableN (0, 4) has no scaling 88 -3 -2.5 -2 -1.5 -1 -0.5 0 -1.5 -1 -0.5 0 Figure 4.9: The scaling coefficient α as a function of the base-10 logarithm of the strength of the noise for the four different noise models. The scaling coefficient α is obtained by performing an exponential fit of the formO(2 αn ), to the median success probability, p GS (t Th f ) vs. n curves (shown in Fig.4.8). The dashed horizontal line atα =−1/3 represents the scaling coefficient below which the speedup, over the best possible classical algorithm to solve the glued-trees problem, is lost. Thus, we see that the long-range models retain the speedup for the range of that we consider, while the short-range models lose the speedup for & 10 −1.75 . with problem size we can conclude that, typically, at the first order of perturbation theory, the perturbed problem hasO(1) gap. A similar argument will establish that the gap isO(1) also for the case of the LS model. Note that no such argument is straightforwardly available for the short-range models. The factthathφ 0 (s)|h LA (s)|φ 0 (s)i is distributedas a scalarGaussian random variable derives from the fact that h LA is drawn from the GOE, which in turn implies that all the matrix elements of h LA are populated. Let us see how the perturbative lifting of the spectral gap helps explain the behavior seen in Fig. 4.7. For ≈ 0, the algorithm succeeds because of the diabatic transitions from the ground state to the first excited state and then back down to the ground state (see Fig. 4.10 for what the noiseless dynamics look like). As we increase , the slight lifting of the gap 89 interferes with these diabatic transitions leading to a somewhat smaller success probability. This can be seen in Fig. 4.11, which shows a typical realization of the noisy spectrum and the noisy dynamics under the LS noise model at = 10 −3 forn = 20: In Fig. 4.11(a), we see the slight lifting of the spectrum, while in Fig. 4.11(b), we see how the diabatic transitions are scrambled due to the noise. Then, as we increase the noise to = 10 −2 , we can see the onset of adiabaticity in Fig. 4.12, which shows a typical realization of the noisy spectrum and the noisy dynamics at = 10 −2 and n = 20: In Fig. 4.12(a) we see that the spectral gap is lifted significantly and in Fig.. 4.12(b) we see that the dynamics are very close to adiabatic. Then as we increase to values greater than 10 −2 , we see the success probability fall off because even if the dynamics were adiabatic, the noisy spectrum and eigenstates have little relationship with the noiseless spectrum and eigenstates. To corroborate this, in Appendix C.4, we show using perturbation theory that the overlap between the unperturbed ground state and the perturbed ground state decays as 1−O( 2 ), which suggests that as we increase, even if the dynamics were adiabatic, the ground state found at the end of the noisy evolution would have low overlap with the EXIT vertex. The perturbative lifting of the gap also helps explain why the long-range models are able to retain the speedup for a larger range of epsilons. Indeed, because the noise induces adiabaticity for a certain range of values of , then as long as the overlap between the unperturbed ground state and the perturbed ground state remains significant, we should expect the long-range noise models to succeed at a higher rate compared to the short-range noise models. 4.6 Conclusion We have analyzed the quantum annealing algorithm for the glued trees problem under four different noise models. The noise models are differentiated along two dichotomies: 90 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Figure 4.10: Populations for the lowest three levels in the noiseless glued trees with n = 20 and t f = 12125. long-range vs. short-range and reflection-symmetric vs. reflection-asymmetric. These noise models are oracular noise models: they add a Gaussian perturbation, of different forms, to the Hamiltonian evolution. We study the success probability—i.e. the probability finding the EXIT vertex—at the end of the Schrödinger evolutions for these different noise models. We find that the long-range noise models induce a perturbative lifting of the spectral gap which causes the dynamics to transition from diabatic to adiabatic. This leads to the long- range models retaining a quantum speedup for a wider range of noise strengths over the short-range noise models. We now list some open questions raised by the work presented in this chapter. Can the perturbation theory argument for noise-induced adiabaticity in the glued-trees problem be made more rigorous? How common is noise-induced adiabaticity in quantum annealing algorithms in general? Can the glued-trees problem be realized in a physical Hamiltonian and are there more physical noise models for it? We leave these questions for future work. 91 0.18 0.2 0.22 0.24 0.26 0.28 0.3 -0.4 -0.35 -0.3 (a) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (b) Figure 4.11: A single instantiation of the long-range symmetric noise for n = 20 and t f = 12125 with = 10 −3 . (a) The lowest three eigenstates for the noiseless and noisy Hamiltonians, zoomed in around the point at which the noiseless Hamiltonian shows the first exponentially small spectral gap. Solid lines represent the noiseless spectrum, while the dashed lines represent the noisy spectrum. We can see that the noisy spectrum has a gap larger than the noiseless spectrum (b) The populations as in the lowest three eigenstates of the noisy Hamiltonian as a function of the anneal parameters. We can see that the diabatic transitions do not happen as cleanly as in the noiseless case (shown in Fig. 4.10). 92 0.18 0.2 0.22 0.24 0.26 0.28 0.3 -0.5 -0.45 -0.4 -0.35 -0.3 (a) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (b) Figure 4.12: A single instantiation of the long-range symmetric noise for n = 20 and t f = 12125 with = 10 −2 . (a) The lowest three eigenstates for the noiseless and noisy Hamiltonians, zoomed in around the point at which the noiseless Hamiltonian shows the first exponentially small spectral gap. Solid lines represent the noiseless spectrum, while the dashed lines represent the noisy spectrum. We can see that the noisy spectrum has a gap larger than the noiseless spectrum and also larger than the gap in the case with = 10 −3 . (b) The populations as in the lowest three eigenstates of the noisy Hamiltonian as a function of the anneal parameter s. We can see that the dynamics are very close to adiabatic. 93 Chapter 5 Conclusion 5.1 Toy models, approximations, and explanations Toy modelsand approximations aretwoof themost powerful toolsin aphysicist’s toolkit. Toy models are highly idealized systems that can capture the essential physical mech- anism in a physical phenomenon. A great advantage of studying toy models is that they are tractable to numerical and analytical methods. But their true importance comes from the fact that they have explanatory power. Consider a complicated real-world physical sys- tem. If we could analyze this system numerically or analytically, we might have the ability to predict and control such a system, but we wouldn’t necessarily have understanding or insight—i.e., we wouldn’t necessarily have explanations of why the system behaves the way it does. But if we construct explanations to elucidate toy models, we may then use these explanations to understand the behavior of the complicated real-world system. Moreover, phenomena discovered while studying toy models can then be used to search for similar phenomena in other models and in real systems. As it turns out, even toy models, with all their simplicity, are rarely fully tractable. Thus, even for very simple systems, in order make progress, the physicist needs to turn to the other powerful tool in their toolkit: approximations. Approximations allow us to compute, numericallyandanalytically, quantitiesthatmightotherwisebetoocomputationallyonerous to obtain. But, just as with toy models, approximations are not just about tractability. They also offer insight and understanding. In order to perform good approximations we need an awareness of the relevant scales in the system and how they interact: the time-scales, energy- scales, length-scales, and so on. Discerning when an approximation succeeds or breaks down 94 allows us to get a picture of when certain dynamical processes are important and when they are not. In this dissertation, we have extensively explored toy models in quantum annealing, performed approximations to their dynamics, and provided explanations for some of this behavior. We now review what we have done. In Chap 2, we studied the perturbed Hamming weight oracle problems. These are toy models with the permutation symmetry permitting them to be numerically and analytically tractable. They provided a model for us to understand quantum tunneling in QA. These toy models also allowed us to discover a new phenomenon in QA, which we call diabatic cascades, and more generally, illustrate the importance of the diabatic regime in QA. In Chap 3, we studied adiabatic Grover search, which we can be thought of as toy model because the dynamics is constrained to a two-dimensional Hilbert space. To this model, we applied the Wentzel-Kramers-Brillouin (WKB) approximation in an effort to get analytical expressions to the Schrödinger evolution as an asymptotic expansion in powers of the inverse anneal time 1/t f . This allowed us to explore the physics in the near-adiabatic regime. It also highlighted how the approximation quality was sensitive to the schedule being employed. In particular, the optimal schedule also provided the best approximation quality. Finally, in Chap 4, we turned to the glued-trees problem. This is a toy problem with a significant amount of symmetry: Its dynamics are restricted to the column-basis and it has a certain reflection symmetry. It highlights how by allowing diabatic transitions to the first- excited state, we can achieve quantum speedups using quantum annealing. We analyzed different phenomenological noise models for the glued-trees problem and discovered that long-range noise allows for better performance than short-range noise. Further, we gave a perturbation theory argument to support the claim that this is because of the noise inducing adiabaticity in the dynamics. 95 5.2 Open questions and directions for future work The development of algorithms and of methods to analyze algorithms has been slower for quantum annealing and adiabatic algorithms compared to the circuit model. Prima facie, this is puzzling because physicists have been working on quantum systems with continuous- time dynamics since the advent of quantum mechanics, and moreover, have been employing tools to analyze continuum systems since the beginning of physics. So it seems surprising that when faced with the new domain of quantum computing, analyses of discrete systems have been much more fruitful. We diagnose the problem as follows. Physicists have tended to work in two regimes. The first is the regime of very small system sizes, such as a single particle or a few particles. The other is the regime of the very, very large: the thermodynamic limit. We have gotten very good at employing our tools to understand physics in these regimes. But computer science occupies an interesting intermediate zone: it is interested in scaling with system size; it asks how finite but large systems behave. This is the question that computer scientists have thought about since the beginning of theoretical computer science from Alan Turing onwards. So, it is to be expected that their tools are better at coming up with and analyzing algorithms. 1 But we think the physicists’ tools still have a lot to contribute. One way to view this dissertation is as an attempt to better advance the use of continuum tools—path-integrals, theWKBapproximation, randommatrices—andapplythemtocomputationalproblemsand to answer computer science questions about scaling, algorithmic performance, and speedups. Webelievethatthesetoolsareverypowerfulandweareonlybeginningtoscratchthesurface of what they are capable of in computer science applications. We now sketch some potential directions for future research along these lines. 1 In this this section we use a caricature of physicists and computer scientists, but we think it’s a useful approximation. 96 1. Tunneling is often understood in the path-integral literature using instantons. But the use of instantons is usually non-rigorous and difficult to control well enough to obtain precise scaling results (see Ref. [71] for an attempt). Can we develop a mathematically rigorous and precise version of the instanton calculus? Can we use it to obtain the scaling for the spectral gap for quantum annealing algorithms? Especially, can we use it for systems that are intractable using other techniques? 2. Generally, in quantum circuit algorithms, once an algorithm is specified, it is relatively easy to estimate its asymptotic scaling: just count the number of gates in the circuit. On the other hand, in adiabatic and annealing algorithms, after the specifying the algorithm, it can be very difficult to estimate its asymptotic scaling. This is because one typically needs to bound the spectral gap or quantify other spectral features in order to get at the scaling. Why is there this discrepancy between the circuit and annealing models? Can we come up with a simple way to estimate the gap scaling in general? If not, why not? Physicists have long studied the spectral gap as part of the theory of quantum phase transitions. Does this theory help illuminate this problem? Recently there was the discovery of that problem of deciding whether a quantum system is gapped or gapless is undecidable [102]. Does this result have any bearing on the problem stated here? 3. We discussed two important diabatic algorithms in this dissertation: diabatic cascades and the glued-trees algorithm. These algorithms rely on striking patterns that arise in the spectra of these systems. What kinds of mathematical tools might we use to study general features of spectra? Is there a classification of spectral structures, that is, some natural of way of organizing avoided level-crossings, their relative locations, and their scaling with system size? One interesting hint here is the theory of Excited State Quantum Phase Transitions (ESQPT) which attempts to classify spectral structures according to the symmetries of the system [103]. Can we connect this theory with 97 quantum annealing? More precisely can we use ESQPTs to obtain estimates on the performance of algorithms? 4. A disadvantage that continuous quantum algorithms suffer from relative to discrete quantum algorithms is that it is often hard in the continuous case to write down, ana- lytically, the state at an intermediate stage of the evolution. In the circuit model, it is straightforward: simply apply the unitary operators and see what the state is. But in the continuous case, even for a problem as simple as Grover search, it is very hard to even write down analytic approximations to the time-evolved state, let alone the exact state. Why is this? Can we do better and come up with some general method to compute or approximate the state during the evolution? What happens when we point the entire arsenal of approximation methods for differential equations towards quan- tum annealing? To illustrate, here is an approach that might be fruitful: The WKB approximation has a deep connection with the theory of renormalization group [104]. Can we employ this connection to better develop a way to getting approximations to quantum annealing dynamics at large system sizes? 5. One way to increase the stock of quantum algorithms within the quantum annealing framework is to just borrow the algorithms already developed in the circuit model. This might be done by building a translation manual between circuit algorithms and annealing algorithms. This has been done, at least in principle, for non-oracle algo- rithms in work that shows the equivalence between the adiabatic and circuit models of quantum computation (first done in [105]). But for oracle algorithms this technique does not apply—at least, not straightforwardly. There has been fascinating work done in Ref. [106] towards the goal of connecting circuit and adiabatic query algorithms, and uses as a crucial tool one of physicists’ oldest workhorses: Ehrenfest’s theorem. This is a line of research that seems worthy of exploration. 98 We end with a clarion call to theoretical physicists and applied mathematicians of all stripes: Your tools are immensely powerful, and if you just turn your attention towards analog quantum computation, you will find a world full of interesting problems waiting for you to explore. 99 Appendix A Appendices to Chapter 2 A.1 Review of the Hamming weight problem and Reichardt’s bound for PHWO problems Here we closely follow Ref. [51]. A.1.1 The Hamming weight problem We review the analysis within QA of the minimization of the Hamming weight function f HW (x) =|x|, which counts the number of 1’s in the bit string x. This problem is of course trivial, and the analysis given here is done in preparation for the perturbed problem. For the adiabatic algorithm, we start with the driver Hamiltonian, H D = 1 2 n X i=1 (I i −σ x i ) = n X i=1 |−i i h−| , (A.1) which has|+i ⊗n as the ground state. The final Hamiltonian for the cost function f HW (x) is H P = 1 2 n X i=1 (I i −σ z i ) = n X i=1 |1i i h1| , (A.2) which has|0i ⊗n as the ground state. 100 We interpolate linearly between H D and H P : H(s) = (1−s)H D +sH P ; s∈ [0, 1] (A.3) = n X i=1 1 2 1−s −(1−s) −(1−s) 1−s i + 0 0 0 s i , (A.4) = 1 2 n X i=1 [I− (1−s)σ x i −sσ z i ]≡H i (s) . (A.5) We note thatH i (s) in Eq. (A.5) is similar to a variant of the Landau-Zener (LZ) Hamiltonian with finite coupling duration [107, 108], for which the Schrödinger equation has an analytical solution, except that there it is assumed that the σ x term is constant and only theσ z terms has a (linear) time dependence over a finite interval. The analytical solution of the problem obtained in Ref. [107] is rather complicated, and for our purposes a simpler approach suffices. Since there are no interactions between the qubits, the adiabatic problem can be solved exactly by diagonalizing the Hamiltonian acting on each qubit separately. For each term, we have the energy eigenvalues E ± (s), E ± (s) = 1 2 (1± Δ(s)); Δ(s)≡ √ 1− 2s + 2s 2 , (A.6) and associated eigenvectors, |v ± (s)i = 1 q 2Δ(Δ∓s) [∓(Δ∓s)|0i + (1−s)|1i] . (A.7) The ground state of H(s) is |ψ GS (s)i =|v − (s)i ⊗n . (A.8) 101 The gap is given by, Gap[H(s)] =H(s)|v + (s)i⊗|v − (s)i ⊗(n−1) −H(s)|v − (s)i ⊗n (A.9a) =E + + (n− 1)E − −nE − (A.9b) =E + −E − (A.9c) = Δ(s) . (A.9d) The gap is minimized at s = 1 2 with minimum value Δ( 1 2 ) = 1 √ 2 . The minimum gap is independent of n and hence does not scale with problem size. Therefore we can predict an adiabatic run time to be given by, t f =O k∂ s Hk Δ 2 ! =O(n) , (A.10) where the n-dependence is solely due tok∂ s Hk (see Appendix-A.4.2). However, this is actually a loose upper bound. We next provide separate numerical and analytical arguments to demonstrate that the actual scaling for AQA isO(n 0.5 ). Numerical argument Suppose the adiabatic algorithm runs long enough so as to attain a desired success prob- ability, p 0 . Let this time be t f . Using the fact that the quantum evolution of the plain Hamming Weight problem is the evolution of n non-interacting qubits, we can express the global ground-state probability in terms of the ground-state probabilities of single qubits. So, if the single qubit ground-state probability for this run-time is p GS (t f ), then we must have p 0 =p GS (t f ) n . 102 Wefindnumerically(seeFig.A.1)thatp GS (t f )hasanenvelopethatisexcellentlyapprox- imated by: p GS (t f ) = 1− 1 t 2 f +O(t −3 f ) , (A.11) for sufficiently large t f . We therefore can write: lnp 0 =n lnp GS (t f )≈n ln 1− 1 t 2 f ! , (A.12) and upon expanding the ln, we extract a tighter scaling for our adiabatic time: t f =O(n 1/2 ) . (A.13) 50 100 150 200 t f 0 0.2 0.4 0.6 0.8 1 1− p GS ×10 -3 QA Fit: 0.999757 t 2 f − 19.4683 t 4 f Figure A.1: Ground state probability for a single qubit for different total time t f , evolving under the Plain Hamming Weight Hamiltonian in Eq. (A.5). Analytical argument Here, we invoke a result due to Boixo and Somma [12]. This result states, 103 Theorem 1 ([12]). To adiabatically prepare a final eigenstate using a Hamiltonian evolution H(s) requires time that scales at least asO L Δ . Here L is the eigenpath length, L≡ Z 1 0 k|∂ s ψ(s)ikds, (A.14) where|ψ(s)i is the eigenpath traversed to reach the final eigenstate. We analytically computeL for the ground-state path in the plain Hamming weight prob- lem, and show that it scales asO( √ n). Since we know that in this case Δ =O(1), we conclude the adiabatic algorithm will require at leastO( √ n) time. Recall that the instantaneous ground state is [Eq. (A.8)]|ψ GS (s)i = N n i=1 |v i − (s)i, where |v i − (s)i = q 1−q(s)|0i i + q q(s)|1i i , with [Eq. (A.7)] q(s) = (1−s) 2 2Δ(Δ +s) . (A.15) Differentiating: d ds |ψ GS (s)i = n X i=1 O j6=i |v j − (s)i⊗ d ds |v i − (s)i , (A.16) so that k|∂ s ψ GS (s))ik 2 ≡h∂ s ψ GS (s))|∂ s ψ GS (s))i (A.17) =nk d ds |v i − (s)ik 2 +n(n− 1)|hv i − (s)| d ds |v i − (s)i| 2 . (A.18) The termk d ds |v i − (s)ik does not have any scaling with n, and the second term vanishes because it is equal to 1 2 d ds hv i − (s)|v i − (s)i = 0, where we use the fact that|v i − (s)i is real-valued and normalized. Thus, taking the square root on both sides and integrating from 0 to 1, we obtain the √ n scaling of L. 104 If we desire to fix the constant in front ofL, a straightforward calculation will show that Z 1 0 k d ds |v i − (s)ikds =π/4 . (A.19) A.1.2 Reichardt’s bound for PHWO problems Here we review Reichardt’s derivation of the gap lower-bound for general PHWO prob- lems, but provide additional details not found in the original proof [51]. We use the same initial Hamiltonian [Eq. (A.1)] and linear interpolation schedule as before, ˜ H(s) = (1−s)H D +s ˜ H P , and choose the final Hamiltonian to be ˜ H P = X x∈{0,1} n ˜ f(x)|xihx| , (A.20) where ˜ f(x) = |x| +p(x) l<|x|<u , |x| elsewhere , (A.21) wherep(x)≥ 0 is the perturbation. Note that here we have not assumed that the perturba- tion, p(x), respects qubit permutation symmetry. We wish to bound the minimum gap of ˜ H(s). Unlike the Hamming weight problemH(s), this problem is no longer non-interacting. Define h k ≡ max |x|=k p(x); h≡ max k h k = max x p(x). (A.22) Lemma 1 ([51]). Letu =O(l) and letE 0 (s) and ˜ E 0 (s) be the ground state energies ofH(s) and ˜ H(s), respectively. Then ˜ E 0 (s)≤E 0 (s) +O(h u−l √ l ). Proof. First note that ˜ H(s)−H(s) =s X x:l<|x|<u p(x)|xihx| . (A.23) 105 Below, we suppress thes dependence of all the terms for notational simplicity. We know that E 0 =hv ⊗n − |H|v ⊗n − i. Using this, h ˜ E 0 | ˜ H| ˜ E 0 i≤hψ| ˜ H|ψi ∀|ψi∈H. (A.24a) =⇒ ˜ E 0 −E 0 ≤hv ⊗n − | ˜ H|v ⊗n − i−E 0 (A.24b) ≤hv ⊗n − | ˜ H−H|v ⊗n − i (A.24c) =s X x:l<|x|<u p(x) hv ⊗n − |xi 2 (A.24d) =s X x:l<|x|<u p(x)q |x| (1−q) n−|x| (A.24e) ≤ X k:l<k<u h k n k ! q k (1−q) n−k , (A.24f) where n k is the number of strings with Hamming weight k, we used the fact that if we measure in the computational basis, the probability of getting outcome x is hv ⊗n − |xi 2 = q(s) |x| (1−q(s)) n−|x| , and q(s) is given in Eq. (A.15). Consider the partial binomial sum (dropping the h k ’s), X k:l<k<u n k ! q k (1−q) n−k . (A.25) Using the fact that the binomial is well-approximated by the Gaussian in the large n limit (note that this approximation requires that q(s) and 1−q(s) not be too close to zero), we can write: X k:l<k<u n k ! q k (1−q) n−k ≈ Z u l dξ 1 √ 2πσ e − (ξ−μ) 2 2σ 2 = 1 σ Z u l dξ φ ξ−μ σ ! = Z (u−μ)/σ (l−μ)/σ dt φ(t) , (A.26) 106 where μ≡ nq, σ≡ q nq(1−q) and φ(t)≡ e −t 2 /2 √ 2π . Note that σ and μ depend on n, and also on s via q(s). The parameters l and u are specified by the problem Hamiltonian, and are therefore allowed to depend on n as long as l(n)<u(n)<n is satisfied for all n. Let us define: B(s,n,l(n),u(n))≡ Z (u(n)−μ(n,s))/σ(n,s) (l(n)−μ(n,s))/σ(n,s) dt e −t 2 /2 √ 2π . (A.27) Weseekanupperboundonthisfunction. Weobservethatq(s)decreasesmonotonicallyfrom 1 2 to 0 ass goes from 0 to 1. Thus, the mean of the Gaussianμ(n,s) =nq(s) decreases from n 2 to 0. Depending on the values of l(n), u(n) and μ(n,s), we thus have three possibilities: (i) l(n) < μ(n,s) < u(n), (ii) μ(n,s) < l(n) < u(n), and (iii) l(n) < u(n) < μ(n,s). Note that (ii) and (iii) are cases where the integral runs over the tails of the Gaussian and so the integral is exponentially small. We focus on (i), as this induces the maximum values of the integral. In this case the lower limit of the integral Eq. (A.27) is negative, while the upper limit is positive. Thus, the integral runs through the center of the standard Gaussian, and we can upper-bound the value of the integral by the area of the rectangle of width u(n)−l(n) σ(n,s) and height 1 √ 2π . Hence B(s,n,l(n),u(n))≤ 1 √ 2π u(n)−l(n) σ(n,s) , (A.28a) = 1 √ 2π u(n)−l(n) q μ(n)(1−q(s)) , (A.28b) ≤ 1 √ 2π u(n)−l(n) q l(n)(1−q(s)) , (A.28c) where we have used the fact that l(n)<μ(n,s) =nq(s). Thus, we obtain the bound: ˜ E 0 −E 0 ≤O h u−l √ l ! . (A.29) 107 Lemma 2 ([51]). If ˜ H−H is non-negative, then the spectrum of ˜ H lies above the spectrum of H. That is, ˜ E j ≥ E j for all j, where ˜ E j and E j denote the jth largest eigenvalue of ˜ H and H, respectively. This can be proved by a straightforward application of the Courant-Fischer min-max theorem (see, for example, Ref. [109]). Combining these lemmas results in the desired bound on the gap: Gap[ ˜ H(s)] = ˜ E 1 − ˜ E 0 , (A.30a) ≥E 1 − ˜ E 0 , (A.30b) =E 1 −E 0 − ( ˜ E 0 −E 0 ), (A.30c) ≥ Δ−O h u−l √ l ! , (A.30d) where in Eq. (A.30b) we used Lemma 2 and in Eq. (A.30d), we used Lemma 1. Now, if we choose a parameter regime for the perturbation such that h u−l √ l =o(1), then the perturbed problem maintains a constant gap. For example, if l = Θ(n) and h(u−l) = O(n 1/2− ), for any > 0, then the gap is constant as n→∞. A.2 (Non-)Locality of PHWO problems Since the PHWO problems, including the plateau, are quantum oracle problems, they cannot generically be represented by a local Hamiltonian. For completeness we prove this claim here and also show why the (plain) Hamming weight problem is 1-local. Let r be a bit string of length n, i.e., r∈{0, 1} n and let σ r ≡σ r 1 1 ⊗σ r 2 2 ⊗···⊗σ rn n , (A.31) 108 with σ 0 i ≡I i and σ 1 i ≡σ z i . This forms an orthonormal basis for the vector space of diagonal Hamiltonians. Thus: H P = X r∈{0,1} n J r σ r , (A.32) with J r = 1 2 n Tr(σ r H P ) (A.33a) = 1 2 n X x∈{0,1} n f(x)hx|σ r |xi (A.33b) = 1 2 n X x∈{0,1} n f(x)(−1) x·r . (A.33c) Note that genericallyJ r will be be non-zero for arbitrary-weight stringsr, leading to|r|-local terms in H P , even as high as n-local. E.g., substituting the plateau Hamiltonian [Eq. (2.3)] into this we obtain: J r = 1 2 n X |x|≤l &|x|≥u |x| (−1) x·r +(u− 1) X l<|x|<u (−1) x·r . (A.34) 109 On the other hand, iff(x) =|x| (i.e., in the absence of a perturbation), the Hamiltonian is only 1-local: H P = X x∈{0,1} n |x||xihx| (A.35a) = 1 X x 1 =0 ··· 1 X xn=0 (x 1 +x 2 +··· +x n )|x 1 ihx 1 | ⊗|x 2 ihx 2 |⊗···⊗|x n ihx n | (A.35b) = n X k=1 (x k |x k ihx k |) O j6=k 1 X x j =0 |x j ihx j | (A.35c) = n X k=1 |1i k h1| O j6=k I j = n X k=1 |1i k h1|. (A.35d) A.3 Derivation of Eq. (2.14) Equation (2.14) is easily derived as follows: the probability of successively failingk times is [1−p GS (t f )] k , sotheprobabilityofsucceedingatleastonceafterk runsis 1−[1−p GS (t f )] k , which we set equal to the desired success probability p d ; from here one extracts the number of runs k and multiplies by t f to get the time-to-solution TTS. Optimizing over t f yields TTS opt , which is natural for benchmarking purposes in the sense that it captures the trade- off between repeating the algorithm many times vs optimizing the probability of success in a single run. The adiabatic regime might be more attractive if one seeks a theoretical guarantee to have a certain probability of success if the evolution is sufficiently slow. 110 A.4 Methods A.4.1 Simulated Annealing SA is a general heuristic solver [37], whereby the system is initialized in a high temper- ature state, i.e., in a random state, and the temperature is slowly lowered while undergo- ing Monte Carlo dynamics. Local updates are performed according to the Metropolis rule [110, 111]: a spin is flipped and the change in energy ΔE associated with the spin flip is calculated. The flip is accepted with probability P Met : P Met = min{1, exp(−βΔE)} , (A.36) where β is the current inverse temperature along the anneal. Note that there could be different schemes governing which spin is to be selected for the update. We consider two such schemes: random spin-selection – where the next spin to be updated is selected at random; and sequential spin-selection – where one runs through all of the n spins in a sequence. Random spin-selection (including just updating nearest neighbors) satisfies detailed-balance and thus is guaranteed to converge to the Boltzmann distribution. Sequential spin-selection does not satisfy strict detailed balance (since the reverse move of sequentially updating in the reverse order never occurs), but it too converges to the Boltzmann distribution [112]. In sequential updating, a “sweep" refers to all the spins having been updated once. In random spin-selection, we define a sweep as the total number of spin updates divided by the total numberofspins. Whenitispossibletoparallelizethespinupdates, theappropriatemetricof time-complexity is the number of sweepsN SW , not the number of spin updates (they differ by a factor ofn) [56]. However, in our problem this parallelization is not possible and hence the appropriate metric is the number of spin updates, and this is what is plotted in Fig. 2.6(b). After each sweep, the inverse temperature is incremented by an amount Δβ according to an annealing schedule, which we take to be linear, i.e., Δβ = (β f −β i )/(N SW − 1). 111 We can use SA both as an annealer and as a solver [113]. In the former, the state at the end of the evolution is the output of the algorithm, and can be thought of as a method to sample from the Boltzmann distribution at a specified temperature. For the latter, we select the state with the lowest energy found along the entire anneal as the output of the algorithm, the better technique if one is only interested in finding the global minimum. We use the latter to maximize the performance of the algorithm. A.4.2 Quantum Annealing Here we consider the most common version of quantum annealing: H(s) = (1−s) n X i=1 1 2 (I i −σ x i ) +s X x∈{0,1} n f(x)|xihx| , (A.37) where s≡ t/t f is the dimensionless time parameter and t f is the total anneal time. The initial state is taken to be|+i ⊗n , which is the ground state of H(0). The initial ground state and the total Hamiltonian are symmetric under qubit permu- tations (recall that f(x) = f(|x|) for our class of problems). It then follows that the time- evolved state, at any point in time, will also obey the same symmetry. Therefore the evo- lution is restricted to the (n + 1)-dimensional symmetric subspace, a fact that we can take advantage of in our numerical simulations. This symmetric subspace is spanned by the Dicke states|S,Mi with S =n/2,M =−S,−S + 1,...,S, which satisfy: S 2 |S,Mi =S (S + 1)|S,Mi (A.38a) S z |S,Mi =M|S,Mi , (A.38b) where S x,y,z ≡ 1 2 P n i=1 σ x,y,z i , S 2 = (S x ) 2 + (S y ) 2 + (S z ) 2 . We can denote these states by: |wi≡ n 2 ,M = n 2 −w = n w ! −1/2 X x:|x|=w |xi, (A.39) 112 where, w∈{0,...,n}. In this basis the Hamiltonian is tridiagonal, with the following matrix elements: [H(s)] w,w+1 = [H(s)] w+1,w = − 1 2 (1−s) q (n−w)(w + 1), (A.40a) [H(s)] w,w =(1−s) n 2 +sf(w). (A.40b) The Schrödinger equation with this Hamiltonian can be solved reliably using an adaptive Runge-Kutte Cash-Karp method [114] and the Dormand-Prince method [115] (both with orders 4 and 5). If the quantum dynamics is run adiabatically the system remains close to the ground state during the evolution, and an appropriate version of the adiabatic theorem is satisfied. For evolutions with a non-zero spectral gap for all s∈ [0, 1], an adiabatic condition of the form t f ≥ const sup s∈[0,1] k∂ s H(s)k Gap(s) 2 (A.41) is often claimed to be sufficient [116] [however, see the discussion after Eq. (21) in Ref. [10]]. In our casek∂ s H(s)k =kH(1)−H(0)k is upper-bounded by n; since we are considering a constant gap, the adiabatic algorithm can scale at most linearly by condition (A.41). This is true for the plateau problems. We showed in the main text that the following version of the adiabatic condition, known to hold in the absence of resonant transitions between energy levels [63], estimates the scaling we observe very well: max s∈[0,1] |hε 0 (s)|∂ s H(s)|ε 1 (s)i| Gap(s) 2 t f , (A.42) where ε 0 (s) and ε 1 (s) are the instantaneous ground and excited states in the symmetric subspace respectively. 113 The permutation symmetry is explicitly enforced only in our numerical simulations of the quantum evolution. Since, of course, we do not have quantum hardware that can implement the problems under consideration, we must explicitly enforce this symmetry in order to be able to perform numerical simulations at large problem sizes. Note that even if we were to simulate the quantum system without explicitly imposing this symmetry, the symmetry would be automatically preserved in the dynamics, and we would draw the same lessons abouttheperformanceofthequantumalgorithm(butourclassicalsimulationswouldquickly become intractable). A.4.3 Spin-Vector Dynamics Starting with the spin-coherent path integral formulation of the quantum dynamics, we can obtain Spin Vector Dynamics (SVD) as the saddle-point approximation (see, for exam- ple, Ref. [59, p.10] or Refs. [57, 58]). It can be interpreted as a semi-classical limit describing coherent single qubits interacting incoherently. In this sense, SVD is a well motivated clas- sical limit of the quantum evolution of QA. SVD describes the evolution of n unit-norm classical vectors under the Lagrangian (in units of~ = 1): L =ihΩ(s)| d ds |Ω(s)i−t f hΩ(s)|H(s)|Ω(s)i, (A.43) where|Ω(s)i is a tensor product of n independent spin-coherent states [117]: |Ω(s)i = n O i=1 " cos θ i (s) 2 ! |0i i + sin θ i (s) 2 ! e iϕ i (s) |1i i # . (A.44) 114 We can define an effective semi-classical potential associated with this Lagrangian: V SC ({θ i },{ϕ i },s)≡hΩ(s)|H(s)|Ω(s)i = (1−s) n X i=1 1 2 (1− cosϕ i (s) sinθ i (s)) (A.45) +s X x∈{0,1} n f(x) Y j:x j =0 cos 2 θ j (s) 2 ! Y j:x j =1 sin 2 θ j (s) 2 ! , with the probability of finding the all-zero state at the end of the evolution (which is the ground state in our case), as Q n i=1 cos 2 θ i (1) 2 . The quantum Hamiltonian obeys qubit per- mutation symmetry: PHP = H where P is a unitary operator that performs an arbitrary permutation of the qubits. This implies that our classical Lagrangian obeys the same sym- metry: L 0 ≡ ihΩ(s)|P d ds P|Ω(s)i−t f hΩ(s)|PH(s)P|Ω(s)i = ihΩ(s)| d ds |Ω(s)i−t f hΩ(s)|H(s)|Ω(s)i =L, (A.46) where the derivative operator is trivially permutation-symmetric. Therefore, the Euler- Lagrange equations of motion derived from this action will be identical for all spins. Thus, if we have symmetric initial conditions, i.e., (θ i (0),ϕ i (0)) = (θ j (0),ϕ j (0))∀i,j, then the time evolved state will also be symmetric: (θ i (s),ϕ i (s)) = (θ j (s),ϕ j (s))∀i,j∀s∈ [0, 1] . (A.47) 115 As we show below, under the assumption of a permutation-symmetric initial condition we only need to solve two (instead of 2n) semi-classical equations of motion: n 2 sinθ(s)θ 0 (s)−t f ∂ ϕ(s) V sym SC (θ(s),ϕ(s),s) = 0 , (A.48a) − n 2 sinθ(s)ϕ 0 (s)−t f ∂ θ(s) V sym SC (θ(s),ϕ(s),s) = 0 , (A.48b) where we have defined the symmetric effective potential V sym SC as: V sym SC (θ(s),ϕ(s),s)≡hΩ sym (s)|H(s)|Ω sym (s)i = (1−s) n 2 (1− cosϕ(s) sinθ(s)) +s n X w=0 f(w) n w ! sin 2w θ(s) 2 ! cos 2(n−w) θ(s) 2 ! , (A.49) and|Ω sym (s)i is simply|Ω(s)i with all the θ’s and ϕ’s set equal. Note that in the main text [see Eq. (2.12)], we slightly abuse notation for simplicity, and use V SC instead of V sym SC . The probability of finding the all-zero bit string at the end of the evolution is accordingly given by cos 2n (θ(1)/2). We would have arrived at the same equations of motion had we used the symmetric spin coherent state in our path integral derivation, but that would have been an artificial restriction. In our present derivation the symmetry of the dynamics naturally imposes this restriction. Note that the object in Eq. (A.45) involves a sum over all 2 n bit-strings and is thus expo- nentially hard to compute; on the other hand, the object in Eq. (A.49) only involves a sum over n terms and is thus easy to compute. Therefore, just as in the quantum case—where due to permutation symmetry the quantum evolution is restricted to the n + 1 dimensional subspace of symmetric states instead of the full 2 n -dimensional Hilbert space—given knowl- edge of the symmetry of the problem we can efficiently compute the SVD potential and efficiently solve the SVD equations of motion. 116 We also remark that the computation of the potential in Eq. (A.45) is significantly simplified if our cost function, f(x), is given in terms of a local Hamiltonian. For example, if H(1) = P i,j J ij σ z i σ z j , then: V SC ({θ i },{ϕ i }, 1) = X i,j J ij cosθ i cosθ j , (A.50) which is easy to compute as it is a sum over poly(n) number of terms. Let us now derive the symmetric SVD equations of motion (A.48). Without any restric- tion to symmetric spin-coherent states, the SVD equations of motion, for the pair θ i ,ϕ i , read: 1 2 sinθ i (s)θ 0 i (s)−t f ∂ ϕ i (s) V SC ({θ i },{ϕ i },s) = 0 , (A.51a) − 1 2 sinθ i (s)ϕ 0 i (s)−t f ∂ θ i (s) V SC ({θ i },{ϕ i },s) = 0 . (A.51b) As can be seen by comparing Eqs. (A.48) and (A.51), it is sufficient to show that: ∂ ∂θ i V SC θ j =θ,ϕ j =ϕ∀j = 1 n ∂ ∂θ V sym SC , (A.52) and an analogous statement holding for derivatives with respect to ϕ. This claim is easily seen to hold true for the term multiplying (1−s) in Eq. (A.45): ∂ ∂θ i n X i=1 1 2 (1− cosϕ i (s) sinθ i (s)) θ j =θ,ϕ j =ϕ∀j = ∂ ∂θ 1 2 (1− cosϕ(s) sinθ(s)) = 1 n ∂ ∂θ V sym SC (θ,φ,s = 0) , (A.53) 117 where in the last line we used Eq. (A.49). Next we focus on the term multiplying s in Eq. (A.45). This term has no ϕ dependence and thus we only consider the θ derivatives. First note that ∂ ∂θ i V SC ({θ i },{ϕ i },s = 1) = X x∈{0,1} n f(x) Y j:x j =0 cos 2 θ j 2 ! Y j:x j =1 sin 2 θ j 2 ! × " −δ x i ,0 sec 2 θ i 2 ! +δ x i ,1 csc 2 θ i 2 !# sinθ i 2 . (A.54) Now, we set all the θ i ’s equal. Let us define p(θ)≡ sin 2 θ 2 . Using this and the fact that f is only a function of the Hamming weight (which is equivalent to the qubit permutation symmetry), we can rewrite the last expression, after a few steps of algebra, as: n X w=0 f(w)p w−1 (1−p) n−w−1 ∂ θ p × " (1−p) n− 1 w− 1 ! −p n− 1 w !# = n X w=0 f(w)p w−1 (1−p) n−w−1 ∂ θ p " 1 n n w ! (w−np) # = 1 n ∂ ∂θ V sym SC (θ,ϕ,s = 1) . (A.55) Similar to the quantum case, we can perform SVD without explicitly imposing the per- mutation symmetry, and obtain the same results. Here too, we are forced to explicitly exploit the symmetry due to the non-local nature of the problem under consideration, which makes directly implementing the SVD oracle (without the symmetry) exponentially hard. For local problems we can efficiently implement the SVD oracle. In the results presented in the main text, it is the implementation of SA that does not share this symmetry. However, while the quantum algorithms and SVD can be implemented without knowledge of the symmetry and still retain their advantage, an implementation of 118 SA that uses the symmetry would require intimate knowledge of the problem. This would be an unfair advantage for SA, not for the quantum evolution. A.4.4 Simulated Quantum Annealing An alternative method to simulated annealing, simulated quantum annealing (SQA, or Path Integral Monte Carlo along the Quantum Annealing schedule) [60, 61] is an annealing algorithm based on discrete-time path-integral quantum Monte Carlo simulations of the transverse field Ising model using Monte Carlo dynamics. At a given timet along the anneal, the Monte Carlo dynamics samples from the Gibbs distribution defined by the action: S[μ] = Δ(t) X τ H P (μ :,τ )−J ⊥ (t) X i,τ μ i,τ μ i,τ+1 (A.56) where Δ(t) = βB(t)/N τ is the spacing along the time-like direction, J ⊥ = − ln[tanh(A(t)/2)]/2 is the ferromagnetic spin-spin coupling along the time-like direction, andμ denotes a spin configuration with a space-like direction (the original problem direction, indexed by i) and a time-like direction (indexed by τ). For our spin updates, we perform Wolff cluster updates [118] along the imaginary-time direction only. For each space-like slice, a random spin along the time-like direction is picked. The neighbors of this spin are added to the cluster (assuming they are parallel) with probability P = 1− exp(−2J ⊥ ) (A.57) When all neighbors of the spin have been checked, the newly added spins are checked. When all spins in the cluster have had their neighbors along the time-like direction tested, the cluster is flipped according to the Metropolis probability using the space-like change in energy associated with flipping the cluster. A single sweep involves attempting to update a single cluster on each space-like slice. 119 t f 5 10 15 p GS 0 0.05 0.1 0.15 0.2 0.25 n= 32 n= 64 n= 96 n= 128 n= 160 n= 192 n= 224 n= 256 (a) n 0 50 100 150 200 250 300 Standard deviation of Gaussian fit 0 1 2 3 4 5 6 7 Numerical Experiment Fit 23.5277 (logn) 1.85705 (b) Figure A.2: Scaling of the p GS peak of QA for the Fixed Plateau with l = 0 and u = 6. (a) p GS vs. t f curves for several problem sizesn. The peak att f ≈ 10 is the cause of the optimal annealing time. (b) For each p GS vs. t f curve, we fit a Gaussian to the peak. Plotted is the standard deviation of the fitted Gaussians as a function of n. An inverse polylogarithmic function provides an excellent fit to this data. As in SA, we can use SQA both as an annealer and as a solver [113]. In the former, we randomly pick one of the states on the Trotter slices at the end of the evolution as the output of the algorithm, while for the latter, we pick the state with the lowest energy found along the entire anneal as the output of the algorithm. We use the latter to maximize the performance of the algorithm. A.5 Behavior of p GS vs. t f curves We found that for many of the PHWO problems studied, the optimal t f lies around t f = 10. This is because there is a peak in the probability of finding the ground state, p GS at this t f . Moreover, we found that this peak becomes increasingly higher as the problem size, n, grows. This is what allows the problem to have anO(1) scaling. Since this peak becomes increasingly sharper with growing n, there may be the worry that one might need 120 an arbitrarily high precision in setting t f ≈ t opt f . We address this concern by showing that in fact the width of thep GS vs. t f curve decreases asO[1/polylog(n)] for the Fixed Plateau. This shows that we only require a polylogarithmically increasing precision in our ability to set t f at the optimal value in order to obtain the speedup. The evidence is summarized in Fig. A.2. The first plot, A.2(a), shows p GS vs. t f curves for several values ofn. The second plot, A.2(b), shows the scaling of the standard deviation of Gaussian fit to the peak at t opt f . This scaling is well matched by polylogarithmic fit. 121 Appendix B Appendices to Chapter 3 B.1 Comparison with the method of Hagedorn and Joye In this section, we recap the asymptotic expansion of Hagedorn and Joye [84], which is a powerful tool for proving adiabatic theorems. In particular, the Hagedorn and Joye method can be used to prove bounds on the error incurred due to their asymptotic expan- sion; in fact, the main goal of Ref. [84] was to show that the adiabatic approximation can provide exponentially small errors if the Hamiltonian is analytic in the time-variable (see also Refs. [119, 64, 120]). Here, we analyze its utility as a computational tool. Hagedorn and Joye (HJ) propose the following method to obtain asymptotic approxima- tions to the time-dependent Schrödinger equation i d|χ(r)i dr =H(r)|χ(r)i. (B.1) Note that the above equation is of the form of Eq. (3.6), with≡ 1 μt f andH(r)≡s 0 (r)H(r). They obtain a theorem which states that for any value of the small parameter , one can write down an approximation for|χ(r)i which takes the form of a power series in . The quality of the approximation (as measured by the 2-norm) scales as e − 1 provided that the number of terms in the series scales as 1/. More precisely: 122 Theorem 2 ([84]). Assume reasonable smoothness and gap conditions on the Hamiltonian. We can then recursively obtain an asymptotic expansion of the form |χ (N) HJ (r,)i =e − i R r 0 E(q)dq (|χ 0 (r)i +|χ 1 (r)i +··· + N |χ N (r)i + N+1 |χ ⊥ N+1 (r)i . (B.2) such that for any r, there exist positive G, C(g), and Γ(g) such that for all g∈ (0,G), the vector|χ (bg/c) HJ (r,)i satisfies k|χ(r,)i−|χ (bg/c) HJ (r,)ik 2 ≤C(g)e −Γ(g)/ , (B.3) for all≤ 1. Here,|χ(r,)i is the Schrödinger evolved wavefunction starting from the initial condition|χ(0,)i =|χ (bg/c) HJ (0,)i. We explore the usefulness of this asymptotic expansion as an approximation tool and thus we do not estimate the number of terms that are necessary to provide an exponentially small error. Instead, we develop the approximation for two orders and compare the resulting asymptotic expansion with the WKB method. Let us develop the terms in the HJ expansion (as given in Ref. [84]). We substitute the asymptotic ansatz |χ HJ i∼e − i R r 0 dqE(q) (|χ 0 (r)i +|χ 1 (r)i +... ) (B.4) into the Schrödinger equation, and equate the terms multiplying the same order of , which results in the following expression for the O( j ) term |χ j (r)i =f j (r)|Φ(r)i +|χ ⊥ j (r)i. (B.5) 123 Here|Φ(r)i is the eigenstate being (approximately) followed, and the other components of the above formula are obtained recursively by using: f 0 (r) = 1; (B.6a) f j−1 (r) =− Z r 0 hΦ(q)|∂ q ψ ⊥ j−1 (q)idq, j≥ 2 (B.6b) = Z r 0 hΦ 0 (q)|ψ ⊥ j−1 (q)idq; (B.6c) |χ ⊥ j (r)i =i[H(r)−E(r)] −1 R (f n−1 (r)|Φ 0 (r)i +P ⊥ (r)∂ r |χ ⊥ j−1 (r)i ; (B.6d) where, in going from Eq. (B.6b) to Eq. (B.6c), we integrated by parts and used hΦ(q)|ψ ⊥ j−1 (q)i = 0. Also, P ⊥ (r)≡ I−|Φ(r)ihΦ(r)| is the instantaneous projector on to the complement of|Φ(r)i; E(r) is the eigenvalue being quasi-adiabatically followed; and [H(r)−E(r)] −1 R is the reduced resolvent, i.e., the inverse of [H(r)−E(r)] restricted to the complement of|Φ(r)i. In order to compare the HJ expansion with the WKB approximation, we will compare the N-th order expansion provided by both methods. Note that the N-th order of the HJ expansion includes terms up toO( N+1 ). This means that we will be comparing the zeroth order of WKB (i.e.,|χ (0) WKB i) with |χ (0) HJ (r)i≡e − i R r 0 E(q)dq |χ 0 (r)i +|χ ⊥ 1 (r)i ; (B.7) and the first order WKB (i.e.,|χ (1) WKB i) with |χ (1) HJ (r)i≡e − i R r 0 E(q)dq |χ 0 (r)i +|χ 1 (r)i + 2 |χ ⊥ 2 (r)i . (B.8) 124 For two-level systems such as the one that we are concerned with, we obtain the fol- lowing simplified expressions, where “GS" and “Exc" denote the ground and excited states respectively and Δ represents the spectral gap: [H(r)−E GS (r)] −1 R = 1 Δ(r) |χ Exc (r)ihχ Exc (r)| (B.9a) =⇒ |χ ⊥ 1 (r)i = i Δ(r) hχ Exc (r)|χ 0 GS (r)i|χ Exc (r)i , (B.9b) f 1 (r) = Z r 0 dqhχ 0 GS (q)|χ ⊥ 1 (q)i , (B.9c) |χ ⊥ 2 (r)i = i Δ(r) (f 1 (r)|χ 0 GS (r)i (B.9d) +|χ Exc (r)ihχ Exc (r)|∂ r χ ⊥ 1 (r)i . We have assumed that the ground state is being followed and hence set|Φi =|χ GS i. We have also used the fact that for real-valued Hamiltonians in two dimensionshχ 0 GS |χ GS i = 0. (Note that this does not mean|χ 0 GS i =|χ Exc i because|χ 0 GS i is generally not normalized and carries a non-trivial phase.) We now restrict to the case of a qubit in a magnetic field. First, consider|χ (0) HJ i (which includes terms up to order ). Figure B.1(a) shows that |χ (0) HJ i provides an approximation that is ‘too adiabatic’. In particular, it fails to capture the oscillations that are captured by the WKB approximation, as seen in Fig. 3.1. Furthermore, from the form of|χ (0) HJ i it is clear that this approximation will predict p GS (t f ) = 1 always: p HJ,0 GS (t f ) = hχ GS (1)|χ (0) HJ i 2 (B.10) =|hχ GS (1)|χ GS (1)i | {z } =1 +hχ GS (1)|χ ⊥ 1 (1)i | {z } =0 | 2 . (B.11) Next, consider|χ (1) HJ i (which includes terms up to order 2 ). Figure B.1(b) shows that this too provides an approximation which fails to capture the oscillations that are present in 125 0.0 0.2 0.4 0.6 0.8 1.0 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 (a) 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.80 0.85 0.90 0.95 (b) Figure B.1: (a) The difference between the predictions of the naive adiabatic approximation (|χ GS (r)i) and the lowest order HJ approximation (|χ (0) HJ i) for the population in the state |χ GS (1)i≡|0i, as a function of the rescaled time parameter r, fort f = 20. The difference is very small, of the order of 10 −3 . (b) The population in the state|mi as a function of time for the HJ expansion using the 0th and 1st orders; the adiabatic solution; and the numerical solution. The adiabatic solution and the HJ method are indistinguishable on the scale of this plot. Clearly, they do not the capture the oscillations displayed by the numerical solution. Here t f = 50. the numerical solution and also in the lowest order WKB solution. Thus, we conclude that the WKB method is more suitable for developing analytic approximations. While we pointed out some of the disadvantages of the HJ method as an approximation technique, we remark that the method is particularly useful to prove scaling results. For example, consider, hχ GS (1)|χ 1 HJ (1)i 2 =|(1 +f 1 (1))| 2 (B.12) = 1 + 2 |f 1 (1)| 2 (B.13) =O(1) + 2 O(1). (B.14) 126 In the first line, we used the fact|χ GS i is orthogonal to any (unnormalized) state that carries the⊥ symbol. In the second line, we used the fact that f 1 (1) =i Z 1 0 dq |hχ 0 GS (q)|χ Exc (q)i| 2 Δ(q) (B.15) is purely imaginary. Thus the HJ expansion captures the 1−O( 1 t 2 f ) scaling of the final ground state probability. 127 Appendix C Appendices to Chapter 4 C.1 Writing in the column basis In this Appendix we show how the matrix elements of the noiseless QA Hamiltonian are obtained in the column basis. I.e., we show how Eqs. (4.4) are obtained when Eq. (4.1) is written in the basis defined in Eq. (4.2). H 0 and H 1 are straightforward: they are represented as −|col 0 ihcol 0 | and −|col 2n+1 ihcol 2n+1 | respectively. This immediately yields Eqs. (4.4a) and (4.4d). Let’s now turn to the adjacency matrix A. Consider first j < n. In this case, in the column basis, A j,j+1 is hcol j |A|col j+1 i =hcol j | X (x,x 0 )∈E |xihx 0 | |col j+1 i (C.1) = 1 q N j N j+1 X x∈col j ;x 0 ∈col j+1 (x,x 0 )∈E 1 (C.2) = N j+1 q N j N j+1 (C.3) = s N j+1 N j = s 2 j+1 2 j = √ 2, (C.4) 128 where we have used the fact that for all columns to the left of the central glue, every vertex has exactly one edge connecting it to the column to its left. We have also used that for j≤n, N j = 2 j . A parallel calculation will go through for j >n + 1. Now, for j =n: hcol n |A|col n+1 i = 1 √ N n N n+1 × 2N n+1 (C.5) = 2× s N n+1 N n = 2, (C.6) where we used that, at the central glue, there are two edges to every vertex. We’ve also used the fact that the number of vertices in the two columns at the glue is equal. Thus, we have: hcol j |A|col j+1 i = 2, j =n √ 2, o.w. . (C.7) For convenience, we will be re-define A such that A new ≡A old / √ 2, giving us: hcol j |A|col j+1 i = √ 2, j =n 1, o.w. , (C.8) which yields Eqs. (4.4b) and (4.4c). C.2 Qubit versions of small glued trees instances If we wanted to implement the glued-trees algorithm in a qubit system, we would need nonlocal and difficult-to-engineer interactions. We do not provide a mathematical proof of this claim, but the basic point can be illustrated using the case of n = 1, the smallest instance of the glued-trees problem which has 6 vertices. Consider the shortest possible naming system, in which each vertex is labelled by a length-3 bit-string. (Set aside, for 129 the moment, the concern about this naming system interfering with the proof of classical hardness.) This means we can implement the QA Hamiltonian using 3 qubits. We’ll name the vertices such that the Hamming distance between two vertices connected by an edge on the graph is as small as possible. This is done in an attempt to reduce the need for many-body terms as much as possible (though, as we will see, this won’t be of much help). For the case of n = 1, this is done as seen in Fig. C.1. 000 001 100 101 010 111 Figure C.1: The smallest instance of the glued-trees problem. The graph is labelled with 3 bits. The naming is chosen so as to minimize the Hamming distance between vertices joined by an edge. Clearly the adjacency matrix A of this graph is A =|v 0 ihv 1 | +|v 0 ihv 2 | +|v 1 ihv 3 | +|v 1 ihv 4 | +|v 2 ihv 3 | +|v 2 ihv 4 | +|v 3 ihv 5 | +|v 4 ihv 5 | +h.c., (C.9) where v 0 ≡ 000,v 1 ≡ 001,v 2 ≡ 100,v 3 ≡ 101,v 4 ≡ 010, and v 5 ≡ 111. 130 Let’s convert this adjacency matrix into a local Hamiltonian. Let{I i ,X i ,Y i ,Z i } denote the Pauli operators on the i-th qubit (i = 1, 2, 3). We use the fact that the tensor products of these Pauli operators make a basis for the vector space of Hamiltonians. Therefore, we can compute Tr[AX 1 ], Tr[AX 1 X 2 ], Tr[AX 1 Y 2 ], and so on, for all the 4 3 = 64 terms to obtain the coefficients of these terms in the Pauli representation of A. Doing this, we get A = 1 4 (2X 1 +X 2 + 2X 3 +X 1 X 2 +X 2 X 3 +X 1 X 3 +Y 1 Y 2 +Y 2 Y 3 −Y 1 Y 3 + 2X 1 Z 2 −X 2 Z 3 −Z 1 X 2 + 2Z 2 Y 3 −X 1 Z 2 X 3 +X 1 X 2 Z 3 +Y 1 Y 2 Z 3 +Y 1 Z 3 Y 3 +Z 1 X 2 X 3 +Z 1 X 2 Z 3 +Z 1 Y 2 Y 3 ). (C.10) Note that there aren’t any Z terms because A doesn’t contain any diagonal matrix ele- ments. We can already see that in order to implement this adjacency matrix we need 3-body interactions, and also “cross-term" interactions, i.e., interactions that couple, say, a Z to an X. To complete the analysis, let us similarly write the Pauli representations of H 0 and H 1 . This yields H 0 = 1 8 (Z 1 +Z 2 +Z 3 +Z 1 Z 2 (C.11) +Z 2 Z 3 +Z 1 Z 3 +Z 1 Z 2 Z 3 ), 131 and H 1 = 1 8 (−Z 1 −Z 2 −Z 3 +Z 1 Z 2 +Z 2 Z 3 (C.12) +Z 1 Z 3 −Z 1 Z 2 Z 3 ). Again we see that we need 3-body interactions. Of course, there’s a possibility that there exists a easily computed naming system for a general instance of the glued-trees problem that generates a Hamiltonian representation for the QA Hamiltonian in such a way that the representation has small, constant locality and required the interactions required are simple, such as XX orYY. But as far as we can tell, this seems to be a hard problem and may even be impossible. C.3 The Gaussian Orthogonal Ensemble The GOE is the measure over the set of N×N real symmetric matrices described by Pr(h)dh =c N exp − 1 2σ 2 Tr(h 2 ) Y i dh ii Y i<j dh ij , (C.13) where σ is the so-called scale factor, c N is a normalization, dh ii and dh ij are the standard Lebesgue measure. If{λ i } are the eigenvalues of h, then Tr(h 2 ) = P i λ 2 i . Therefore, the peak of this distribution is centered around matrices with eigenvalues close to 0. The GOE is invariant under conjugation by orthogonal matrices: i.e., Eq. (C.13) holds in one basis, then it is also holds in another basis related by an orthogonal transformation to the first basis. A standard way to sample from the GOE is to first build a matrixM all of whose entries are sampled fromN (0, 1). Then, a GOE matrix h can be obtained by taking h = M+M T √ 2 . These and other results on the GOE are available in any standard textbook, e.g., in [121]. 132 C.4 Perturbative decay of overlap between the noisy and noiseless ground states In this appendix, we calculate how the overlap between the noisy and noiseless ground- states decays as a function of for the case of the long-range noise models. This is done using perturbation theory, so we expect to be valid for small . Consider first the perturbative expansion for the perturbed ground state. | ˜ φ 0 i =c()|φ 0 i + X k>0 |φ k i h k0 E 0 −E k +O( 2 ), (C.14) where c() is free constant whose value depends on the sort of normalization we choose to pick (see Chapter 5 of Ref. [122] for a detailed discussion). Imposingh ˜ φ 0 | ˜ φ 0 i = 1, and assuming c()∈R, we get c()≈ 1− 2 X k>0 h 2 k0 (E 0 −E k ) 2 1 2 . (C.15) Since, c() =hφ 0 | ˜ φ 0 i, this is quantity we care about, namely, the overlap between the noisy and the noiseless grounstates. We will now estimate this quantity. Note that h k0 ∼N (0, 1) and soh 2 k0 is distributed as a chi-squared distribution with one degree of freedom χ 2 (x)dx = x 1/2 e −x/2 √ 2Γ(1/2) dx, x≥ 0 0, x< 0. (C.16) with Γ(·) being the standard gamma function. Let g 0k ≡ 1 (E 0 −E k ) 2 . We want to understand how P k>0 g 0k χ 2 is distributed. 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Abstract (if available)
Abstract
This dissertation studies analog quantum optimization, in particular, quantum annealing (QA). QA is a limited model of quantum computation that might offer speedups for optimization problems. We study several QA oracle algorithms using semiclassical methods. First, we evaluate the claim that QA derives its computational power from tunneling. We study permutation-symmetric problems and show that tunneling is neither necessary nor sufficient for a speedup. We also discover in many of these algorithms a novel mechanism which we call a diabatic cascade, arising due to a lining-up of avoided level-crossings in the spectrum of the Hamiltonian. Second, we analyze the adiabatic Grover search problem using a quasiadiabatic variation on the WKB method. We find that the WKB approximation is able to capture features of the dynamics that a naive adiabatic approximation is unable to capture. While the approximation is able to recover the Grover speedup in the case of the optimal schedule, the approximation breaks down for other schedules. Finally, we turn to the glued-trees problem: a search problem on two randomly glued trees for which QA is known to provide an exponential speedup via a diabatic pathway. We analyze additive random-matrix noise models for this problem. We find that models which induce interactions between distant nodes of the graph are able to retain a quantum speedup for a larger range of noise strengths when compared to models that only induce interactions between neighboring vertices of the graph. We also find that a certain symmetry essential to the working of the closed-sytem algorithm can be broken without significantly affecting performance.
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Creator
Muthu Krishnan, Siddharth
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Core Title
Tunneling, cascades, and semiclassical methods in analog quantum optimization
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College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
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Physics
Publication Date
07/20/2018
Defense Date
05/07/2018
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analog quantum optimization,OAI-PMH Harvest,quantum annealing,quantum computing,quantum mechanics
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Lidar, Daniel (
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), Brun, Todd (
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), Campos Venuti, Lorenzo (
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), Spedalieri, Federico (
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muthukri@usc.edu,siddharth.muthukrishnan@gmail.com
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Tags
analog quantum optimization
quantum annealing
quantum computing
quantum mechanics