Adiabatic Quantum Computing

Home , PostBQP, QMA, Quantum logic gate, Swap test

Tameem Albash Information Sciences Institute, University of Southern California, Marina del Rey, CA 90292 Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA Daniel A. Lidar Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA

Adiabatic quantum computing (AQC) started as an approach to solving optimization problems, and has evolved into an important universal alternative to the standard circuit model of quantum computing, with deep connections to both classical and quantum complexity theory and condensed matter physics. In this review we give an account of most of the major theoretical developments in the field, while focusing on the closed- system setting. The review is organized around a series of topics that are essential to an understanding of the underlying principles of AQC, its algorithmic accomplishments and limitations, and its scope in the more general setting of computational complexity theory. We present several variants of the adiabatic theorem, the cornerstone of AQC, and we give examples of explicit AQC algorithms that exhibit a quantum speedup. We give an overview of several proofs of the universality of AQC and related Hamiltonian quantum complexity theory. We finally devote considerable space to Stoquastic AQC, the setting of most AQC work to date, where we discuss obstructions to success and their possible resolutions. arXiv:1611.04471v2 [quant-ph] 2 Feb 2018 CONTENTS algorithm 10 2. Quadratic quantum speedup 11 I. Introduction2 3. Multiple marked states 11 B. Adiabatic Deutsch-Jozsa algorithm 12 II. Adiabatic Theorems5 1. Unitary interpolation 12 A. Approximate versions6 2. Linear interpolation 13 3. Interpretation 13 B. Rigorous versions6 C. Adiabatic Bernstein-Vazirani algorithm 14 1. Inverse cubic gap dependence with generic H(s)7 D. The glued trees problem 15 2. Rigorous inverse gap squared7 E. Adiabatic PageRank algorithm 16 3. Arbitrarily small error8 1. Google matrix and PageRank 16 4. Lower bound8 2. Hamiltonian and gap 17 3. Speedup 17 III. Algorithms9 A. Adiabatic Grover9 IV. Universality of AQC 18 1. Setup for the adiabatic quantum Grover A. The circuit model can efficiently simulate AQC 19 2

B. AQC can efficiently simulate the circuit model: A. Technical calculations 56 history state proof 19 1. Upper bound on the adiabatic path length L 56 C. Fermionic ground state quantum computation 21 2. Proof of the inequality given in Eq. (25) 56 D. Space-time Circuit-to-Hamiltonian Construction 22 E. Universal AQC in 1D with 9-state particles 23 B. A lower bound for the adiabatic Grover search problem 57 F. Adiabatic gap amplification 24 C. Technical details for the proof of the universality of AQC V. Hamiltonian quantum complexity theory and universal using the History State construction 57 AQC 25 1. |γ(0)i is the ground state of Hinit 57 A. Background 25 2. |ηi is a ground state of Hfinal 58 L 1. Boolean Satisfiability Problem: k-SAT 25 3. Gap bound in the space spanned by {|γ(`)i}`=0 58 2. NP, NP-complete, and NP-hard 25 a. Bound for s < 1/3 58 3. The k-local Hamiltonian Problem 26 b. Bound for s ≥ 1/3 59 4. Motivation for Adiabatic Quantum Computing 26 4. Gap bound in the entire Hilbert space 60 B. MA and QMA 27 C. The general relation between QMA completeness and D. Proof of the Amplification Lemma (Claim1) 61 universal AQC 27 D. QMA-completeness of the k-local Hamiltonian E. Perturbative Gadgets 62 problem and universal AQC 27 1. Degenerate Perturbation Theory àla (Bloch, 1958) 62 2. Perturbative Gadgets 63 VI. Stoquastic Adiabatic Quantum Computation 29 A. Why it might be easy to simulate stoquastic References 65 Hamiltonians 30 B. Why it might be hard to simulate stoquastic Hamiltonians 31 I. INTRODUCTION 1. Topological obstructions 31 2. Non-topological obstructions 32 C. QMA-complete problems and universal AQC using Quantum computation (QC) originated with Benioff’s stoquastic Hamiltonians with excited states 33 proposals for quantum Turing machines (Benioff, 1980, D. Examples of slowdown by StoqAQC 33 1982) and Feynman’s ideas for circumventing the diffi- 1. Perturbed Hamming Weight Problems with Exponentially Small Overlaps 34 culty of simulating quantum mechanics by classical com- 2. 2-SAT on a Ring 35 puters (Feynman, 1982). This led to Deutsch’s proposal 3. Weighted 2-SAT on a chain with periodicity 36 for universal QC in terms of what has become the “stan- 4. Topological slowdown in a dimer model or local dard” model: the circuit, or gate model of QC (Deutsch, Ising ladder 37 5. Ferromagnetic Mean-field Models 38 1989). Adiabatic quantum computation (AQC) is based 6. 3-Regular 3-XORSAT 38 on an idea that is quite distinct from the circuit model. 7. Sherrington-Kirkpatrick and Two-Pattern Whereas in the latter a computation may in principle Gaussian Hopfield Models 39 evolve in the entire Hilbert space and is encoded into a se- E. StoqAQC algorithms with a scaling advantage over simulated annealing 40 ries of unitary quantum logic gates, in AQC the computa- 1. Spike-like Perturbed Hamming Weight Problems 40 tion proceeds from an initial Hamiltonian whose ground 2. Large plateaus 40 state is easy to prepare, to a final Hamiltonian whose F. StoqAQC algorithms with undetermined speedup 41 ground state encodes the solution to the computational 1. Number partitioning 41 2. Exact Cover and its generalizations 41 problem. The adiabatic theorem guarantees that the sys- 3. 3-Regular MAXCUT 42 tem will track the instantaneous ground state provided 4. Ramsey numbers 43 the Hamiltonian varies sufficiently slowly. It turns out 5. Finding largest cliques in random graphs 43 that this approach to QC has deep connections to con- 6. Graph isomorphism 43 densed matter physics, computational complexity theory, 7. Machine learning 44 G. Speedup mechanisms? 44 and heuristic algorithms. 1. The role of tunneling 44 In its first incarnation, the idea of encoding the solu- 2. The role of entanglement 46 tion to a computational problem in the ground state of a VII. Circumventing slowdown mechanisms for AQC 47 quantum Hamiltonian appeared as early as 1988, in the A. Avoiding poor choices for the initial and final context of solving classical combinatorial optimization Hamiltonians 47 problems, where it was called quantum stochastic opti- B. Quantum Adiabatic Brachistochrone 48 mization (Apolloni et al., 1989).1 It was renamed quan- C. Modifying the initial Hamiltonian 50 et al. D. Modifying the final Hamiltonian 50 tum annealing (QA) in (Apolloni , 1988) and rein- E. Adding a catalyst Hamiltonian 51 vented several times (Amara et al., 1993; Finnila et al., F. Addition of non-stoquastic terms 52 G. Avoiding perturbative crossings 52 H. Evolving non-adiabatically 54

1 VIII. Outlook and Challenges 55 Even though (Apolloni et al., 1989) was published in 1989, it was submitted in 1988, before (Apolloni et al., 1988), which ref- Acknowledgments 56 erenced it. 3

1994; Kadowaki and Nishimori, 1998; Somorjai, 1991).2 shown to allow a general purpose quantum computation These early papers emphasized that QA was to be under- to be embedded in the ground state of a quantum system stood as an algorithm that exploits simulated quantum (Feynman, 1985; Kitaev et al., 2000). A related ground (rather than thermal) fluctuations and tunneling, thus state embedding approach was independently pursued in providing a quantum-inspired version of simulated an- (Mizel et al., 2001, 2002), around the same time as the nealing (SA) (Kirkpatrick et al., 1983). The first direct original development of the QAA. To define AQC, we comparison between QA and SA (Kadowaki and Nishi- first need the concept of a k-local Hamiltonian, which is mori, 1998) suggested that QA can be more powerful. a Hermitian matrix H acting on the space of p-state par- r A very different approach was taken via an experimen- ticles that can be written as H = i=1 Hi where each Hi tal implementation of QA in a disordered quantum fer- acts non-trivially on at most k particles, i.e., H = h 11 i ⊗ romagnet (Brooke et al., 1999, 2001). This provided the where h is a Hamiltonian on at mostP k particles, and 11 impetus to reconsider QA from the perspective of quan- denotes the identity operator. tum computing, i.e., to consider a dedicated device that solves optimization problems by exploiting quantum evo- Definition 1 (Adiabatic Quantum Computation). A k- lution. Thus was born the idea of the quantum adiabatic local adiabatic quantum computation is specified by two algorithm (QAA) (Farhi et al., 2001, 2000) [also referred k-local Hamiltonians, H0 and H1, acting on n p-state to as adiabatic quantum optimization (AQO) (Reichardt, particles, p 2. The ground state of H0 is unique and ≥ 2004; Smelyanskiy et al., 2001)], wherein a physical quan- is a product state. The output is a state that is ε-close tum computer solves a combinatorial optimization prob- in `2-norm to the ground state of H1. Let s(t) : [0, tf ] 7→ lem by evolving adiabatically in its ground state. The [0, 1] (the “schedule”) and let tf be the smallest time such term adiabatic quantum computation we shall use here that the final state of an adiabatic evolution generated by was introduced in (van Dam et al., 2001), though the H(s) = (1 s)H0 +sH1 for time tf is ε-close in `2-norm 3 − context was still optimization. to the ground state of H1. Adiabatic quantum algorithms for optimization problems typically use “stoquastic” Hamiltonians, character- Several comments are in order. (1) A uniqueness re- ized by having only non-positive off-diagonal elements quirement was imposed on the ground state of H1 in in the computational basis. Adiabatic quantum compu- (Aharonov et al., 2007), but this is not necessary. E.g., tation with non-stoquastic Hamiltonians is as powerful in the setting where H1 represents a classical optimiza- as the circuit model of quantum computation (Aharonov tion problem, multiple final ground states do not pose a et al., 2007). In other words, non-stoquastic AQC and problem as any of the final states represents a solution all other models for universal quantum computation to the optimization problem. (2) Sometimes it is bene- can simulate one another with at most polynomial re- ficial to consider adiabatic quantum computation in an source overhead. For this reason the contemporary use excited state (see, e.g., Sec. VI.C). (3) As already noted of the term AQC typically refers to the general, non- in (Aharonov et al., 2007), it is useful to allow for more stoquastic setting, thus extending beyond optimization general “paths” between H0 and H1, e.g., by introducing to any computational. When discussing the case of sto- an intermediate “catalyst” Hamiltonian that vanishes at quastic Hamiltonians we will use the term “stoquastic s = 0, 1 (see, e.g., Sec. VII.E). AQC” (StoqAQC).4 A crucial question that will occupy us throughout this For most of this review we essentially adopt the def- review is the cost of running an algorithm in AQC. In inition of AQC from (Aharonov et al., 2007), as this the circuit model the cost is equated with the number of definition allows for the proof of the equivalence with gates, so one cost definition would be to the count the the circuit model, and is thus used to establish the uni- number of gates needed to simulate the equivalent adia- versality of AQC. Interestingly, this proof builds on one batic process. This cost definition presupposes that the of the first QC ideas due to Feynman, which was later circuit model is fundamental, which may be unsatisfac- tory. In AQC one might be tempted to just use the run time tf , but in order for this quantity to be meaningful it 2 It was called “quasi-quantal method” in (Somorjai, 1991), is necessary to define an appropriate energy scale for the “imaginary-time algorithm” in (Amara et al., 1993), and quan- Hamiltonian. In (Aharonov et al., 2007) the cost of the tum annealing in (Finnila et al., 1994; Kadowaki and Nishimori, adiabatic algorithm was defined to be the dimensionless 1998). The latter term has become widely accepted. quantity 3 The first documented use of the term “adiabatic quantum computation” was in (Averin, 1998), but the context was an adiabatic implementation of a quantum logic gate in the circuit model. cost = tf max H(s) , (1) 4 Quantum annealing, like StoqAQC, currently usually involves s k k stoquastic Hamiltonians; we differentiate between them in the following sense: we restrict StoqAQC purely to the case of closed in order to prevent the cost from being made arbitrar- system evolutions, whereas QA refers to a (not necessarily adia- ily small by changing the time units, or distorting the batic) evolution in an open system. scaling of the algorithm by multiplying the Hamiltoni- 4 ans by some size-dependent factor.5 From hereon we will The goal of this article is to review the field of AQC focus on the run time tf , which should be compared to from its inception, with a focus on the closed system case. the circuit depth of analogous circuit model algorithms, That is, we omit the fascinating topic of AQC in open whereas the full cost in Eq. (1) should be compared to systems coupled to an environment. This includes all the circuit gate count. experimental work on AQC, and all work on quantum The run time tf of an adiabatic algorithm scales at error correction and suppression methods for AQC, as worst as 1/∆3, where ∆ is the minimum eigenvalue gap these topics deserve a separate review (Albash and Lidar, between the ground state and the first excited state of 2017) and including them here would limit our ability to the Hamiltonian of the adiabatic algorithm (Jansen et al., do justice to the many years of work on AQC in closed 2007). If the Hamiltonian is varied sufficiently smoothly, systems, an extremely rich topic with many elegant re- one can improve this to O(1/∆2) up to a polylogarith- sults. For the same reasons we also omit the blossom- mic factor in ∆ (Elgart and Hagedorn, 2012). While ing and closely related fields of holonomic QC (Zanardi these are useful sufficient conditions, they involve bound- and Rasetti, 1999), topological QC (Nayak et al., 2008), ing the minimum eigenvalue gap of a complicated many- and adiabatic state preparation for quantum simulation body Hamiltonian, a notoriously difficult problem. This (Babbush et al., 2014b). To achieve our goal we orga- is one reason that AQC has generated so much interest nized this review around a series of topics that are essen- among physicists: it has a rich connection to well studied tial to an understanding of the underlying principles of problems in condensed matter physics. For example, be- AQC, its algorithmic accomplishments and limitations, cause of the dependence of the run time on the gap, the and its scope in the more general setting of computa- performance of quantum adiabatic algorithms is strongly tional complexity theory. influenced by the type of quantum phase transition the We begin by reviewing the adiabatic theorem in Sec.II. same system would undergo in the thermodynamic limit The adiabatic theorem forms the backbone of AQC: it (Latorre and Orus, 2004). provides a sufficient condition for the success of the com- Nevertheless, a number of examples are known where putation, and in doing so provides the run time of a com- the gap analysis can be carried out. For example, adia- putation in terms of the eigenvalue gap ∆ of the Hamil- batic quantum computers can perform a process analo- tonian and the Hamiltonian’s time-derivative. In fact gous to Grover search (Grover, 1997), and thus provide there is not one single adiabatic theorem, and we review a quadratic speedup over the best possible classical algo- a number of different variants that provide different run rithm for the Grover search problem (Roland and Cerf, time requirements, under different smoothness and dif- 2002). Other examples are known where the gap analysis ferentiability assumptions about the Hamiltonian. can be used to demonstrate that AQC provides a speedup Next, we review in Sec.III the handful of explicit al- over classical computation, including adiabatic versions gorithms for which AQC is known to give a speedup over of some of the keystone algorithms of the circuit model. classical computation. The emphasis is on “explicit”, However, much more common is the scenario where either since Sec.IV provides several proofs for the universal- the gap analysis reveals no speedup over classical compu- ity of AQC in terms of its ability to efficiently simulate tation, or where a clear answer to the speedup question the circuit model, and vice versa. This means that every is unavailable. In fact, the least is known about adiabatic quantum algorithm that provides a speedup in the cir- quantum speedups in the original setting of solving clas- cuit model [many of which are known (Jordan, 2016b)] sical combinatorial optimization problems. This remains can in principle be implemented with up to polynomial an area of very active research, partly due to the origi- overhead in AQC. That the number of explicit AQC algo- nal (still unmaterialized) hope that the QAA would de- rithms is still small is therefore likely to be a reflection of liver quantum speedups for NP-complete problems (Farhi the relatively modest amount of effort that has gone into et al., 2001), and partly due the availability of commer- establishing such results compared to the circuit model. cial quantum annealing devices such as those manufac- However, there is also a real difficulty, in that performing tured by D-Wave Systems Inc. (Johnson et al., 2011), the gap analysis in order to establish the actual scaling designed to solve optimization problems using stoquastic (beyond the polynomial-time equivalence) is, as already Hamiltonians. mentioned above, in many cases highly non-trivial. A second non-trivial aspect of establishing a speedup by AQC is that when such a speedup is polynomial, relying on universality is insufficient, since the polynomial over- 5 Unless stated otherwise, we shall always use k·k to denote the head involved in implementing the transformation from operator norm for operators: the circuit model to AQC can then swamp the speedup. kAk = sup {kA|ψik : |ψi ∈ H with hψ|ψi = 1} A good example is the case of Grover’s algorithm, where (i.e., the largest singular value of the operator A), and the Eu- a direct use of the equivalence to the circuit model would clidean vector norm for vectors. Which one is used will be clear not suffice; instead, what is required is a careful analy- by context. sis and choice of the adiabatic schedule s(t) in order to 5 realize the quantum speedup. should not necessarily be a cause for pessimism. Some In Sec.V we go beyond universality into Hamiltonian of the obstacles in the way of a quantum speedup can quantum complexity theory. This is an active contempo- be overcome or circumvented, as we discuss in Sec.VII. rary research area, that started with the introduction of In all cases this involves modifying some aspect of the the complexity class QMA (“quantum Merlin-Arthur”) Hamiltonian, either by optimizing the schedule s(t), or by as the natural quantum generalization of the classical adding certain terms to the Hamiltonian such that small complexity classes ”nondeterministic polynomial time” gaps are avoided. This can result in a non-stoquastic (NP) and MA (Kitaev et al., 2000). The theory of QMA- Hamiltonian whose final ground state is the same as that completeness deals with decision problems that are effi- of the original Hamiltonian, with an exponentially small ciently checkable using quantum computers. It turns out gap (often corresponding to a first order quantum phase that these decision problems can be formulated naturally transition) changing into a polynomially small gap (of- in terms of k-local Hamiltonians, of the same type that ten corresponding to a second order phase transition). appear in the proofs of the universality of AQC. Thus Another type of modification is to give up adiabatic evo- universality and Hamiltonian quantum complexity stud- lution itself, and allow for diabatic transitions. While ies are often pursued hand-in-hand, and a reduction of k this results in giving up the guarantee of convergence to as well as the dimensionality p of the particles appearing the ground state provided by the adiabatic theorem, it in these constructions is one of the main goals. For ex- can be a strategy that results in better run time scaling ample, already k = 2 and p = 2 leads to both universal for the same Hamiltonian than an adiabatic one. AQC and QMA-complete Hamiltonians in 2D, while in We conclude with an outlook and discussion of future 1D p > 2 is needed for both.6 directions in Sec.VIII. Various technical details are pro- We turn our attention to StoqAQC in Sec.VI. This vided in the Appendix. is the setting of the vast majority of AQC work to date. The final Hamiltonian H1 is assumed to be a classical Ising model Hamiltonian, typically (but not always) rep- II. ADIABATIC THEOREMS resenting a hard optimization problem such as a spin The origins of the celebrated quantum adiabatic glass. The initial Hamiltonian H0 is typically assumed to x approximation date back to Einstein’s “Adiabatenhy- be proportional to a transverse field, i.e., i σi , whose ground state is the uniform superposition state in the pothese”: “If a system be affected in a reversible adi- computational basis. AQC with stoquasticP Hamiltonians abatic way, allowed motions are transformed into al- is probably less powerful than universal quantum compu- lowed motions” (Einstein, 1914). Ehrenfest was the tation, but examples can be constructed which show that first to appreciate the importance of adiabatic invari- it may nevertheless be more powerful than classical com- ance, guessing—before the advent of a complete quan- putation. Moreover, if we relax the definition of AQC to tum theory— that quantum laws would only allow mo- allow for computation using excited states, it turns out tions which are invariant under adiabatic perturbations that stoquastic Hamiltonians can even be QMA-complete (Ehrenfest, 1916). The more familiar, modern version and support universal AQC. To do justice to this mixed of the adiabatic approximation was put forth by Born and complicated picture, we first review examples where and Fock already in 1928 for the case of discrete spec- it is known that StoqAQC does not outperform classi- tra (Born and Fock, 1928), after the development of the cal computation (essentially because the eigenvalue gap Born-Oppenheimer approximation for the separation of ∆ decreases rapidly with problem size but classical algo- electronic and nuclear degrees of freedom a year earlier rithms do not suffer a slowdown), then discuss examples (Born and Oppenheimer, 1927). Kato put the approxi- where StoqAQC offers a quantum scaling advantage over mation on a firm mathematical foundation in 1950 (Kato, simulated annealing in the sense that it outperforms clas- 1950) and arguably proved the first quantum adiabatic sical simulated annealing but not necessarily other classi- theorem. cal algorithms, and finally point out examples where it is The adiabatic approximation states, roughly, that for currently not known whether StoqAQC offers a quantum a system initially prepared in an eigenstate (e.g., the ground state) ε (0) of a time-dependent Hamiltonian speedup, but one might hope that it does. We also dis- | 0 i cuss the role of potential quantum speedup mechanisms, H(t), the time evolution governed by the Schrödinger in particular tunneling and entanglement. equation The somewhat bleak picture regarding StoqAQC ∂ ψ(t) i | i = H(t) ψ(t) (2) ∂t | i

6 We say that H is a dD(d-dimensional) Hamiltonian if the par- (we set ~ 1 from now on) will approximately keep the actual state≡ ψ(t) of the system in the corresponding ticles are arranged on a d-dimensional grid and the summands | i of H couple only pairs of nearest neighbor particles. Note that instantaneous ground state (or other eigenstate) ε0(t) being dD implies that the Hamiltonian is 2-local. of H(t), provided that H(t) varies “suﬃciently slowly”.| i 6

Quantifying the exact nature of this slow variation is the where s t/t [0, 1] is the dimensionless time, and ≡ f ∈ subject of the Adiabatic Theorem (AT), which exists in H(s) is tf -independent. This includes the “interpo- many variants. In this section we provide an overview lating” Hamiltonians of the type often considered in of these variants of the AT, emphasizing aspects that AQC, i.e., H(s) = A(s)H0 + B(s)H1 [where A(s) and are pertinent to AQC. We discuss the “folklore” adia- B(s) are monotonically decreasing and increasing, re- 8 batic condition, that the total evolution time tf should spectively] and excludes cases with multiple timescales. be large on the timescale set by the square of the inverse The Schrödingerequation then becomes gap, and the question of how to ensure a high fidelity between the actual state and the ground state. We then 1 ∂ ψtf (s) | i = iH(s) ψtf (s) , (4) discuss a variety of rigorous versions of the AT, empha- tf ∂s − | i sizing different assumptions and consequently different performance guarantees. Throughout this discussion, it which is the starting point for all rigorous adiabatic the- is important to keep in mind that ultimately the AT pro- orems. vides only an upper bound on the evolution time required A more careful adiabatic condition subject to this for- to achieve a certain fidelity between the actual state and mulation is given by (Amin, 2009): the target eigenstate of H(t). 1 ε (s) ∂ H(s) ε (s) max |h i | s | j i| 1 j = i . (5) t s [0,1] ε (s) ε (s) 2 ∀ 6 f ∈ | i − j | A. Approximate versions The conditions (3) and (5) give rise to the widely used criterion that the total adiabatic evolution time should be Let εj(t) (j 0, 1, 2,... ) denote the instantaneous eigenstate| ofi H(∈t) { with energy} ε (t) such that ε (t) large on the timescale set by the minimum of the square j j ≤ ε (t) j, t, i.e., H(t) ε (t) = ε (t) ε (t) and j = 0 of the inverse spectral gap ∆ij(s) = εi(s) εj(s). In most j+1 ∀ | j i j | j i − denotes the (possibly degenerate) ground state. Assume cases one is interested in the ground state, so that ∆ij(s) that the initial state is prepared in one of the eigenstates is replaced by εj(0) . | i ∆ min ∆(s) = min ε1(s) ε0(s) . (6) The simplest as well as one of the oldest traditional ver- ≡ s [0,1] s [0,1] − sions of the adiabatic approximation states that a system ∈ ∈ initialized in an eigenstate εj(0) will remain in the same However, arguments such as those leading to Eqs. (3) and instantaneous eigenstate ε| (t) (upi to a global phase) for (5) are approximate, in the sense that they do not result | j i all t [0, tf ], where tf denotes the final time, provided in strict inequalities and do not result in bounds on the (Messiah,∈ A., 1962): closeness between the actual time-evolved state and the desired eigenstate. We discuss this next. εi ∂tεj εi ∂tH εj max |h | i| = max |h | | 2 i| 1 i = j . t [0,tf ] ε ε t [0,tf ] ε ε ∀ 6 ∈ | i − j| ∈ | i − j| (3) B. Rigorous versions This version has been critiqued (Du et al., 2008; Marzlin and Sanders, 2004; Tong et al., 2005; Wu et al., 2008) on The first rigorous adiabatic condition is due to Kato the basis of arguments and examples involving a sepa- (Kato, 1950), and was followed by numerous alterna- rate, independent timescale. Indeed, if the Hamiltonian tive derivations and improvements giving tighter bounds includes an oscillatory driving term then the eigenstate under various assumptions, e.g., (Ambainis and Regev, population will oscillate with a timescale determined by 2004; Avron and Elgart, 1999; Cheung et al., 2011; El- this term, that is independent of tf , even if the adiabatic gart and Hagedorn, 2012; Ge et al., 2015; Hagedorn and criterion (3) is satisfied.7 Joye, 2002; Jansen et al., 2007; Lidar et al., 2009; Nen- A more careful statement of the adiabatic condi- ciu, 1993; O’Hara and O’Leary, 2008; Reichardt, 2004; tion that excludes such additional timescales is thus re- Teufel, 2003). All these rigorous results are more severe quired. The first step is to assume that the Hamilto- in the gap condition than the traditional criterion, and nian H (t) in the Schrödinger equation ∂ ψ (t) /∂t = they involve a power of the norm of time derivatives of the tf | tf i iH (t) ψ (t) can be written as H (st ) = H(s), Hamiltonian, rather than a transition matrix element. − tf | tf i tf f We summarize a few of these results here, and refer the reader to the original literature for their proofs. For

7 For example, it is easily checked that when H(t) = aσz + b sin(ωt)σx, the adiabatic condition (3) reduces to |bω| a2. However, even if this condition is satisfied the population can oscillate between the two eigenstates: at resonance (when ω ≈ 2a) 8 For example, a case such as H(t) = aσz + b sin(ωt)σx is now the system undergoes Rabi oscillations with period π/|b|, a excluded since after a change of variables we have H(s) = aσz + x timescale that is independent of tf . b sin(ωtf s)σ and evidently H(s) still depends on tf . 7 simplicity we always assume that the system is initialized in finite-dimensional spaces.10 Then for any s [0, 1], in its ground state and that the gap is the ground state ∈ m(0) H(1)(0) m(s) H(1)(s) gap (6). We also assume that for all s [0, 1] the Hamil- P (s) P (s) + ∈ tf 2 2 tonian H(s) has an eigenprojector P (s) with eigenenergy − ≤ tf ∆ (0) tf ∆ (s) 9 2 ε0(s), and that the gap never vanishes, i.e., ∆ > 0. The 1 s m H(2) 7m√m H(1) ground state, and hence the projector P (s), is allowed + 2 + 3 dx (7) tf 0 ∆ ∆ ! to be (even infinitely) degenerate. P (s) represents the Z “ideal” adiabatic evolution. The numerator depends on the norm of the first or second time derivative of H(s), rather than the matrix Let Ptf (s) = ψtf (s) ψtf (s) . This is the projector onto the time-evolved| solutionih of| the Schrödingerequa- element that appears in the traditional versions of the tion, i.e., the “actual” state. Adiabatic theorems are adiabatic condition. usually statements about the “instantaneous adiabatic Ignoring the m-dependence for simplicity, this result shows that the adiabatic limit can be approached arbi- distance” Ptf (s) P (s) between the projectors associated withk the actual− andk ideal evolutions, or the “final- trarily closely if (but not only if) time adiabatic distance” P (1) P (1) . Typically, 2 k tf − k H(2)(s) H(1)(s) adiabatic theorems give a bound of the form O(1/tf ) tf max max , max , s [0,1] ∆2(s) s [0,1] ∆3(s) for the instantaneous case, and a bound of the form ( ∈ ∈

n (1) O(1/tf ) for any n N for the ﬁnal-time case. After H (s) ∈ max k k . (8) squaring, these projector-distance bounds immediately s [0,1] ∆2(s) become bounds on the transition probability, deﬁned as ∈ 2 Similar techniques based on Kato’s approach can be ψt⊥f (s) ψtf (s) , where ψt⊥f (s) = Qtf (s) ψtf (s) , with Q|h = I | P . i| | i | i used to prove a rigorous adiabatic theorem for open quan- − tum systems, where the evolution is generated by a non- Hermitian Liouvillian instead of a Hamiltonian (Venuti et al., 2016).

1. Inverse cubic gap dependence with generic H(s) 2. Rigorous inverse gap squared

Kato’s work on the perturbation theory of linear oper- A version of the AT that yields a scaling of tf with ators (Kato, 1950) introduced techniques based on resol- the inverse of the gap squared (up to a logarithmic cor- vents and complex analysis that have been widely used rection) was given in (Elgart and Hagedorn, 2012). All in subsequent work. Jansen, Ruskai, and Seiler (JRS) other rigorous AT versions to date have a worse gap de- proved several versions of the AT that build upon these pendence (cubic or higher). The proof introduces as- techniques (Jansen et al., 2007), and that rigorously es- sumptions on H(s) that go beyond those of Theorem1. tablish the gap dependence of tf , without any strong Namely, it is assumed that H(s) is bounded and infinitely assumptions on the smoothness of H(s). Their essen- differentiable, and the higher derivatives cannot have a tial assumption is that the spectrum of H(s) has a band magnitude that is too large, or more specifically, that associated with the spectral projection P (s) which is sep- H(s) belongs to the Gevrey class Gα: arated by a non-vanishing gap ∆(s) from the rest. Here Definition 2 (Gevrey class). H(s) Gα if dH(s)/ds = we present one their theorems: 0 s [0, 1] and there exist constants∈C, R > 0, such that6 for∀ all∈ k 1, ≥ Theorem 1. Suppose that the spectrum of H(s) re- max H(k)(s) CRkkαk . (9) stricted to P (s) consists of m(s) eigenvalues separated s [0,1] ≤ ∈ by a gap ∆(s) = ε1(s) ε0(s) > 0 from the rest of − An example is H( s) = [1 A(s)]H0 + A(s)H1, where the spectrum of H(s), and that H(s) is twice continu- A(s) = c s exp[ 1/(x x2−)]dx if s (0, 1), and A(s) = (1) (2) ously differentiable. Assume that H, H , and H are 0 if s / [0,−∞1]. The− constant− c is chosen∈ so that A(1) = 1. bounded operators, an assumption that is always fulfilled ∈ R (k) (k) 2k For this family H (s) = A (s) H1 H0 Ck , so that H(s) G2. k − k ≤ ∈ The AT due to (Elgart and Hagedorn , 2012) can now be stated as follows:

9 There is a weaker form of the AT, where one does not require a non-vanishing gap (Avron and Elgart, 1999). In this case, as in k 10 We use the notation H(k)(s) ≡ ∂ H(x)| throughout. Theorem2, the estimate on the error term is o(1) as tf → ∞. ∂x s 8

Theorem 2. Assume that H(s) is bounded and belongs (1) 11 while the scaling of tf with V is encoded into HV . to the Gevrey class Gα with α > 1, and that ∆ h, An example of a function whose first V derivatives van- where h H(0) = H(1) . If ≡ k k k k ish at the boundaries s = 0, 1 is the regularized β func- R s xV (1 x)V dx K tion A(s) = 0 − (Rezakhani et al., 2010b). It 6α R 1 xV (1 x)V dx tf ln(∆/h) (10) 0 ≥ ∆2 | | is possible to further− reduce the error quadratically in for some ∆-independent constant K > 0 (with units of tf using an interference effect that arises from imposing an additional boundary symmetry condition (Wiebe and energy), then the distance Ptf (s) P (s) is o(1) s [0, 1]. k − k ∀ ∈ Babcock, 2012). Note that an important difference between Theorems2 This result is remarkable in that it rigorously gives and3 is that the former applies for all times s [0, 1] an inverse gap squared dependence, which is essen- (“instantaneous AT”), while the latter applies only∈ at the tially tight due to existence of a lower bound of the final time s = 1 (“final-time AT”), which typically gives 2 form tf = O(∆− / ln ∆ ) for Hamiltonians satisfying rise to tighter error bounds. | | rank H(1) dim( )(Cao and Elgart, 2012). However, Also note that Landau and Zener already showed that H the error bound is not tight, and we address this next. the transition probability out of the ground state is C∆2t O(e− f )(C. Zener, 1932; Landau, L. D., 1932) [see (Joye, 1994) for a rigorous proof for analytic Hamiltoni- 3. Arbitrarily small error ans], thus combining an inverse gap square dependence with an exponentially small error bound. However, this Building on work originating with (Nenciu, 1993) [see result only holds for two-level systems. also (Hagedorn and Joye, 2002)], (Ge et al., 2015) proved a version of the AT that results in an exponentially small error bound in tf . The inverse gap dependence is cubic. 4. Lower bound Assume for simplicity that ε0(s) = 0 and choose the phase of ε0(s) so that ε˙0(s) ε0(s) = 0, where the dot Let H(s), with s [0, 1], be a given continuous Hamil- | i h | i ∈ denotes ∂s. tonian path and ε(s) the corresponding non-degenerate | i Theorem 3. Assume that all derivatives of the Hamilto- eigenstate path (eigenpath). In the so-called black-box model the only assumption is to be able to evolve with nian H(s) vanish at s = 0, 1, and moreover that it satisfies the following Gevrey condition: there exist constants H[s(t)] for some schedule s(t) (here s is allowed to be a general function of t), without exploiting the unknown C, R, α > 0 such that for all k 1, ≥ structure of H(s). Define the path length L as: (k!)1+α max H(k)(s) CRk . (11) 1 s [0,1] ≤ (k + 1)2 L = ε˙(s) ds , (13) ∈ k| ik 0 Then the adiabatic error is bounded as Z

1 where dot denotes ∂s. Assuming, without loss of general- 3 C c ∆ t 1+α iθ − 2 C2 f ity, that the phase of ε(s) is chosen so that ε(s) ε˙(s) = min ψtf (1) e ε0(1) c1 e (12) | i h | i θ | i − | i ≤ ∆ 0, then L is the only natural length in projective Hilbert 3 space (up to irrelevant normalization factors). 8π2 1 3 5 where c1 = eR 3 and c2 = 4eR2 4π2 . It was shown in (Boixo and Somma, 2010) that there is a lower bound on the time required to prepare ε(1) C2 | i Thus, as long as tf 3 , the adiabatic error is expo- from ε(0) with bounded precision: ∆ | i nentially small in tf . The idea of using vanishing boundary derivatives dates tf > O(L/∆) . (14) back at least to (Garrido, L. M. and Sancho, F. J., 1962). ˙ 12 It was also used in (Lidar et al., 2009) for a diﬀerent Since an upper bound on L is maxs H(s) /∆, one obtains the estimate t O(max H˙ (sk) /∆k2), reminis- class of functions than the Gevrey class: functions that f ∼ s k k are analytic in a strip of width 2γ in the complex time cent of the approximate versions of the adiabatic con- plane and have a ﬁnite number V of vanishing boundary dition [e.g., Eq. (5)]. The proof of the lower bound is derivatives, i.e., H(v)(0) = H(v)(1) = 0 v [1,V ]. The ∀ ∈ γ+1 V adiabatic error is then upper-bounded by (V + 1) q− 2 q (1) 3 11 as along as tf γ V maxs HV (s) /∆ , where q > 1 This corrects an omission in (Lidar et al., 2009), where the de- ≥ (1) is a parameter that can be optimized given knowledge pendence of H on V was ignored since the supremum of (1) (1) H (s) was taken over s ∈ [0, 1] instead of over the region of of HV . Thus, the adiabatic error can be made ar- analyticity of H(s), as noted in (Ge et al., 2015). 12 bitrarily small in the number of vanishing derivatives, See Appendix A.1 as well as Appendix G of (Boixo et al., 2009a).

9 essentially based on the optimality of the Grover search This definition allows for the existence of other clas- algorithm. sical algorithms that are already better than the The lower bound is nearly achievable using a “digital”, quantum algorithm. The notion of a limited quan- non-adiabatic method proposed in (Boixo et al., 2010), tum speedup will turn out to be particularly useful that does not require path continuity or differentiability. in the context of StoqAQC. The time required scales as O[(L/∆) log(L/)], where is a specified bound on the error of the output state ε(1) . A refinement of this classification geared at experimental L is the angular length of the path and is suitably defined| i quantum annealing was given in (Mandrà et al., 2016). to generalize Eq. (13) to the non-differentiable case. Using this classification, this section collects most Armed with an arsenal of adiabatic theorems we are of the adiabatic quantum algorithms known to give now well equipped to start surveying AQC algorithms. a provable quantum speedup (Grover, Deutsch-Jozsa, Bernstein-Vazirani, and glued trees), or just a quantum speedup (PageRank).13 III. ALGORITHMS Many other adiabatic algorithms have been proposed, and we review a large subset of these in Sec.VI. In a In this section we review the algorithms which are few of these cases there is a scaling advantage over clas- known to provide quantum speedups over classical al- sical simulated annealing, while in some cases there are gorithms. However, to make the idea of a quantum definitely faster classical algorithms. speedup precise we need to draw distinctions among different types of speedups, as several such types will arise in the course of this review. Toward this end we adopt A. Adiabatic Grover a classification of quantum speedup types proposed in (Rønnow et al., 2014). The classification is the following, The adiabatic Grover algorithm (Roland and Cerf, in decreasing order of strength. 2002) is perhaps the hallmark example of a provable quantum speedup using AQC, so we review it in detail. A “provable” quantum speedup is the case where • As in the circuit model Grover algorithm (Grover, 1997), there exists a proof that no classical algorithm informally the objective is to find the marked item (or can outperform a given quantum algorithm. The possibly multiple marked items) in an unsorted database best known example is Grover’s search algorithm of N items by accessing the database as few times as (Grover, 1997), which, in the query complexity set- possible. More formally, one is allowed to call a function ting, exhibits a provable quadratic speedup over f : 0, 1 n 0, 1 (where N = 2n is the number of bit the best possible classical algorithm (Bennett et al., strings){ } with7→the { promise} that f(m) = 1 and f(x) = 0 1997). x = m, and the goal is to find the unknown index m in ∀the6 smallest number of calls. This is an oracular prob- A “strong” quantum speedup was originally defined lem (Nielsen and Chuang, 2000), in that the algorithm • in (Papageorgiou and Traub, 2013) by comparing can make queries to an oracle that recognizes the marked a quantum algorithm against the performance of items. The oracle remains a black box, i.e., the details of the best classical algorithm, whether such a clas- its implementation and its complexity are ignored. This sical algorithm is explicitly known or not. This allows for an uncontroversial determination of the com- aims to capture computational complexity consid- plexity of the algorithm in terms of the number of queries erations allowing for the existence of yet-to-be dis- to the oracle. covered classical algorithms. Unfortunately, the For a classical algorithm, the only strategy is to query performance of the best possible classical algorithm the oracle until the marked item is found. Whether the is unknown for many interesting problems (e.g., for classical algorithm uses no memory, i.e., the algorithm factoring). does not keep track of items that have already been A “quantum speedup” (unqualified, without adjec- checked, or uses an exponential amount of memory (in n) • tives) is a speedup against the best available classi- to store all the items that have been checked, the clas- cal algorithm [for example Shor’s polynomial time sical algorithm will have an average number of queries factoring algorithm (Shor, 1994)]. Such a speedup that scales linearly in N. may be tentative, in the sense that a better classical In the AQC algorithm we denote the marked item by algorithm may eventually be found. the binary representation of m. The oracle is defined in Finally, a “limited quantum speedup” is a speedup • obtained when compared specifically with classical algorithms that ‘correspond” to the quantum algo- 13 The glued trees case is, strictly, not an adiabatic quantum algorithm in the sense that they implement the same rithm, since it explicitly makes use of excited states. Also, in the algorithmic approach, but on classical hardware. PageRank case the evidence for a quantum speedup is numerical. 10 terms of the final Hamiltonian H1 = 11 m m , where The remaining N 2 eigenstates of H(s) have eigenvalue m is the marked state associated with the− | markedih | item. 1 throughout the evolution.− The minimum gap occurs at In| thisi representation, the binary representations give the s = 1/2 and scales exponentially with n: eigenvalues under σz, i.e., σz 0 = + 0 and σz 1 = 1 . | i | i | i −| i The marked state is the ground state of this Hamiltonian 1 n/2 ∆min = ∆(s = 1/2) = = 2− . (20) with energy 0, and all other computational basis states √N have energy 1. (This can be viewed as a special case of Lemma1 below.) 1. Setup for the adiabatic quantum Grover algorithm In our discussion of the adiabatic theorem we saw that without special assumptions on s(t) except that We use the initial Hamiltonian H = 11 φ φ , where it is twice differentiable, the adiabatic condition is 0 − | ih | φ is the uniform superposition state, inferred from Eq. (7), which requires setting tf | i 2 1 2 3 2 maxs ∂sH(s) /∆ (s)+ 0 ∂sH(s) /∆ (s)ds, where N 1 we havek accountedk for the boundaryk conditionsk and used 1 − n φ = i = + ⊗ , (15) R | i √ | i | i the positivity of the integrand to extend the upper limit N i=0 14 X to 1. Differentiating Eq. (17) yields where = 1 ( 0 1 ). We take the time-dependent |±i √2 | i ± | i 1 1 1 1 1 Hamiltonian to be an interpolation: N √N N ∂sH(s) = − − − , (21)  1 1 1 1q 1  H(s) = [1 A(s)] H0 + A(s)H1 (16) √N − N − N −   = [1 A(s)] (11 φ φ ) + A(s)(11 m m ) , q − − | ih | − | ih | 1 which has eigenvalues 1 , so that ∂sH where s = t/t [0, 1] is the dimensionless time, t is ± − N k k ≤ f f 1. The other integrand in Eq. (7), involving the total computation∈ time, and A(s) is a “schedule” q ∂2H(s) /∆2(s), vanishes after differentiating Eq. (21). that can be optimized. For simplicity, we first consider a s The ground state degeneracy m(s) = 1 throughout. linear schedule: A(s) = s. Note that H1 is n-local. 3/2 s 3 N N (1 2s) Since 1/∆ (x)dx = − , which is If the initial state is initialized in the ground state of 0 2 2√N(1 2s)2+4(1 s)s − − − H(0), i.e., ψ(0) = φ , then the evolution of the system a monotonicallyR increasing function of s that approaches is restricted| toi a two-dimensional| i subspace, defined by 2 N = ∆min− as s 1, the adiabatic condition becomes 1 N 1 the span of m and m = − i . In this → ⊥ √N 1 i=m | i | i − 6 | i two-dimensional subspace H(s) can be written as: 1 P 1 1 3 tf 2 max + ds = . (22) s ∆2(s) ∆3(s) ∆2 1 ∆(s) cos θ(s) sin θ(s) Z0 min [H(s)] = 112 2 , m , m⊥ 2 × − 2 sin θ(s) cos θ(s) | i | i − (17) This suggests the disappointing conclusion that the quan- where: tum adiabatic algorithm scales in the same way as the classical algorithm. 4 ∆(s) = (1 2s)2 + s(1 s) , (18a) However, by imposing the adiabatic condition globally, r − N − i.e., to the entire time interval tf , the evolution rate is 1 1 constrained throughout the whole computation, while the cos θ(s) = 1 2(1 s) 1 , (18b) ∆(s) − − − N gap only becomes small around s = 1/2. Thus, it makes sense to use a schedule A(s) that adapts and slows down 2 1 1 sin θ(s) = (1 s) 1 . (18c) near the minimum gap, but speeds up away from it (van ∆(s) − √N r − N Dam et al., 2001; Roland and Cerf, 2002) [this is related to the idea of rapid adiabatic passage, which has a long The eigenvalues and eigenvectors in this subspace are history in nuclear magnetic resonance (Powles, 1958)]. then given by: By doing so the quadratic quantum speedup can be re- 1 1 covered, as we address next. ε (s) = (1 ∆(s)) , ε (s) = (1 + ∆(s)) , 0 2 − 1 2 (19a) θ(s) θ(s) ε0(s) = cos m + sin m⊥ , (19b) | i 2 | i 2 | i 14 Whenever we use the symbol we mean that the larger quantity θ(s) θ(s) should be larger by some large multiplicative constant, such as ε (s) = sin m + cos m⊥ . (19c) | 1 i − 2 | i 2 | i 100. 11

2. Quadratic quantum speedup first rewrite Eq. (24) in dimensional time units as ∂tA = 2 c0 ∆ [A(t)], with the boundary conditions A(0) = 0 and Consider again the adiabatic condition (7), which we A(tf ) = 1;. To solve this differential equation we rewrite can rewrite as: t A(t) 2 it as t = 0 dt = A(0) dA/[c0∆ (A)]. After integration 1 2 2 we obtain ∂sH(s) ∂s H ∂sH R R tf 2 max k 2 k + 2 + k 3 k ds , s ∆ (s) 0 ∆ ∆ ! Z N 1 (23) t = tan− √N 1 (2A(t) 1) 2c0√N 1 − − where now H and ∆ depend on a schedule A(s). Let us − h 1 now use the ansatz (Jansen et al., 2007; Roland and Cerf, + tan− √N 1 . (27) − 2002) i Evaluating Eq. (27) at tf gives: p ∂sA = c ∆ [A(s)] ,A(0) = 0 , p, c > 0 . (24) N 1 π tf = tan− √N 1 √N, (28) c √N 1 − → 2c0 This schedule slows down as the gap becomes smaller, 0 − as desired. The normalization constant c = which is the expected quadratic quantum speedup. 1 p A(1) p 0 ∆− [A(s)]∂sAds = A(0) ∆− (u)du [using u = A(s)] One may be tempted to conclude that tf can be made is chosen to ensure that A(1) = 1. arbitrarily small since so far c0 is arbitrary and can be R R It follows that: chosen to be large. However, the adiabatic error bound (26) shows that this is not the case: while it is not tight, 1 2 2 ∂s H[A(s)] ∂sH[A(s)] it suggests that if t scales as √N then c must scale + k k ds f 0 ∆2[A(s)] ∆3[A(s)] as 1/ log(2N) in order to keep the adiabatic error small. Z0 ! 1 Thus, the general conclusion is that increasing c0 results p 3 16 4c ∆ − (u)du (25) in a larger adiabatic error. ≤ Z0 Inverting Eq. (27) for A(t) [or, equivalently, solving Eq. (24) for p = 2] gives the locally optimized schedule (the proof is given in Appendix A.2). Finally, the bound- ∂ H(s) p 2 ary term in Eq. (23) yields 2 max k s k 4c∆ − . 1 1 1 s ∆2(s) ≤ min A(s) = + tan (2s 1) tan− √N 1 , The case p = 2 serves to illustrate the main point. 2 2√N 1 − − − h i In this case the boundary term is 4c and evaluating the (29) integrals yields where we replaced t/tf [with tf given by Eq. (28)] with s. As expected, this schedule rises rapidly near s = 0, 1 1 N π and is nearly flat around s = 1/2, i.e., it slows down near 2 1 √ √ c = ∆− (u)du = tan− N 1 N the minimum gap. 0 √N 1 − → 2 Z − Since the choice in Eq. (24) is not unique, we may √N 1√N+N 1 1 log − − √N 1√N (N 1) wonder if there exists a schedule that gives an even better ∆ 1(u)du = − − − log(2N)/2 , − h i scaling. Given that Grover’s algorithm is known to be 0 2 N 1 → Z N− optimal in the circuit model setting (Bennett et al., 1997; q Zalka, 1999), this is, unsurprisingly, not the case, and where the asymptotic expressions are for N 1. Sub- a general argument to that effect which applies to any stituting this into Eq. (25) yields the adiabatic condition Hamiltonian quantum computation was given by (Farhi and Gutmann, 1998). We review this argument in the t 2π√N[1 + log(2N)] , (26) f AQC setting, in AppendixB. which is a sufficient condition for the smallness of the adiabatic error, and nearly recovers the quadratic speedup 3. Multiple marked states expected from Grover’s algorithm. The appearance of the logarithmic factor latter is ac- The present results generalize easily to the case where 15 tually an artifact of using bounds that are not tight. we have M 1 marked states, for which Grover’s algo- √ The quadratic speedup, i.e., the scaling of tf with N, rithm is known≥ to also give a quadratic speedup in the can be fully recovered by solving for the schedule from Eq. (24) in the p = 2 case (Roland and Cerf, 2002). We √ 16 Note that the scaling conclusion tf ∼ N reported in (Roland and Cerf, 2002) is based on the interpretation of Eq. (24) as a heuristic “local” adiabatic condition and does not constitute a 15 A detailed analysis of the adiabatic Grover algorithm along with proof that the adiabatic error is small. The evidence√ that the tighter error bounds than we have given here was presented in rigorous bound (26) is not tight and that tf ∼ N suffices to (Rezakhani et al., 2010b). achieve a small adiabatic error for the schedule (29) is numerical. 12 circuit model (Biham et al., 1999; Boyer et al., 1998). where is the index set of the marked states. Let: The final Hamiltonian can be written as: M 1 m = i i . (31) | ⊥i √N M | ih | i/ − X∈M Instead of evolving in a two-dimensional subspace, the system evolves in an M +1 dimensional subspace spanned by ( m m , m ), and instead of Eq. (17), the {| i} ∈M | ⊥i H = 11 m m , (30) Hamiltonian can be written in this basis as: 1 − | ih | m X∈M

1 1 s √N M (1 s) 1 N N− ... (1 s) N− − 1 s− − 1 1 s − − √N M  N− (1 s) 1 N N− ... (1 s) N−  H(s) = − − − − − − . (32) . .. .  . . .   √N M √N M N M   (1 s) − (1 s) − . . . s + (1 s) 1 −   − − N − − N − − N   

This Hamiltonian can be easily diagonalized, and one (Sarandy and Lidar, 2005) and an implementation us- finds that there are M 1 eigenvalues equal to 1 s, and ing a linear interpolation was given in (Wei and Ying, two eigenvalues − − 2006). These algorithms match the speedup obtained in the circuit model [for an earlier example where this is not 1 1 4M the case see (Das et al., 2002)], and we proceed to review λ = (1 2s)2 + s(1 s) , (33) both. We note that, just like the adiabatic Grover’s al- ± 2 ± 2r − N − gorithm, the adiabatic Deutsch-Jozsa algorithm requires that determine the relevant minimum gap: ∆(s) = n-local Hamiltonians. We also note that both the uni- (1 2s)2 + 4M s(1 s). The remaining N M 1 tary interpolation and linear interpolation strategies we − N − − − qeigenvalues of the unrestricted Hamiltonian are equal to describe here are not unique to the Deutsch-Jozsa prob- 1, Comparing the M = 1 case Eq. (18a) to the present lem, and apply equally well to any depth-one quantum case, the only difference is the change from 1/N to M/N. circuit. Thus they should be viewed in this more general Therefore our discussion from earlier goes through with context, and are used here with a specific algorithm for only this modification. illustrative purposes. In closing, we note that an experimentally realizable version of the adiabatic Grover search algorithm using a single bosonic particle placed in an optical lattice was 1. Unitary interpolation recently proposed in (Hen, 2017). The initial Hamiltonian is chosen such that its ground state is the uniform superposition state φ [Eq. (15)] n n | i B. Adiabatic Deutsch-Jozsa algorithm and N = 2 , i.e., H(0) = ω i=1 i , where ω is the energy scale. The Deutsch-Jozsa|−i h−| problem can Given a function f : 0, 1 n 0, 1 which is promised be solved by a single computationP of the function f { } 7→ { } to be either constant or balanced [i.e., f(x) = 0 on half through the unitary transformation U x = ( 1)f(x) x the inputs and f(x) = 1 on the other half], the Deutsch- (x 0, 1 n)(Collins et al., 1998), so| thati in− the x| i Jozsa problem is to determine which type the function is. (computational)∈ { } basis U is represented by the diagonal{| i} f(0) f(2n 1) There exists a quantum circuit algorithm that solves the matrix U = diag[( 1) , ..., ( 1) − ]. An adia- problem in a single f-query (Deutsch and Jozsa, 1992). batic implementation− requires a− final Hamiltonian H(1) n 1 Classically, the problem requires 2 − + 1 f-queries in such that its ground state is ψ(1) = U ψ(0) . This n 1 | i | i the worst case, since it is possible that the first 2 − can be accomplished via a unitary transformation of queries return a constant answer, while the function is H(0), i.e., H(1) = UH(0)U †. Then the final Hamil- actually balanced. It is important to note that the quan- tonian encodes the solution of the Deutsch problem in tum advantage requires a deterministic setting, since the its ground state, which can be extracted via a measure- classical error probability is exponentially small in the ment of the qubits in the + , basis (note that number of queries. this is compatible with the{| definitioni |−i} of AQC, Def.1, An adiabatic implementation of the Deutsch-Jozsa which does not restrict the measurement basis). A suit- algorithm using unitary interpolation was given in able unitary interpolation between H(0) and H(1) can 13

be defined by H(s) = U˜(s)H(0)U˜ †(s), where U˜(s) = determine the adiabatic run time for the adiabatic Hamil- π ˜ exp i 2 sU , for which U(1) = iU. Since a unitary trans- tonian H(s) = (1 s)H0 + sH1. The following Lemma formation of H(0) preserves its spectrum, it does not (Aharonov and Ta-Shma− , 2003) comes in handy: change the ground state gap, which remains ω. The run Lemma 1. Let α and β be two states in some sub- time of the algorithm can be determined from the adia- | i | i space of an N-dimensional Hilbert space , and let Hα = batic condition (8), and what remains is the numerator: H 11 α α , Hβ = 11 β β . For any convex combination H(1)(s) = i π [U, H(s)] π H(0) = π [and simi- −| ih | −| ih | 2 ≤ k k Hη = (1 η)Hα + ηHβ, where η [0, 1], the ground state larly for H(2)(s) ]. This yields t 1/ω. This result is − ∈ f gap ∆(Hη) α β . independent of n so the adiabatic run time is O(1). ≥ |h | i| Proof. Expand β = a α + b α⊥ where α α⊥ = 0, | i | i | i h | i and complete α , α⊥ to an orthonormal basis for . {| i | i} H 2. Linear interpolation Writing Hη in this basis yields 2 η b ηab∗ Unitary interpolations, introduced in (Siu, 2005), are Hη = | | 2 11(N 2) (N 2) . (36) ηa∗b η a + 1 η ⊕ − × − somewhat less standard. Therefore we also present the | | − standard linear interpolation method as an alternative. The eigenvalues of this matrix are all 1 for the identity matrix block and the difference between the eigenvalues Consider the usual initial Hamiltonian H0 = 11 φ φ − | ih | 2 over n qubits, where once again φ is the uniform su- in the 2 2 block is ∆(Hη) = 1 4η(1 η) b . This | i is minimized× for η = 1/2, where it equals− −a . | | perposition state. Let the final Hamiltonian be H1 = p | | 11 ψ ψ , where − | if h | Applying this lemma, we see that ∆[H(s)] φ ψ = 1/√2. Since H(1)(s) = H H ≥2 N/2 1 N/2 1 |h | f i| k 1 − 0k ≤ µf − 1 µf − and H(2)(s) = 0, it follows from the adiabatic condi- ψ f = 2i + − 2i + 1 , (34) | i N/2 | i N/2 | i tion (8) that t is independent of n, i.e., the adiabatic i=0 i=0 f X X run time is O (1) as in the circuit model depth. and wherep p

1 3. Interpretation µ = ( 1)f(x) . (35) f N − x 0,1 n As mentioned above, a classical probabilistic algorithm ∈{X} that simply submits random queries to the oracle will

Clearly, µf = 1 or 0 if f is constant or balanced, respec- fail with a probability that is exponentially small in the tively. Therefore ψ is a uniform superposition over all | if number of queries. One might thus be concerned that the even (odd) index states if f is constant or balanced, re- adiabatic algorithms above are no better (Hen, 2014a), spectively, and a measurement of the ground state of H1 since they are probabilistic, in the sense that there is a in the computational basis reveals whether f is constant non-zero probability of ending in an excited state. How- or balanced, depending on whether the observed state ever, for the linear interpolation adiabatic algorithm re- belongs to the even or odd sector, respectively. However, viewed, measuring the energy of the final state returns we note that one may object to the reasonableness of the 0 in the ground state or 1 in an excited state. In the final Hamiltonian H . Namely, preparing the state ψ 1 | if latter ”inconclusive” case, the algorithm needs to be re- involves precomputing the quantity µf , which directly peated until an energy of 0 is found and only then is a encodes whether f is constant or balanced and so may computational basis measurement performed. If an even 17 be thought to represent an oracle that is too powerful. (or odd) index state is measured, the corresponding con- Indeed, H1 in this construction is not of the standard stant (or balanced) result is guaranteed to be correct. form x f(x) x x , wherein each oracle call f(x) corre- Moreover, an excited state outcome can (and should) be sponds to a query| ih about| a single basis state x . There- P | i made exponentially unlikely using a smooth schedule as fore, there is no classical analogue to this oracle in the per Theorem3. Thus, the adiabatic algorithms above computational basis. improve upon a classical probabilistic algorithm in the Setting this concern in the present version of the algo- following sense: In the adiabatic case, to know with cer- rithm due to (Wei and Ying, 2006) aside, it remains to tainty that the function is constant or balanced (an even or odd index measurement result) happens with proba- t bility p = 1 q where q e− f , where the run time t − ∼ f 17 is independent of n. Therefore, the expected number of The only efficient way known to compute a quantity similar to µf involves running the Deutsch-Jozsa algorithm in the gate model, runs r to certainty in the adiabatic case is where 1 P (−1)f(x), without the absolute value, ap- N x 0,1 n ∞ ∈{ } r 1 1 t pears as the amplitude of the state |0i n (Nielsen and Chuang, r = pq − r = 1 + q 1 + e− f . (37) ⊗ h i 1 q ≈ ∼ 2000). r=1 X − 14

On the other hand, classically, to know with certainty Thus for each state w in subsystem A, there is an inde- that the function is constant requires N = 2n/2 + 1 runs pendently evolving state| i in subsystem B. In particular, or queries (all yielding identical outcomes). note that adiabaticity in subsystem B does not depend Finally, we note that there exists a non-adiabatic on the size of system A. Hamiltonian quantum algorithm that solves the Deutsch- The adiabatic algorithm (Hen, 2014b) encodes the ac- Jozsa problem in constant time with a deterministic guar- tion of fa(w) in a Hamiltonian acting on two subsystems antee of ending up with the right answer (i.e., in the A and B comprising n qubits and 1 qubit respectively: ground state) (Hen, 2014a). This algorithm is based on finding a fine-tuned schedule s(t). H1 = hw , (44a) w 0,1 n ∈{X } 1 h w w 11 + ( 1)fa(w)σz . (44b) C. Adiabatic Bernstein-Vazirani algorithm w ≡ −2| iAh | ⊗ B − B The initial Hamiltonian is chosen to be The Bernstein-Vazirani problem (Bernstein and Vazi- 1 x rani, 1993) is to find an unknown binary string a H0 = (11A (11B σ )) 2 ⊗ − B 0, 1 n with as few queries as possible of the function∈ { } 1 x (or oracle) = w A w (11B σ ) . (45) 2 | i h | ⊗ − B w 0,1 n f (w) = w a 0, 1 , (38) ∈{X } a ∈ { } Any state of the form where denotes the bitwise inner product modulo 2, Ψ(0) = cw w A + B (46) n | i | i ⊗ | i and w 0, 1 as well. In the quantum circuit model, w 0,1 n it can∈ be { shown} that a can be determined with O(1) ∈{X } queries (Bernstein and Vazirani, 1993) whereas classical is a ground state of H0, with eigenvalue 0. We assume algorithms require n queries (the classical algorithm tries that the initial state is prepared as the uniform superpo- n/2 sition state, i.e., c = 2− w. all n w’s with a single 1 entry to identify each bit of a). w ∀ This is a polynomial quantum speedup. The total Hamiltonian is thus given by: Before presenting the adiabatic algorithm we point out H(s) = (1 s)H0 + sH1 = w A w Hw(s) , the following useful observation. For an initial state: − | i h | ⊗ w 0,1 n ∈{X } (47) Ψ(0) = c w ψ (0) , c 2 = 1 | i w| iA⊗| w iB | w| w 0,1 n w 0,1 n where ∈{X } ∈{X } (39) 1 s s H (s) = − (11 σx ) 11 + ( 1)fa(w)σz . that undergoes an evolution according to the time- w 2 B − B − 2 B − B dependent Hamiltonian of the form (48) The adiabatic algorithm proceeds, after preparation of H(s) = w w H (s) , (40) | iAh | ⊗ w the initial state, by adiabatic evolution to the final state: w 0,1 n ∈{X } Ψ(tf ) = (49) we have: | i 1 1 n/2 w A exp itf ε0,w(s)ds fa(w) B , Ψ(t) = cw w A ψw(t) B , (41) 2 | i ⊗ − | i | i | i ⊗ | i w 0,1 n Z0 w 0,1 n ∈{X } ∈{X } where ε0,w(s) is the instantaneous ground-state energy where of Hw(s), and we have used the general argument from Eqs. (39)-(42). Finally an x measurement on subsystem ψ (t) = ψ (s) (42) w B tf ,w B B is performed. Since we can write | i | i s 1 = Texp itf dσHw(σ) ψw(0) B , fa(w) − | i fa(w) = + + ( 1) , (50) Z0 | i √2 | i − |−i where Texp denotes the time-ordered exponential. To see the state collapses to either of the following states with this, simply expand the formal solution: equal probability: s 1 Ψ+ = w A + B = Ψ(0) , (51a) Ψ(t) = Texp itf dσH(σ) Ψ(0) = (43) | i 2n/2 | i ⊗ | i | i | i − 0 | i w 0,1 n ∈{ } Z s X 1 fa(w) cw w A Texp itf dσHw(σ) ψw(0) B . Ψ = w A ( 1) B . (51b) | i ⊗ − | i | −i 2n/2 | i ⊗ − |−i w 0,1 n Z0 w 0,1 n ∈{X } ∈{X } 15

Note that since fa(w) counts the number of 1-agreements any given vertex one can only see the adjacent vertices. between a and w, we can write: More formally, an oracle outputs the adjacent vertices of a given input vertex (note that the roots of the trees n 1 − are the only vertices with adjacency two, so it is easy ( 1)fa(w) w = ( 0 + ( 1)ak 1 ) , − | iA | ik − | ik A to check if the right root was found). The problem is, w 0,1 n k=0 ∈{X } O given the name of the left root and access to the oracle, (52) to find the name of the right root in the smallest number so that of queries. Classical algorithms require at least a sub- n 1 − exponential in n number of oracle calls, but there exists 1 ak Ψ = ( 0 k + ( 1) 1 k)A B . (53) a polynomial-time quantum algorithm based on quan- | −i 2n/2 | i − | i ⊗ |−i Ok=0 tum walks for solving this problem (Childs et al., 2003). If the measurement gives +1 [i.e., Eq. (51a)], then the A polynomial-time quantum almost-adiabatic algorithm measured state is the initial state and no information is was given in (Somma et al., 2012). The qualifier “almost” gained and the process must be repeated. If the mea- is important: the algorithm is not adiabatic during the surement gives 1, the resulting state in the A subspace entire evolution, since it explicitly requires a transition encodes all the− bits of a, since if the k-th qubit is in the from the ground state to the first excited state and back. We now review the algorithm, which (so far) provides + k state, then ak = 0 and if it is in the k state, then | i |−i m the only example of a (sub-)exponential almost-adiabatic ak = 1. The probability of failure after m tries is 2− so it is exponentially small and n-independent. quantum speedup. The run time of the algorithm is also n-independent, Let us denote the bit-string corresponding to the first root by a0 and the second root by aN 1. Define the since only a single qubit (system B) is effectively evolv- − ing.. diagonal (in the computational basis) Hamiltonians In conclusion, the adiabatic Bernstein Vazirani algo- H0 = a0 a0 ,H1 = aN 1 aN 1 , (54) rithm finds the unknown binary string a in O(1) time, −| ih | −| − ih − | matching the circuit model depth. Using a similar tech- and the states nique, (Hen, 2014b) presented a quantum adiabatic ver- 1 sion of Simon’s exponential-speedup period finding al- cj = ai , (55) | i N | i gorithm (Simon, 1997) [a precursor to Shor’s factor- j i j th column ing algorithm (Shor, 1997)], again matching the circuit ∈ − X p model depth scaling. An important aspect of these which are a uniform superposition over the vertices in j quantum adiabatic constructions is that they go beyond the j-th column with Nj = 2 for 0 j n and Nj = 2n+1 j ≤ ≤ the general-purpose (and hence suboptimal) polynomial- 2 − for n + 1 j 2n + 1. Note that c0 = a0 ≤ ≤ | i | i equivalence prescription of universality proofs that map and c2n+1 = aN 1 . Let us define the Hamiltonian A | i | − i circuit-based algorithms into quantum-adiabatic ones associated with the oracle as having the following non- (see SectionIV). That equivalence does not necessar- zero matrix elements: ily preserve a polynomial quantum speedup, whereas the √2 j = n construction in (Hen, 2014b) discussed here does. c A c = (56) h j| | j+1i 1 otherwise We then pick as our interpolating Hamiltonian: D. The glued trees problem

H(s) = (1 s)αH0 s(1 s)A + sαH1 (57) Consider two binary trees, each of depth n. Each − − − n tree has 2j = 2n+1 1 vertices, for a total of where α (0, 1/2) is a constant (independent of n) and j=0 − N = 2n+2 2 vertices, each labelled by a randomly cho- s(t) is the∈ schedule. Note that a unitary evolution ac- P sen 2n-bit− string. The two trees are randomly glued as cording to this Hamiltonian will keep a state within the shown in Fig.1. More specifically, choose a leaf on the subspace spanned by cj if the state is initially within left end at random and connect it to a leaf on the right that subspace. Since{| thei} instantaneous ground state at end chosen at random. Then connect the latter to a leaf s = 0 ( a0 ) is in this subspace, it suffices to only consider on the left chosen randomly among the remaining ones, this subspace.| i Because of the form of the Hamiltonian, and so on, until every leaf on the left is connected to the eigenvalue spectrum is symmetric about s = 1/2. two leaves on the right (and vice versa). This creates a In this subspace, at s = α/√2 (and by symmetry at random cycle that alternates between the leaves of the 1 s ), the energy gap× between the ground state and two trees. The problem is, starting from the left root, the− first× excited state closes exponentially in n. This to find a path to the right root in the smallest possible is depicted in Fig.2, where s1, s2 represent the region number of steps, while traversing the tree as in a maze. around s and s3, s4 represent the region around 1 s . I.e., keeping a record of one’s moves is allowed, but at In the regions× s [0, s ), s [s , s ), and s [s , 1],− the× ∈ 1 ∈ 2 3 ∈ 4 16

...... s1 s2 s3 s4 j =0 j =1 j = n j = n +1 j =2n +1 0 2(s)

-0.1

-0.2 n/2 3 10 2 10 c/n 1(s) / -0.3 Eigenvalues 0(s) -0.4 3 c0/n 21 c/n3 -0.5 10 FIG. 1 A glued tree with n = 4. The labeling j from Eq. (55) 0 0.2 0.4 0.6 0.8 1 is depicted on top of the tree. s

FIG. 2 The ground state (λ0(s), blue solid curve), first ex- energy gap between the ground state and first excited cited state (λ1(s), red dashed curve) and second excited state 3 state is lower-bounded by c/n . The gap between the (λ2(s) yellow dot-dashed√ curve) of the glued-trees Hamilto- 3 first and second excited states is lower-bounded by c0/n nian (57) for α = 1/ 8 and n = 6. Inside the region [s1, s2] and [s , s ], the gap between the ground state and first excited throughout the evolution. Both c, c0 > 0. 3 4 state ∆ closes exponentially with n. In the region [s , s ], The proposed evolution exploits the symmetry and gap 10 2 3 the gap between the ground state and first excited state ∆10 structure of the spectrum as follows. A schedule is chosen and the gap between the first excited state and second excited 3 that guarantees adiabaticity only if the energy gap scales state ∆21 are bounded by n− . Similarly, in the region [s4, 1], 3 as 1/n . Then, during s [0, s1), the desired evolution the gap between the ground state and first excited state ∆10 ∈ 3 is sufficiently adiabatic that it follows the instantaneous is bounded by n− . ground state. During s [s1, s2), the evolution is non- adiabatic (since the gap∈ scales as 1/en) and a transition to the first excited state occurs with high probability. tial quantum speedup. Note that unlike the previous al- During s [s2, s3), the evolution is again sufficiently gorithms we reviewed in this section, which all provided adiabatic that∈ it follows the instantaneous first excited a provable quantum speedup, the current algorithm pro- state. During s [s3, s4), the evolution is again non- vides a “regular” quantum speedup, in the sense that adiabatic and a transition∈ from the first excited state it outperforms all currently known classical algorithms, back to the ground state occurs with high probability. but better future classical algorithms have not been ruled During s [s4, 1], the evolution is again adiabatic and out. follows the∈ instantaneous ground state. 1 1 6 Since t = ds(ds/dt)− n , we conclude that f 0 ∼ aN 1 can be found in polynomial time. 1. Google matrix and PageRank | − i R PageRank can be seen as the stationary distribution of E. Adiabatic PageRank algorithm a random walker on the web-graph, which spends its time on each page in proportion to the relative importance of We review the adiabatic quantum algorithm from that page (Langville and Meyer, 2006). (Garnerone et al., 2012) that prepares a state contain- To model this define the transition matrix P1 associ- ing the same ranking information as the PageRank vec- ated with the (directed) adjacency matrix A of the graph tor. The latter is a central tool in data mining and information retrieval, at the heart of the success of the 1/d(i) if (i, j) is an edge of A; P (i, j) = (58) Google search engine (Brin and Page, 1998). Using the 1 0 else, adiabatic algorithm, the extraction of the full PageRank vector cannot, in general, be done more efficiently than where d(i) is the out-degree of the ith node. when using the best classical algorithms known. How- The rows having zero matrix elements, corresponding ever, there are particular graph-topologies and specific to dangling nodes, are replaced by the vector ~e/n whose tasks of relevance in the use of search engines (such as entries are all 1/n, where n is the number of pages or finding just the top-ranked entries) for which the quan- nodes, i.e., the size of the web-graph. Call the resulting tum algorithm, combined with other known quantum (right) stochastic matrix P2. However, there could still protocols, may provide a polynomial, or even exponen- be subgraphs with in-links but no out-links. Thus one 17

deﬁnes the “Google matrix” G as Equations (61)-(63) completely characterize the adiabatic quantum PageRank algorithm, apart from the G := αP T + (1 α)E, (59) 2 − schedule s(t). where E ~v ~e . The “personalization vector” ~v is a By numerically simulating the dynamics generated by probability≡ distribution; | ih | the typical choice is ~v = ~e/n. h(s), (Garnerone et al., 2012) showed that for typical The parameter α (0, 1) is the probability that the random graph instances generated using the “preferential walker follows the link∈ structure of the web-graph at each attachment model” (Barabasi and Albert, 1999; Bollob´as step, rather than hop randomly between graph nodes ac- et al., 2001) and “copying model” (Kleinberg et al., 1999) cording to ~v (Google reportedly uses α = 0.85). By con- (both of which yield sparse random graphs with small- struction, G is irreducible and aperiodic, and hence the world and scale-free features), the typical run time of the Perron-Frobenius theorem (Horn and Johnson, 2012),18 adiabatic quantum PageRank algorithm scales as ensures the existence of a unique eigenvector with all pos- b 1 b t (log log n) − (log n) , (64) itive entries associated to the maximal eigenvalue 1. This f ∼ eigenvector is precisely the PageRank ~p. Moreover, the where b > 0 is some small integer that depends on the modulus of the second eigenvalue of G is upper-bounded details of the graph parameters. The numerically com- by α (Nussbaum, 2003). This is important for the conver- b puted gap scales as (log n)− , which (Garnerone et al., gence of the power method, the standard computational 2012) found to be due to the power law distribution of technique employed to evaluate ~p. It uses the fact that the out-degree nodes d(i).19 for any probability vector ~p0

k ~p = lim G ~p0. (60) k 3. Speedup →∞ The power method computes ~p with accuracy ν in a time We next discuss two tasks for which this adiabatic that scales as O[sn log(1/ν)/ log(1/α)], where s is the quantum ranking algorithm offers a speedup. sparsity of the graph (maximum number of non-zero en- The best currently known classical Markov Chain tries per row of the adjacency matrix). The rate of con- Monte Carlo (MCMC) technique used to evaluate the vergence is determined by α. full PageRank vector requires a time (in the bulk syn- chronous parallel computational model (Valiant, 1990)) 2. Hamiltonian and gap which scales as O[log(n)] (Das Sarma et al., 2015). The algorithm launches log n random walks from each node Consider the following non-local final Hamiltonian as- of the graph in parallel (for a total of n log(n) walkers), sociated with a generic Google matrix G (in this subsec- with each node communicating O[log(n)] bits of data to tion we use H and h for local and non-local Hamiltonians, each of its connected neighbors after each step. After respectively): O[log(n)] steps, the total number of walkers that have visited a node is used to estimate the PageRank of that h1 = h(G) (11 G)† (11 G) . (61) node. In the absence of synchronization costs [synchro- ≡ − − nization and communication are known to be important Since h(G) is positive semi-definite, and 1 is the maxi- issues for networks with a large number of processors mal eigenvalue of G associated with ~p, it follows that the (Awerbuch, 1985; Bisseling, 2004; Kumar et al., 2003; ground state of h(G) is given by π ~p/ ~p . The initial | i ≡ k k Rauber and Rünger, 2010)], the classical cost can be Hamiltonian has a similar form, but it is associated with taken to be O[n log(n)2], i.e., the number of parallel pro- the Google matrix Gc of the complete graph cesses multiplied by the duration of each process.20 At the conclusion of the adiabatic evolution gener- h = h(G ) (11 G )† (11 G ) . (62) 0 c ≡ − c − c ated by the Hamiltonian in Eq. (63), the PageRank The ground state of h0 is the uniform superposition state vector ~p = pi is encoded into the quantum PageR- n { } n ψ(0) = j /√n. The basis vectors j span the ank state π = √πi i of a (log n)-qubit system, j=1 | i i=1 | i 2 |n-dimensionali Hilbert| i space of log n qubits| i The inter- where i denotes the i-th node in the graph G. The 2 | i P polating adiabaticP Hamiltonian is

h(s) = (1 s)h + sh . (63) − 0 1 19 The gap becomes too small for a quantum advantage, i.e., scales as 1/poly(n), for graphs with only in-degree power-law distribution or when the out-degrees are equal to the in-degrees. This 18 This theorem states that if all elements of a real symmetric was studied in more detail in (Frees et al., 2013). square matrix A are non-negative, then the largest eigenvalue 20 This analysis improves upon the estimates of the classical cost of A is real; furthermore, the components of the corresponding presented in (Garnerone et al., 2012), and accounts for the cri- eigenvector can be chosen to be all non-negative. tique presented in (Moussa, 2013). 18

probability of measuring node i is π = p2/ ~p 2. One IV. UNIVERSALITY OF AQC i i k k can estimate πi by repeatedly sampling the expectation z value of the operator σi in the final state. The num- What is the relation between the computational power ber of measurements M needed to estimate πi is given of the circuit model and the adiabatic model of quan- by the Chernoff-Hoeffding bound (Hoeffding, 1963), al- tum computing? It turns out that they are equivalent, lowing one to approximate πi with an additive error ei up to polynomial overhead. It is well known that the 1 and with M = poly(ei− ). A nontrivial approximation circuit model is universal for quantum computing, i.e., requires e p and, these are typically O(1/n). that there exist sets of gates acting on a constant num- i ≤ i The fact that the amplitudes of the quantum PageR- ber of qubits each that can efficiently simulate a quantum Turing machine (Deutsch, 1985; Yao, 1993). A set ank state are √πi = pi/ ~p , rather than √pi , is a virtue: the number{ of samplesk k} needed to estimate{ } the of gates is said to be universal for QC if any unitary op- 2γi 1 eration may be approximated to arbitrary accuracy by a rank πi with additive error ei πi scales as O[n − ], so ∼2γ 1 21 quantum circuit involving only those gates (Nielsen and the total quantum cost is O[n i− polylog(n)]. Thus, for the combined task of state preparation and rank Chuang, 2000). The analog of such a set of gates in AQC estimation, there is a polynomial quantum speedup is a Hamiltonian. An operational definition of universal 2γi 1 AQC is thus to efficiently map any circuit to an adiabatic whenever γi < 1, namely O[n − polylog(n)] vs. O[n polylog(n)]; simulations reported in (Garnerone computation using a sufficiently powerful Hamiltonian. et al., 2012) show that this is indeed the case for the top- Formally: ranked log(n) entries, and in applications one is most Definition 3 (Universal Adiabatic Quantum Computa- often interested in the top entries. We emphasize that tion). A time-dependent Hamiltonian H(t), t [0, tf ], is this holds in the average (not worst) case, and is not a universal for AQC if, given an arbitrary quantum∈ circuit provable speedup; the evidence for the scaling is numer- U operating on an arbitrary initial state ψ of n p-state | i ical, and it is unknown whether a classical algorithm for particles and having depth L, the ground state of H(tf ) the preparation of π rather than ~p may give a similar is equal to U ψ with probability greater than > 0, the scaling to the quantum scaling, though if that is the case number of particles| i H(t) operates on is poly(n) t, and 2 2 ∀ one could consider quantum preparation of πi / π , tf = poly(n, L). etc. { k k } The stipulation that the ground state of H(t ) is equal Another context for useful applications is comparing f to the final state at the end of the circuit ensures that successive PageRanks, or more generally “q-sampling” the circuit and the adiabatic computation have the same (Aharonov and Ta-Shma, 2003). Suppose one per- output. We note that it is possible and useful to relax turbs the web-graph. The adiabatic quantum algorithm the ground state requirement and replace it with another can provide, in time O[polylog(n)], the pre- and post- eigenstate of H(t) (see, e.g., Sec. VI.C). The requirement perturbation states π and π˜ as input to a quantum that the number of particles and time taken by the adi- circuit implementing| thei SWAP-test| i (Buhrman et al., abatic computation are polynomial in n and L ensures 2001). To obtain an estimate of the fidelity π π˜ 2 that the resources used do not blow up. one needs to measure an ancilla O(1) times, the|h num-| i| We begin, in Sec. IV.A, by showing that the circuit ber depending only on the desired precision. In contrast, model can efficiently simulate AQC. The real challenge deciding whether two probability distributions are close is to show the other direction, i.e., that AQC can ef- classically requires O[n2/3 log n] samples from each (Batu ficiently simulate the circuit model, which is what we et al., 2000). Whenever some relevant perturbation of the devote the rest of this section to. Along the way, this previous quantum PageRank state is observed, one can establishes the universality of AQC. We present several decide to run the classical algorithm again to update the proofs, starting in Sec. IV.B with a detailed review of classical PageRank. the history state construction of (Aharonov et al., 2007), who showed in addition that six-state particles in two dimensions suffice for universal adiabatic quantum computation. This was improved in (Kempe et al., 2006), using perturbation-theory gadgets, who showed that qubits 21 It was observed numerically in (Garnerone et al., 2012) that pi ∝ γi 2 can be used instead of six-state particles, and that adi- 1/n , where γi ∈ (0.6, 1], and that k~pk2 ∝ 1/n. Let ei denote 2 2 2 abatic evolution with 2-local Hamiltonians is quantum the additive error corresponding to πi = pi /k~pk2 ∼ npi . It follows from the Chernoff-Hoeffding inequality that the number universal. A 2-local model of universal AQC in 2D, which of samples M(x) from the distribution x, where x = π = {πi} we review in Sec. IV.C, was proposed in (Mizel et al., (output of the quantum algorithm), required for a given, fixed additive estimation error, is proportional to the inverse of the 2007), using fermions. Universal AQC using qubits on a additive error: M(π) ∼ 1/ei. Assuming ei ∼ πi, it follows that two-dimensional grid was accomplished in (Oliveira and 2 2γi 1 M(π) ∼ 1/πi ∼ 1/(npi ) ∼ n − . The total cost required to Terhal, 2008). Further simplifications of universal AQC prepare the sample in the quantum case is O[polylog(n)]. in 2D were presented in (Breuckmann and Terhal, 2014; 19

Gosset et al., 2015; Lloyd and Terhal, 2016), using the Therefore, accounting for the M terms in the product space-time circuit model, which we review in Sec. IV.D. and observing that the error in Eq. (67) is subdominant, The ultimate reduction in spatial dimensionality was ac- the total error is (van Dam et al., 2001): complished in (Aharonov et al., 2009), who showed that M universal AQC is possible with 1D 9-state particles, as 2 U(t , 0) U 00 O poly(n)t /M . (71) we review in Sec. IV.E. Finally, in Sec. IV.F we review k f − mk ∈ f m=1 a construction that allows one to quadratically amplify Y the gap of any Hamiltonian used in AQC (satisfying a This means we can approximate U(tf , 0) with a product frustration-freeness property), though this requires the 2 of 2M unitaries provided that M scales as tf poly(n). computation to take place in an excited state. Depending on the form of H0 and H1, they may need further decomposition in order to write the terms in Eq. (69) in terms of few-qubit unitaries. E.g., for the A. The circuit model can efficiently simulate AQC x standard initial Hamiltonian H0 = i σi , which is a sum of commuting single qubit operators,− we can write That the circuit model can efficiently simulate the adi- e i∆t(1 t/tf )H0/K as a product of n one-qubitP unitaries. abatic model is relatively straightforward and was first − − Likewise, assuming that H1 is 2-local, we can write shown in (Farhi et al., 2000). Assume for simplicity a m∆t i ∆tH1/M linear schedule, i.e., an AQC Hamiltonian of the form e− tf as a product of up to n2 two-qubit uni- t t H(t) = (1 )H0 + H1. The evolution of a quan- taries within the same order of approximation as Eq. (69). − tf tf tum system generated by the time-dependent Hamilto- Thus, U(tf , 0) can be approximated as a product of uni- nian H(t) is governed by the unitary operator: tary operators each of which acts on a few qubits. The scaling of tf required for adiabatic evolution is inherited tf by the number of few-qubit unitary operators in the as- U(t , 0) = Texp i dtH(t) . (65) f − sociated circuit version of the algorithm. 0 Z A more efficient method was proposed in (Boixo If tf satisfies the condition for adiabaticity, U(tf , 0) will et al., 2009b), building upon the ideas explained in map the ground state at t = 0 to the ground state at tf . Sec. II.B.4. This “eigenpath traversal by phase random- Therefore it suffices to show that the circuit model can ization” method applies the Hamiltonian H(tj) in piece- simulate U(tf , 0). To do so, we approximate the evolu- wise continuous manner at random times tj. Each in- tion by a product of unitaries involving time-independent iH(tj ) terval [tj, tj+1] corresponds to a unitary e− , which Hamiltonians H0 H(m∆t): m ≡ then needs to be decomposed into one- and two-qubit M M gates, as above. The randomization introduces an effec- i∆tH0 U(t , 0) U 0(t , 0) = U 0 = e− m , (66) tive eigenstate decoupling in the Hamiltonian eigenba- f 7→ f m m=1 m=1 sis (similarly to the effect achieved by projections in the Y Y Zeno effect), so that if the initial state is the ground state, where ∆t = t /M. The error incurred by this approxi- f the evolution will follow the ground state throughout as mation is (van Dam et al., 2001): required for AQC. The algorithmic cost of this randomization method is defined as the average number of times U(t , 0) U 0(t , 0) O t poly(n)/M , (67) k f − f k ∈ f the unitaries are applied, and it can be shown that the q cost is O[L2/(ε∆)], where ε is the desired maximum error where we used of the final state compared to the target eigenstate, and L 1 is the path length [Eq. (13)]. Since L maxs H˙ (s) /∆ H(t) H0mt/t H1 H0 (68) ≤ k k k − d f ek ≤ M k − k as we saw in Sec. II.B.4, the worst-case bound on the cost ˙ 2 3 O(poly(n)/M) . is maxs H(s) /(ε∆ ), up to logarithmic factors. ∈ k k We now wish to approximate each individual term in the product in Eq. (66) using the Baker-Campbell-Hausdorff B. AQC can efficiently simulate the circuit model: history formula (Klarsfeld and Oteo, 1989) by: state proof

m∆t m∆t i∆t 1 H0 i∆t H1 − − tf − tf The goal is, given an arbitrary n-qubit quantum cir- Um0 Um00 = e e , (69) 7→ cuit, to design an adiabatic computation whose ﬁnal which incurs an error eA+B eAeB O ( AB ) due to ground state is the output of the quantum circuit de- k − k ∈ k k the neglected leading order commutator term [A, B]/2, scribed by a sequence of L one or two-qubit unitary gates, i.e., U1,U2,...UL. This adiabatic simulation of the circuit

2 should be eﬃcient, i.e., it may incur at most polynomial tf overhead in the circuit depth L. In this subsection we re- Um0 Um00 O 2 H0H1 . (70) k − k ∈ M k k! view the proof presented in (Aharonov et al., 2007). This 20

L was the first complete proof of the universality of AQC, Hinput: Ensures that if the clock state is 0 , then and many of the ideas and techniques introduced therein • the computation qubits are in the 0n state.| i inspired subsequent proofs, remaining relevant today. | i n Let us assume that the n-qubit input to the circuit is H = 1 1 0 0 (77) the 0 0 state. After the `-th gate, the state of the input | iih i| ⊗ | 1ich 1| i=1 quantum| ··· circuiti is given by α(`) . To proceed, we use X the “circuit-to-Hamiltonian”| constructioni (Kitaev et al., H : Ensures that the propagation from ` 1 to ` 2000), where the final Hamiltonian will have as its ground ` • corresponds to the application of U . − state the entire history of the quantum computation. ` This “history state” is given by: H = 11 0 0 0 0 U 1 0 0 0 1 ⊗ | 1 2ich 1 2| − 1| 1 2ich 1 2| L 1 U † 0102 c 1102 + 11 1102 c 1102 (78a) η = γ(`) (72a) − 1 | i h | ⊗ | i h | | i √L + 1 | i H2 ` L 1 = 11 1` 10`0`+1 c 1` 10`0`+1 X`=0 ≤ ≤ − ⊗ | − i h − | ` L ` U 1 1 0 1 0 0 γ(`) α(`) 1 0 − (72b) ` ` 1 ` `+1 c ` 1 ` `+1 | i ≡ | i ⊗ | ic − | − i h − | ` L ` U`† 1` 10`0`+1 c 1` 11`0`+1 where 1 0 − c denotes the “Feynman clock” (Feynman, − | − i h − | | i + 11 1` 11`0`+1 c 1` 11`0`+1 (78b) 1985) register composed of L + 1 qubits. The notation ⊗ | − i h − | means that we have ` ones followed by L ` zeros to HL = 11 1L 10L c 1L 10L − ⊗ | − i h − | denote the time after the `-th gate. We wish to con- UL 1L 11L c 1L 10L − | − i h − | struct a Hamiltonian Hinit with ground state γ(0) and a Hamiltonian H with ground state η . Let:| i U1† 1L 10L 1L 11L final | i − | − ih − | + 11 1L 11L c 1L 11L (78c) − − Hinit = Hc init + Hinput + Hc (73a) ⊗ | i h | − 1 Note that the first and last terms leave the state H = H + H + H (73b) final 2 circuit input c unchanged. The second term propagates the com- L putational state and clock register forward, while Hcircuit = H` . (73c) the third term propagates the computational state and clock register backward. X`=1 The full time independent Hamiltonian H(s) is given It turns out that the state γ(0) = α(0) 0L is the by (Aharonov et al., 2007): ground state of H with eigenvalue| i | 0, andi ⊗ | η i is the init | i H(s) = (1 s)H + sH (74) ground state of Hfinal with eigenvalue 0. Let 0 be the − init final L S s subspace spanned by γ(`) `=0. The state α(0) is the = Hinput + Hc + (1 s)Hc init + Hcircuit . {| i} | i − − 2 input to the circuit, so it can be taken to be the 0 0 state, i.e., the initial ground state is an easily prepared| ··· i The various terms are chosen so that the ground state state. Since the initial state γ(0) , the dynamics always has energy 0: 0 generated by H(s) keep the state| ini ∈. S It turns out that S0 Hc: This term should ensure that the clock’s state the ground state is unique for s [0, 1]. By mapping • ` L ` ∈ is always of the form 1 0 − . Therefore, we en- the Hamiltonian within to a stochastic matrix, it is | ic 0 ergetically penalize any clock-basis state that has possible to find a polynomialS lower bound on the gap the sequence 01: from the ground state within : S0 L 1 − 1 1 2 Hc = 0`1`+1 c 0`1`+1 (75) ∆(H ) . (79) | i h | S0 ≥ 4 6L X`=1 where 0`1`+1 c denotes a 0 on the `-th clock qubit It is also possible to bound the global gap (i.e., not | i and 1 on the (`+1)-th clock qubit. Any illegal clock restricted to the 0 subspace) as state will have an energy 1. Any legal clock state S will have energy 0. ≥ ∆(H) Ω(1/L3) . (80) ≥ L Hc init: Ensures that the initial clock state is 0 c. A measurement of the final state will find the final out- • − | i come of the quantum circuit γ(L) with probability 1 . Hc init = 11 c 11 (76) | i L+1 − | i h | This can be amplified by inserting identity operators at Note that we only need to specify the first clock the end of the circuit, hence causing the history state qubit to be in the zero state. For a legal clock to include a greater superposition of the final outcome state, Eqs. (75) and (76) imply that the rest are in of the circuit. Together, these results show that there is the zero state as well. an efficient implementation of any given quantum circuit 21 using the adiabatic algorithm with H(s). Here and else- It is convenient to group creation operators into row where “efficient” means up to polynomial overhead, i.e., vectors Cq,`† = (aq,`† bq,`† ). Then for each single-qubit gate where tf scales as a polynomial in L. U (1) we introduce a term The proof techniques used to obtain these results are q,` instructive and of independent interest, so we review ad- (1) (1) (1) Hq,` (s) = Cq,`† sCq,`† 1(Uq,` )† Cq,` sCq,` 1Uq,` ditional technical details in AppendixC. − − − − To conclude this section, we briefly mention addi- (81) tional results supporting the equivalence of the circuit into the circuit Hamiltonian Hcircuit. The off-diagonal and adiabatic approach, in terms of state preparation. terms represent hopping or tunneling of the q-th electron from site ` 1 to ` (and v.v.), while U (1) acts on In (Aharonov and Ta-Shma, 2003) it was shown (The- − q,` orem 2) that any quantum state that can be efficiently the electron’s spin. The diagonal terms Cq,`† Cq,` and generated in the circuit model can also be efficiently gen- (1) Cq,`† 1Cq,` 1 ensure that Hq,` 0. The parameter erated by an adiabatic approach, and vice versa, for the s −[0, 1] controls− the interpolation≥ from the initial, sim- same initial state. The proof relies on two important lem- ple∈ to prepare ground state at s = 0 when there is no mas, the “sparse Hamiltonian lemma” and the “jagged tunneling and every electron is frozen in place, to the adiabatic path” lemma. The former gives conditions un- (1) full realization of all the gates U when s = 1. der which a Hamiltonian is efficiently simulatable in the q,` One can similarly define CNOT Hamiltonian terms be- circuit model, and the latter provides conditions under tween electrons or fermions, whose form can be found which a sequence of Hamiltonians, defining a path, can in (Mizel et al., 2007) [see also (Breuckmann and Ter- have a non-negligible spectral gap. hal, 2014)]. These 2-local terms can be understood as a sum of an identity and NOT term. For such two-qubit gates, the fermions corresponding to the control and tar- C. Fermionic ground state quantum computation get qubits both tunnel forward or backward and the internal spin-state of the target fermion changes depending A model of ground state quantum computation on the internal state of the control fermion. An impor- (GSQC) using fermions was independently proposed in tant additional ingredient is the addition of a penalty (Mizel et al., 2001) [see also (Mizel, 2004; Mizel et al., term that imposes an energy penalty on states in which 2002) and (Mao, 2005a,b)] around the same time as AQC. one qubit has gone through the CNOT gate without the In GSQC, one executes a quantum circuit by producing other. Instead of the Feynman clock used in the history a ground state that spatially encodes the entire temporal state construction, there are many local clocks, one per trajectory of the circuit, from input to output. It was qubit. The synchronization mechanism takes place via shown in (Mizel et al., 2007) how to adiabatically reach the CNOT Hamiltonian. Moreover, the entire construc- the desired ground state, thus providing an alternative tion naturally involves only 2-local interactions between to history state type constructions for universal AQC. fermions in 2D. One of the differences between the GSQC and history While the fermionic GSQC model proposed in (Mizel states constructions is that instead of relying on Feyn- et al., 2007) was shown there to be universal for AQC, its man “global clock particle” idea, particles are synchro- gap analysis was incomplete.22 This was fixed in (Childs nized locally (via CNOT gates), an idea that traces back et al., 2013), which proved the “Nullspace Projection to (Margolus, 1990) and was later adopted in some of the Lemma” that was implicitly assumed in (Mizel et al., space-time circuit-to-Hamiltonian constructions (Breuck- 2007). This lemma is interesting in its own right, so we mann and Terhal, 2014). reproduce it here: Consider a quantum circuit with n qubits and depth L. We associate 2(L + 1) fermionic modes with every Lemma 2 (Nullspace Projection Lemma (Childs et al., qubit q, via creation operators aq,`† and bq,`† , where ` = 2013)). Let ∆(A) denote the smallest nonzero eigenvalue 0,...,L. One can view these 2n(L + 1) modes as the of the positive semidefinite operator A. Let H0 and H1 state-space of n spin-1/2 fermions, where each fermion can be localized at sites on a 1D (time)-line of length L + 1. To illustrate this with a concrete physical system, 22 imagine a two-dimensional array of quantum dots with On p.4 of (Mizel et al., 2007) it was claimed that “hZ|H|Zi ≥ EO(1/N 2)” (N is L in our notation), implying a lower bound L + 1 columns and two rows per qubit, corresponding to on the spectral gap of the total Hamiltonian H. This claim the 0 and 1 basis states of that qubit. A total of n was based on (Mizel et al., 2002), but was in fact not proven electrons| i are| placedi in the array. The state of each qubit there. Here E is the energy scale of the CNOT terms, the total determines the spin state of the corresponding electron, Hamiltonian is H = H0 + H1, where H0 contains all of the single qubit terms and H1 includes all of the CNOT terms, and which in term determines which of the two rows it is |Zi denotes the known ground state of H0. As pointed out in in, while the clock of each qubit is represented by which (Breuckmann and Terhal, 2014), the missing step is essentially column the electron is in. to exclude zero-energy, invalid time-configurations. 22

be positive semidefinite and assume the nullspace S of H0 7 8 7 4 11 n is non-empty. Also assume that ∆(H1 S) c > 0 and 4 11 2 | ≥ 2 8 14 2 8 14 ∆(H0) d > 0. Then ≥ 1 5 12 16 1 5 12 16 w 3 9 15 3 9 15 cd 6 13 6 13 ∆(H0 + H1) . (82) ≥ c + d + H 10 10 n k 1k 1 2 As shown in (Breuckmann and Terhal, 2014), the 1 8 t fermionic GSQC model can be unitarily mapped onto (a) (b) (c) the space-time circuit-to-Hamiltonian model for qubits in 2D, where the gap analysis is more convenient. Using FIG. 3 (a) A 2n = 8 qubit quantum circuit, where each grey the same mapping, (Breuckmann and Terhal, 2014) also square (n2 = 16 in total) corresponds to a 2-qubit gate. (b) showed that the fermionic model of (Mizel et al., 2007) is An equivalent representation of the quantum circuit in (a) in in fact QMA-complete. We thus proceed to discuss the terms of a rotated grid. The red dashed line corresponds to an space-time model next. allowed string configuration for the particles. (c) The circuit is constrained such that the majority of the gates are identity except in a k × k subgrid (shown in black), located such that D. Space-time Circuit-to-Hamiltonian Construction its left vertex is at the center of the rotated grid. A successful computation requires the t position of the 2k particles with w positions that cross the interaction region, to lie to the right Here we briefly review another construction that re- of the interaction region. See also Fig. 1 in (Gosset et al., alizes universal adiabatic quantum computation (Gosset 2015). et al., 2015; Lloyd and Terhal, 2016). This builds on the so-called space-time circuit-to-Hamiltonian construction (Breuckmann and Terhal, 2014), which in turn is based and 1 correspond to an edge going down and right. The on the Hamiltonian computation construction of (Janz- Hamming weight of such configurations must be n, since ing, 2007). We consider the 2n-qubit quantum circuit they start and end in the middle of the grid and so must with n2 two-qubit gates, arranged as shown in Fig. 3(a). go left and right an equal number of times. This form is sufficient for universal quantum computation We are now ready to describe the Hamiltonian: (Janzing, 2007). An equivalent representation of the circuit is given in Fig. 3(b), where the n2 gates are arranged H(λ) = Hstring + Hcircuit(λ) + Hinput . (83) in a rotated n n grid. Each plaquette p is associated H : This term ensures that the ground state has with a gate U ,× of which the majority are identity gates. input p the• internal state of all particles set to s = 0 when the Only a k k subgrid of the n n grid with k = √n/16 string lies on the left-hand side of the grid by energet- has non-identity× gates, with the× subgrid located as shown ically penalizing all states (on the left-hand side) with in Fig. 3(c). This region is referred to as the interaction s = 1. It is given by: region. The circuit is mapped to a Hamiltonian H(λ), with 2n λ [0, 1]. The Hamiltonian describes the evolution of Hinput = nt,1[w] . (84) ∈ particles that live on the edges of the rotated n n grid. w=1 t n X X≤ The positions of the particles are given in terms× of the H : This term ensures that the ground state is coordinates (t, w) 1,..., 2n 2 as shown in Fig. 3(b). string in• the subspace of connected strings. Consider a sin- Each particle has two∈ { internal} degrees of freedom in or- gle vertex v in the grid with incident edges labeled by der to encode the qubits of the circuit. Let a [w] de- t,s (t, w), (t + 1, w), (t, w + 1), (t + 1, w + 1). We can asso- note the annihilation operator which annihilates a parti- ciate a Hamiltonian Hv to each vertex, cle with internal state s 0, 1 on the edge (t, w). The string ∈ { } v number operator is defined as nt,s[w] = at,s† [w]at,s[w], Hstring = nt[w] + nt+1[w] + nt[w + 1] + nt+1[w + 1] which counts the number of particles (which will be ei- 2 (nt[w] + nt+1[w]) (nt[w + 1] + nt+1[w + 1]) ther 0 or 1) at position (t, w) with state s. Let n [w] = − t (85) nt,0[w] + nt,1[w]. We focus on configurations of particles that form con- (for vertices at the boundary of the grid with two or v nected segments starting at the top and ending at the three incident edges, the definition of Hstring needs to v bottom [an example is shown in Fig. 3(b)], referred to as be modified accordingly) such that Hstring = v Hstring. consistent connected string configurations. For a fixed w For connected string configurations, the energy due to (i.e., a horizontal line on the rotated grid), there is only this Hamiltonian is zero, while disconnected stringsP with one occupied edge. We can describe such configurations L string segments have a higher energy 2L 2. p 2 − in terms of 2n bits, denoted by z. Specifically, let the Hcircuit(λ) = p Hgate(λ)+√1 λ Hinit: Define for bit value 0 correspond to an edge going down and left each• plaquette p with borders given− by the edges (t, w), P { 23

(t + 1, w), (t, w + 1), (t + 1, w + 1) E. Universal AQC in 1D with 9-state particles } Hp (λ) = n [w]n [w + 1] + n [w]n [w + 1] + λHp gate t t t+1 t+1 prop The constructions of universal AQC we have reviewed (86) so far are all spatially two-dimensional (2D). It was un- where clear for some time whether universal AQC is possible in p H = β, δ U α, γ a† a [w] 1D, with some suggestive evidence to the contrary, such prop − h | p| i t+1,β t,α α,β,γ,δX as the impressive success of density matrix renormal- ization group (DMRG) techniques in calculating ground a† [w + 1]a [w + 1] + h.c. (87) × t+1,δ t,γ state energies and other properties of a variety of 1D p The term Hprop allows for a pair of particles located on quantum systems (Schollwöck, 2005). Moreover, classical the left (right) edges of a plaquette to hop together such 1D systems are generally “easy”; e.g., a 1D restriction of that they both are located on the right (left) edges of a MAX-2-SAT with p-state variables can be solved by dy- plaquette, with their internal states changed according namic programming and hence is in the complexity class to Up (Up†). Note that this move preserves the connect- P. In addition, the area law implies that 1D systems with p edness of the string. Furthermore, the term p Hgate(0) a constant spectral gap can be efficiently simulated clas- is minimized by a configuration lying either entirely on sically (Hastings, 2009). All this implies that adiabatic the left border (corresponding to the bit stringPz = 0n1n) evolution with 1D Hamiltonians is not useful for univer- or entirely on the right border (z = 1n0n), which in con- sal QC unless certain conditions are met, in particular a junction with Hinit, given by spectral gap that tends to zero. This was accomplished in (Aharonov et al., 2009), who H = n [w = 1] + n [w = 2n] , (88) init n+1 n+1 proved that it is possible to perform universal AQC using ensures that the ground state of H(0) is such that all a 1D quantum system of 9-state particles. The striking particles lie along the left boundary of the grid. Including qualitative difference between the quantum and the clas- the effect of Hcircuit(0) and Hinput, the ground state of sical 1D versions of the same problem seems surprising. H(0) is given by 02n 0n1n with eigenvalue 1. This is However, the k-local Hamiltonian problem allows for the an easily prepared| groundi| state.i encoding of an extra dimension (time), by making the It can be shown that the ground state of H(λ) ground state a superposition of states corresponding to [Eq. (83)] along λ [0, 1] is unique and the energy gap different times. This means that the correct analogue of above the ground state∈ is lower bounded by 1/poly(n) for the quantum 1D local Hamiltonian problem is 2D classi- all λ [0, 1] [Theorem 1 in Ref. (Gosset et al., 2015)]. To cal MAX-k-SAT, which is NP-complete. measure∈ the output of the quantum circuit, we measure The proof presented in (Aharonov et al., 2009) builds the t positions of the 2k particles for the w values that heavily on the history state construction reviewed in cross the interaction region (recall that there will always Sec. IV.B. However, there are a couple of important dif- be one particle per horizontal w line). If we find that ferences. As in the history state construction, the start- all 2k particles lie to the right of the interaction region, ing point is a quantum circuit Ux acting on n qubits then their internal states must encode the output of the (where x is the classical input to the function imple- quantum circuit. For the choice k = √n/16, this occurs mented by the circuit in the universal AQC case). A with a probability lower bounded by a positive constant. 1D p-state Hamiltonian is designed which will verify cor- Together, these properties allow for an efficient (up to rect propagation according to this circuit. Then this is polynomial overhead) simulation of the quantum circuit used as the final Hamiltonian for the adiabatic evolution. using the adiabatic algorithm generated by H(λ). The problem with directly realizing this in the 1D case However, this implementation requires 4-body inter- is that only the particles nearest to the clock would be actions [see for example the product term in Eq. (85)]. able to take advantage of it in order to check correct In (Lloyd and Terhal, 2016), improvements to the above propagation in time. To overcome this, the circuit Ux construction were presented with only 2-local interac- is first modified into a new circuit U˜x with a distributed tions using a first order perturbation gadget and a clock. The history state construction relies on the ability quadratic increase in the number of qubits from the orig- to copy qubits from one column to the next in order to inal quantum circuit. The use of only first order per- move to the next block of gates in the computation, so turbation theory is particularly significant, since effec- a new strategy is needed in 1D. For the modified circuit tive interactions obtained in k-th order degenerate per- U˜x, the qubits are instead placed in a block of n adjacent turbation theory with perturbative coupling g and gap particles. One set of gates is performed, and then all of ∆ of the unperturbed Hamiltonian scale in strength as the qubits are moved over n places to advance time in the k 1 g(g/∆) − , leading to a correspondingly small gap of original circuit Ux. More states per particle are needed the effective Hamiltonian. In addition, multiple uses of to accomplish this than in the 2D case. The second main higher-order perturbation theory can increase qubit over- new idea that is needed is related to ensuring that the head and complexity. state of the system had a valid structure. In the 2D case 24 local constraints were used to check that there are no two We now review the construction in (Somma and Boixo, qubit states in adjacent columns. However, using only lo- 2013) in some detail. To place it in context, note that the cal constraints, there is no way to check that there are ex- universality results we reviewed thus far can be summa- actly n qubit data states in an unknown location in a 1D rized as follows: Any quantum circuit specified by uni- system, since there are only a constant number of local tary gates U1,...,UQ can be simulated by an adiabatic rules available, which are therefore unable to count to an quantum evolution involving frustration-free Hamiltoni- L arbitrarily large n. Instead, it is ensured that, under the ans: H(s) = k=1 ak(s)Πk(s). The ground state of the transition rules of the system, any invalid configurations final Hamiltonian H(1) has large overlap with the out- will evolve in polynomial time into a configuration which put state of theP quantum circuit. Moreover, L is poly- can be detected as illegal by local rules. Thus, for every nomial in Q, and Πk denotes nearest-neighbor, two-body state which is not a valid history state, either the prop- interactions between spins of corresponding many-body agation is wrong, which implies an energy penalty due systems in one- or two-dimensional lattices. The inverse to the propagation Hamiltonian, or the state evolves to minimum gap of H(s) is polynomial in Q, and hence so an illegal configuration which is locally detectable, which is the duration of the adiabatic simulation. implies an energy penalty due to the local check of illegal Now, consider a frustration-free Hamiltonian configurations. For additional technical details required L to complete the proof see (Aharonov et al., 2009). A 20- H(s) = a Π (s) , (89) state translation-invariant modification of the construc- k k k=1 tion from (Aharonov et al., 2009) for universal AQC in X 1D was given in (Nagaj and Wocjan, 2008), improving where each Πk(s) is a projector for all s [0, 1], and is on a 56-state construction by (Janzing et al., 2008). a local operator. Denote the eigenvalues∈ of this Hamil- tonian by λj , where λ1 = 0 is the ground state energy. Then, take{ the} Hamiltonian F. Adiabatic gap amplification L H¯ (s) = √a Π (s) ( k 0 + 0 k ) , (90) k k ⊗ | ih | | ih | In all universality constructions the run time of the k=1 adiabatic simulation of a quantum circuit depends on X the inverse minimum gap of the simulating Hamiltonian. where 0 and k are ancilla registers defined over one | i | i Therefore it is of interest to develop a general technique and log2(L) qubits, respectively. It can be shown that for amplifying this gap, as was done in (Somma and H¯ (s) has the desired properties, i.e., if ψ(s) was the | i Boixo, 2013). ground state of H, then ψ(s) 10 0 is a (degener- | i| ¯··· i Consider a Hamiltonian H with ground state φ . The ate) zero-eigenvalue eigenstate of H and the eigenvalues | i of H¯ (s) are λ [the proof is given in Appendix B goal is to construct a new Hamiltonian H0 that has φ {± j} as an eigenstate (not necessarily the ground state) but| i of (Somma and Boixo, 2013)]. Thus, the gap has been p ¯ with a larger spectral gap. A quadratic spectral gap am- quadratically amplified, and one can evolve with H to plification is possible when H is frustration-free [see also transform eigenstates at s = 0 to eigenstates at s = 1 and (Bravyi and Terhal, 2009)]: simulate the original quantum circuit with a quadratic speedup over the simulation involving H. ¯ Definition 4. (frustration freeness) A Hamiltonian H In general H(s) will be log2(L)-local due to the ap- N N is frustration free if it can be written as a sum∈ pearance of k . To avoid these many-body interac- C C | i over× positive semi-definite operators: H = L a Π , tions one can represent k using a unary encoding, i.e., k=1 k k | i with a [0, 1] and L = polylog(N). Further, if φ k 0 ... 010 ... 0 (with 1 at the k-th position). In this k | i 7→ | i is the ground∈ state of H then it is a groundP state (i.e.,| i single-particle subspace the new Hamiltonian becomes zero eigenvector) of every term in the decomposition of L H, i.e., Π φ = 0 k. + + k H¯ (s) = √a Π (s) (σ σ− + σ−σ ) , (91) | i ∀ k k ⊗ k 0 k 0 k=1 (Somma and Boixo, 2013) took Πk as projectors. X x y The quadratic amplification is optimal for frustration- where σ± = (σ iσ )/2 are Pauli raising and lowering ± free Hamiltonians in a suitable black-box model, and operators. Note that since each Πk(s) interacts with the no spectral gap amplification is possible, in general, if same qubit 0 of the new register, if the original H was the frustration-free property is removed. An important geometrically local, then H¯ is not, i.e., it has a central caveat is that the construction replaces ground state evo- spin geometry. lution by evolution of a state that lies in the middle of One more issue that needs to be dealt with is the de- the spectrum; thus it does not fit the strict definition of generacy of the zero eigenvalue. To remove this degen- AQC (Def.1). We will have another occasion to relax eracy from contributions within the single-particle sub- 1 √ z ¯ the definition in the same sense, in Sec. VI.C. space one can add a penalty term 4 ∆(11 + σ0 ) to H(s), 25

which penalizes all states with qubit 0 in 0 ; the rele- Hamiltonian quantum complexity theory is an ex- vant spectral gap in the single-particle subspace| i is then tremely rich subject that is rapidly advancing and has still of order √∆. To remove additional degeneracy already been reviewed a number of times, so we will only from the many-particle subspaces one can add penal- touch upon it and highlight some aspects that are rele- ties for states that belong to such subspaces. Adding vant to AQC. Perhaps the most direct connection is the L z Z = (L 2)11 k=0 σk achieves this since it acts as a fact that 2-local Hamiltonians of a form that naturally penalty− that grows− with the Hamming weight a of states appears in AQC, are QMA-complete. Additionally, some in the a-particleP subspace. Thus of the technical tools that played an important role in QMA-completeness locality reductions, such as pertur- L 1 bative gadgets, have also found great use in proofs of the H0(s) = √a Π (s) ( k 0 + 0 k ) L1/d k k ⊗ | ih | | ih | universality of AQC with different Hamiltonians. "k=1 X The reviews (Aharonov and Naveh, 2002; Gharibian, 1 + √∆(11 + σz) + Z, (92) 2013; Gharibian et al., 2015), are excellent resources for 4 0 additional perspectives and details on Hamiltonian quantum complexity theory. has ψ0 10 0 as a unique eigenstate of eigenvalue 0, and| all otheri| ··· eigenvaluesi are at distance at least √∆/L1/d if d 2.23 This is the desired quadratic gap amplification A. Background result.≥ How far can gap amplification methods go? It was 1. Boolean Satisfiability Problem: k-SAT shown in (Schaller, 2008) that for the one-dimensional transverse-field quantum Ising model, and for the prepa- Consider a Boolean formula Φ that depends on n lit- ration of cluster states (Raussendorf and Briegel, 2001), erals xi 0, 1 (with 0 and 1 representing False and it is possible to use a series of straight-line interpolations True, respectively)∈ { } or their negations. The problem is to in order to generate a schedule along which the gap is decide whether there exists an assignment of values to always greater than a constant independent of the sys- the literals that satisfies the Boolean formula, i.e., such tem size, thus avoiding the quantum phase transition. that Φ = 1. If there exists such an assignment then the However, there exists an efficient method to compute the formula is satisfiable, otherwise it is unsatisfiable. ground state expectation values of local operators of 2D The Boolean formula is typically written in conjunc- lattice Hamiltonians undergoing exact adiabatic evolu- tive normal form: it is written in terms of a conjunc- tion, and this implies that adiabatic quantum algorithms tion (AND - ) of r clauses, where each clause contains based on such local Hamiltonians, with unique ground the disjunction∧ (OR - ) of k literals (variables) or their states, can be simulated efficiently if the spectral gap negation (NOT - ). A∨ literal and its negation are often does not scale with the system size (Osborne, 2007). referred to as positive¬ and negative literals. The Boolean formula is written as: V. HAMILTONIAN QUANTUM COMPLEXITY THEORY Φ = C C C (93) AND UNIVERSAL AQC 1 ∧ 2 ∧ · · · ∧ r where C = x x x and x is the j-th positive In this section we review Hamiltonian quantum com- i i1 i2 ik ij or negative literal∨ in· · the · ∨ i-th clause. The question of plexity theory from the perspective of QMA complete- Boolean satisfiability, or k-SAT, is whether there exists ness. This theory naturally incorporates decision prob- a choice X = (x , . . . , x ) such that Φ(X) = 1. Note lems of the type that motivate AQC. Essentially, it con- 1 n that it only requires O(kr) steps to check whether X is cerns a problem involving the ground state of a local a satisfying assignment, yet there are 2n possible choices Hamiltonian, whose ground state energy is promised to for X. either be below a threshold a or above another threshold For k = 3, the Boolean satisfiability problem, called b > a, and where b a is polynomially small in the sys- 3-SAT, is NP-complete. Let us explain what this means. tem size. In some cases− this problem is easy, and in other cases it turns out to be so hard that we do not hope to solve it efficiently even on a quantum computer. Char- 2. NP, NP-complete, and NP-hard acterizing which types of local Hamiltonians fall into the latter category is the subject of QMA-completeness. Informally, problems in NP are those whose verification can be done efficiently (e.g., checking whether a Boolean formula is satisfied). An important conjecture, called the 23 The factor L1/d in Eq. (92) is introduced so that the eigenvalues Exponential Time Hypothesis (Impagliazzo and Paturi, coming from the many-particle subspaces will not mix with the 2001), states that there are problems in NP that take eigenvalues of the single-particle subspace. exponentially long to solve. 26

Formally, a decision problem Q is in NP if and only if Two real numbers a and b specified with poly(n) there is an efficient algorithm V , called the verifier, such • bits of precision, such that that for all inputs η (e.g., in the case of SAT, this would 1 be the clauses) of the problem: b a > . (94) − poly(n) if Q(η) = 1, then there exists a witness X such that • V (η, X) = 1. The output (0 or 1) answers the question: Is the smallest if Q(η) = 0, then for all witnesses X we have eigenvalue of H smaller than a (output is 1), or are all • V (η, X) = 0. eigenvalues larger than b (output is 0)? We are promised that the ground state eigenvalue cannot be between a 24 In both cases, we typically take X = poly( η ), where and b. η is the number of bits in the binary| | string| associated| We may map 3-SAT to the 3-local Hamiltonian Prob- | | with the input η. The verifier is efficient in the sense that lem as follows. For every clause Ci (which involves three its cost scales as poly( X ). In SAT the witness X would literals), we can define a 3-local projector Hi onto all the | | be our test assignment. unsatisfying assignments of Ci. Because Hi is a projec- A decision problem Q is NP-complete if: tor, it has eigenvalues 0 and 1, where the 0 eigenvalue is associated with satisfying assignments and the 1 eigen- Q is in NP value with unsatisfying assignments. Therefore: • Every problem in NP is reducible to Q in polyno- r • H X = H X = q X (95) mial time. | i i| i | i i=1 X Here reducibility means that given a problem A in where q is the number of unsatisfied assignments by X. NP and a problem B that is NP-complete, A can be Thus 3-SAT is equivalent to the following 3-local Hamil- solved using a hypothetical polynomial-time algorithm tonian problem: is the smallest eigenvalue of H zero (the that solves for B. A commonly used reduction is the 3-SAT problem is satisfiable) or is it at least 1 (the 3-SAT polynomial-time many-to-one reduction (Karp, 1972), problem is unsatisfiable)? whereby the inputs of A are mapped into the inputs to B such that the output of B matches the output of A. The hypothetical algorithm then solves B to get the answer to A. 4. Motivation for Adiabatic Quantum Computing A decision problem Q is NP-hard if every problem in NP is reducible to Q in polynomial time. (Note that Adiabatic evolution seems well-suited to tackling the unlike the NP-complete case, Q does not need to be in k-local Hamiltonian problem. By initializing an n- NP). Clearly, NP-complete NP-hard. qubit system in an easily prepared ground state, we can ⊆ (in principle) evolve the system with a time-dependent Hamiltonian whose end point is the k-local Hamiltonian. 3. The k-local Hamiltonian Problem If the evolution is adiabatic, then we are guaranteed to be in the ground state of the k-local Hamiltonian with high The history state construction of Sec. IV.B relies on probability. By measuring the state of the system, we can a 5-local Hamiltonian. Such a Hamiltonian belongs to determine the energy eigenvalue of the state (which hope- an important class of decision problems known as the fully is the ground state energy) and hence determine k-local Hamiltonian Problem, of which a complete com- the answer to an NP-complete problem such as 3-SAT. plexity classification was given in (Cubitt and Monta- This motivated early work on the quantum adiabatic al- naro, 2016) subject to restrictions on the set of local gorithm (Farhi et al., 2001). terms from which the Hamiltonian can be composed [see Another possibility is to try to use AQC as the veri- also (Bravyi and Hastings, 2014)]. Recall that a k-local fier. However, the quantum algorithm only gives us the Hamiltonian is a Hermitian matrix that acts non-trivially answer probabilistically, so we must first define a proba- on at most k p-state particles. bilistic analog of NP and then a quantum version. These The k-local Hamiltonian Problem is defined on n new complexity classes are MA and QMA (Kitaev et al., qubits, with the following input: 2000).

r A k-local Hamiltonian H = i=1 Hi with r = • poly(n). Each H is k-local and satisﬁes H = i k ik poly(n) and its non-zero entriesP are speciﬁed by 24 The quantity b − a is sometimes called the “promise gap” and is poly(n) bits. distinct from the spectral gap. 27

B. MA and QMA C. The general relation between QMA completeness and universal AQC Informally, MA can be thought of as a probabilistic analog of NP, allowing for two-sided errors. Formally, The class of efficiently solvable problems on a quantum a decision problem Q is in MA iff there is an efficient computer is bounded error quantum polynomial time probabilistic verifier V such that for all inputs η of the (BQP) (Bernstein and Vazirani, 1993), which consists problem: of the class of decision problems solvable by a uniform if Q(η) = 1, then there exists a witness X such that family of polynomial-size quantum circuits with error • Pr(V (η, X) = 1) 2 (completeness). probability bounded below 1/2. Because of the poly- ≥ 3 nomial equivalence between AQC and the circuit model, if Q(η) = 0, then for all witnesses X we have BQP is also the class of efficiently solvable problems on • Pr(V (η, X) = 1) 1 (soundness). a universal adiabatic quantum computer. Its classical ≤ 3 Again we take X = poly( η ). MA is typically viewed as analog is the class bounded-error probabilistic polyno- an interaction| between| two| | parties, Merlin and Arthur. mial time (BPP), and as expected BPP BQP (Bern- stein and Vazirani, 1997). In addition, BQP⊆ QCMA Merlin provides Arthur with a witness X, on which ⊆ Arthur runs V . If Q(η) = 0, Merlin should never be (Aharonov and Naveh, 2002). Another interesting char- able to fool Arthur with a witness X into believing that acterization is that BQP=QMAlog, where QMAlog is Q(η) = 1 with probability > 1/3. the same as QMA except that the quantum proof has O(log η ) qubits instead of poly( η )(Marriott and Wa- Note that there is nothing special about the proba- | | | | bilities (2/3, 1/3). We can generalize our description to trous, 2005). MA(c, s): This motivates the study of QMA, and in particular QMA completeness, as a tool for understanding uni- Claim 1. MA(c, c 1/ η g) MA(2/3, 1/3) =MA(1 g g versality. Indeed, it is often the case that whenever e η , e η ), where−g is| | a constant⊆ and c > 0 and c − −| | −| | adiabatic universality can be proven for some class of 1/ η g < 1. − | | Hamiltonians, then the local Hamiltonian problem with The proof of this “amplification lemma” [see, e.g., (roughly) the same class can be shown to be QMA- (Goldreich, 2008; Marriott and Watrous, 2005; Nagaj complete and vice versa. Note, however, that there is no et al., 2009)] is interesting since it invokes the Chernoff formal implication from either of those problems to the bound, a widely used tool. We thus present it in Ap- other (Aharonov et al., 2009). On the one hand, prov- pendixD for pedagogical interest. ing QMA-completeness is in general substantially harder The complexity class QMA can be viewed as the quan- than achieving universal AQC, where we can choose the tum analogue of MA. Thus, QMA is informally the class initial state to be any easily prepared state that will help of problems that can be efficiently checked on a quan- us solve the problem, so we can choose to work in any con- tum computer given a “witness” quantum state related venient subspace that is invariant under the Hamiltonian. to the answer to the problem. Formally, define a quan- Indeed, in the history-state construction, we introduce tum verifier V (a quantum circuit) that takes η and a penalty terms to guard against illegal clock states [recall 2 poly( η ) quantum witness state X (C )⊗ | | as inputs and Eq. (75)]. For QMA, the states we work with are chosen | i ∈ probabilistically outputs a binary number. The decision adversarially from the full Hilbert space, and we must be problem Q is said to be in QMA if and only if there exists able to check, using only local Hamiltonian terms, that an efficient (polynomial time) V for all inputs η of the they are of the correct (clock-state) form. On the other problem that satisfies: hand, proving adiabatic universality involves analyzing if Q(η) = 1, then there exists a witness X such the spectral gap of the continuous sequence of Hamiltoni- • that Pr(V (η, X ) = 1) 2 (completeness).| i ans over the entire duration of the computation, whereas | i ≥ 3 QMA-completeness proofs are only concerned with one if Q(η) = 0, then for all witnesses X we have Hamiltonian. • Pr(V (η, X ) = 1) 1 (soundness). | i | i ≤ 3 The amplification lemma applies here as well. The definition of QMA also allows V to have poly( η ) ancilla D. QMA-completeness of the k-local Hamiltonian problem qubits each initialized in the 0 state (Gharibian| | et al., and universal AQC 2015). | i One can also define the class QCMA, which is similar To prove that a promise problem is QMA-complete, to QMA except that X is a classical state (Aharonov one needs to prove that it is contained in QMA and that it and Naveh, 2002). Since| i the quantum verifier can force is QMA-hard. The k-local Hamiltonian Problem belongs Merlin to send him a classical witness by measuring the to QMA for any constant k, and in fact even for k = witness before applying the quantum algorithm, we have: O (log n)(Kitaev et al., 2000). For pedagogical proofs see MA QCMA QMA. (Aharonov and Naveh, 2002; Gharibian, 2013; Gharibian ⊆ ⊆ 28 et al., 2015). Translationally invariant 1D systems for which • The first example of a QMA-hard problem was the k- finding the ground state energy is complete for 25 local Hamiltonian problem for k 5 (Kitaev et al., 2000), QMAEXP (Gottesman and Irani, 2013). so that in particular the 5-local≥ Hamiltonian problem is QMA-complete. This was reduced to 3-local (Kempe and The 1D case is interesting since, as remarked in Regev, 2003; Nagaj and Mozes, 2007) and then to 2-local Sec. IV.E, the 1D restriction of MAX-2-SAT with p-state (Kempe et al., 2006). Note that the 1-local Hamiltonian variables is in P, yet (Aharonov et al., 2009) showed that problem is in the complexity class P, since one can simply for 12-state particles, the problem of approximating the optimize for each 1-local term independently. Various ground state energy of a 1D system is QMA-complete. simplifications of QMA-completeness for the 2-local case This result was improved to 11-state particles in (Nagaj, followed. In order to describe these, we first need to 2008), and then to 8-state particles in (Hallgren et al., define a class of Hamiltonians: 2013), who also pointed out a small error in (Aharonov et al., 2009) that could be fixed by using 13-state parti- x y z cles. Whether and at which point a further Hilbert space H1 = J XiXj + J YiYj + J ZiZj ij ij ij dimensionality reduction becomes impossible remains an (i,j) X∈E interesting open problem. x y z + hi Xi + hi Yi + hi Zi , (96) The reduction from 5-local to 2-local is done using i X∈V perturbative gadgets (Biamonte and Love, 2008; Bravyi et al., 2008a; Cao et al., 2015; Jordan and Farhi, 2008; where and are the vertex and edge sets of a graph = Kempe et al., 2006; Oliveira and Terhal, 2008). The goal ( , ),V and allE local fields hα and couplings J α (Gα V E { i } { ij} ∈ of the gadget is to approximate some target Hamiltonian x, y, z ) are real. The Heisenberg model corresponds to T { x }y z x y z H of n qubits (e.g., the 5-local Hamiltonian from the Jij = Jij = Jij, the XY model to Jij = Jij and Jij = 0, x y α history state construction at any time s) by a gadget and the Ising model to Jij = Jij = 0. When Jij < 0 (> 0) Hamiltonian HG acting on the same n qubits as well as the interaction between qubits i and j is ferromagnetic an additional poly(n) ancilla qubits. The gadget Hamil- (antiferromagnetic). When we write “fully” below we tonian is typically written as mean that all interactions have the same sign. Unless explicitly mentioned otherwise we assume that the local HG = HA + λV , (97) fields are all zero. Most of the simplifications of QMA-completeness are where HA is an unperturbed Hamiltonian (also called the special cases of Eq. (96): penalty Hamiltonian), acting only on the ancilla space, and where λV is a perturbation that acts between the Geometrical locality: nearest-neighbor interactions qubits of HT and the ancilla qubits. Using perturbation • with being a 2D square lattice (Oliveira and Ter- theory, which we review in AppendixE, one can show G hal, 2008) or a triangular lattice (Piddock and Mon- that the lowest 2n eigenvalues of HG differ from those of tanaro, 2015). HT by at most and the corresponding eigenstates have an overlap of at least 1 . − Simple interactions in 2D: ZZXX and ZX model Completeness of the 2-local Hamiltonian problem • (Biamonte and Love, 2008) [defined in Eqs. (99) means that every problem in QMA is reducible to the 2- and (100) below], fully ferromagnetic and fully an- local Hamiltonian decision problem in polynomial time. tiferromagnetic Heisenberg model with local fields Since this reduction involves perturbative gadgets that (Schuch and Verstraete, 2009), antiferromagnetic preserve the spectrum of the original 5-local Hamiltonian, Heisenberg and XY models without local fields this means that the 2-local Hamiltonian derived from the (Piddock and Montanaro, 2015). 5-local Hamiltonian appearing in the universality proof of Sec. IV.B will also have an energy gap that is an inverse Some of the simplifications of QMA completeness use polynomial in the circuit length, and that the computa- other types of Hamiltonians: tion remains in the ground subspace with illegal clock states gapped away by the (now 2-local) penalty Hamil- Interacting fermions in 2D and the space-time con- • tonian. In the remainder of this subsection we briefly struction (Breuckmann and Terhal, 2014). discuss a particularly simple form of 2-local Hamiltoni- ans that is universal for AQC. Multi-state particles in 1D (Aharonov et al., 2009; • Hallgren et al., 2013; Nagaj, 2008).

Non-translationally invariant 1D systems (all two- 25 • QMAEXP is the same as QMA but with exponential size (in the particle terms identical but position-dependent input) witness and veriﬁcation circuit, whereas both are polyno- one-particle terms) (Kay, 2008). mial for QMA. 29

The QMA completeness of general 2-local Hamiltoni- matrices, such as arise in the theory of classical Markov ans can be extended to show that a more restricted set of chains. 2-local Hamiltonians composed of real-valued sums of the Restricting to any basis still leaves some freedom following pairwise products of Pauli matrices are QMA- in the definition. For example, a Hamiltonian H = x complete (Biamonte and Love, 2008): i σi + HZ , where HZ is diagonal in the computational− basis, is clearly stoquastic. However, applying a IX,XI,IZ,ZI,ZX,XZ,ZZ,XX . (98) P z { } unitary transformation U = i σi to the Hamiltonian x The basic two steps to do this are: (1) Using the result gives H0 = σ + HZ , which according to Def.5 is not i Q of (Bernstein and Vazirani, 1997) that any quantum cir- stoquastic in the computational basis. Applying a local P cuit can be represented using real-valued unitary gates unitary basis transformation should not change the com- operating on real-valued wavefunctions in the proof of plexity of the problem. Therefore, for clarity we fix the the QMA-completeness of the 5-local Hamiltonian of the basis such that the standard initial Hamiltonian always previous subsection, the Hamiltonian terms are all real- carries a minus sign, i.e., σx. From this point for- − i i valued. This therefore extends QMA-completeness to 5- ward, we restrict our discussion of stoquasticity to the P local real Hamiltonians. (2) The same gadgets used in standard computational basis. With this in mind, the (Kempe et al., 2006; Oliveira and Terhal, 2008) can be class of stoquastic Hamiltonians includes the fully ferro- used to reduce the locality from five to two. magnetic Heisenberg and XY models, and the quantum This can be further simplified to show that “ZZXX” transverse field Ising model [recall Eq. (96)]. Hamiltonians” that are linear combinations with real co- Given the restriction of the Hamiltonian, one may ask efficients of only whether there is a complexity class for which the k-local stoquastic Hamiltonian problem is complete. This led IX,XI,IZ,ZI,ZZ,XX (99) to the introduction of the class StoqMA, for which the { } k 2-local stoquastic Hamiltonian is StoqMA-complete are QMA-complete. This is done by showing, using per- (Bravyi≥ et al., 2006). This can be further refined to turbation theory, that such Hamiltonians can be used the result that the transverse Ising model on degree-3 to approximate the σz σx and σx σz terms. Simi- graphs is StoqMA-complete (Bravyi and Hastings, 2017). larly, perturbation theory⊗ can be used⊗ to show that “ZX” Rather than give the formal (and rather involved) defini- Hamiltonians” that are linear combinations with real co- tion of StoqMA, we note that the only difference between efficients of only StoqMA and MA is that a stoquastic verifier in StoqMA IX,XI,IZ,ZI,ZX,XZ (100) is allowed to do the final measurement in the + , { } basis, whereas a classical coherent verifier in{| MAi |−i} can are QMA-complete (Biamonte and Love, 2008; Bravyi only do a measurement in the standard 0 , 1 basis.26 and Hastings, 2014; Cubitt and Montanaro, 2016). Unlike MA and QMA, the threshold probabilities{| i | i} in Sto- qMA have an inverse polynomial rather than constant separation; this prevents amplification of the gap between VI. STOQUASTIC ADIABATIC QUANTUM the threshold probabilities based on repeated measure- COMPUTATION ments with majority voting. Finally, it is known that MA StoqMA QMA (Bravyi et al., 2006). In this section we focus on the special class of “sto- ⊆ ⊆ To capture the important class of problems that are quastic Hamiltonians” [originally introduced in (Bravyi characterized by stoquastic evolution with the constraint et al., 2008b)], that often arise in the context of quantum of adiabatic evolution, we first introduce the following optimization. definition of a model of computation: Definition 5 (stoquastic Hamiltonian (Bravyi and Ter- Definition 6 . Stoquastic adiabatic quantum hal, 2009)). A Hamiltonian H is called stoquastic with (StoqAQC) computation (StoqAQC) is the special case of AQC (Def- respect to a basis iff H has real nonpositive off-diagonal matrix elements inB the basis . inition1) restricted to k-local (k fixed) stoquastic Hamil- B tonians. For example, a Hamiltonian is stoquastic in the computational basis iff Because we defined StoqAQC as a special case of AQC, the computation must proceed in the ground state. How- n x H x0 0 x, x0 0, 1 x = x0 . (101) ever, recall that the algorithm for the glued trees problem h | | i ≤ ∀ ∈ { } 6 The computational basis is often singled out since it plays the role of the basis in which the final Hamiltonian is measured, which sometimes coincides with the basis in 26 MA has an alternative quantum definition as a restricted version which that Hamiltonian is diagonal. The term “stoquas- of QMA in which the verifier is a coherent classical computer tic” was introduced due to the similarity to stochastic (Bravyi et al., 2008b). 30

(Sec. III.D) is not subject to this ground state restriction and hence is not in StoqAQC. In Sec. VI.C we consider PostBQP another model of stoquastic computation that is not sub- PostBPP ject to the ground state restriction. QMA StoqAQC has generated considerable interest since experimental implementations of stoquastic Hamiltonians StoqMA are quite advanced (Bunyk et al., 2014; Weber et al., BQP 2017). To characterize its computational power, we in- MA troduce a natural promise problem based on StoqAQC, BStoqP and modeled after the k-local Hamiltonian problem:27 NP Definition 7 (StoqAQCEval). The StoqAQCEval prob- BPP lem is defined on n qubits, with the following input: a continuous family of (k 2)-local (k fixed) sto- • ≥ r P quastic Hamiltonians H(s) = i=1 Hi(s) with r = poly(n) and parameterized by s [0, 1]. For all i P∈ and all s, the non-zero entries of Hi(s) are spec- FIG. 4 Known relations between complexity classes relevant ified by poly(n) bits of precision, and Hi(s) = for AQC. The BStoqP class defined here (Def.8) lies in the k k poly(n). The ground state energy gap ∆[H(s)] sat- intersection StoqMA and BQP, and includes BPP. isfies ∆[H(s)] 1/poly(n) for all s. ≥ two real numbers a and b specified with poly(n) bits et al., 2008b). It is clear that BPP BStoqP, since 5-local • of precision, and b a > 1/poly(n). StoqAQCEval is BPP-hard (using⊆ a a classical reversible − The output (0 or 1) answers the question: Is the smallest circuit for universal AQC, with stoquastic gate terms and eigenvalue of H(s = 1) smaller than a (output is 1), or a 5-local stoquastic clock Hamiltonian). Finally, we know that BStoqP BQP, since StoqEvalAQC is in BQP by us- are all eigenvalues larger than b (output is 0)? Just as in ⊆ the local Hamiltonian problem, we are promised that the ing the same proof that the adiabatic model in general outcome that the ground state energy is between a and b can be simulated by the circuit model. is not possible. This allows us to (informally) define the complexity A. Why it might be easy to simulate stoquastic class that captures StoqAQC: Hamiltonians

Definition 8 (BStoqP). BStoqP is the set of problems In this subsection we briefly summarize the that are polynomial-time reducible to StoqAQCEval. complexity-theoretic evidence obtained so far that The StoqAQCEval problem is clearly in StoqMA, be- suggests that the StoqAQC setting is less powerful than cause the k 2-local stoquastic Hamiltonian prob- universal quantum computation. Let us start with a lem is StoqMA-complete≥ (Bravyi et al., 2006). Hence lemma that characterizes the “classicality” of ground BStoqP StoqMA, as depicted in Fig.4, which summa- states of stoquastic Hamiltonians. ⊆ rizes the relations between many of the complexity classes Lemma 3. The ground state ψ of a stoquastic Hamil- 28 we have discussed. NP and MA are unlikely to be sub- tonian H can always be expressed| i using only real nonneg- sets of BStoqP, since StoqAQC would not be expected ative amplitudes: ψ = x 0,1 n ax x , where ax 0 to solve NP-complete problems in polynomial time. The x. | i ∈{ } | i ≥ tightest inclusion in a classical complexity class we know ∀ P of is in AM,29 since the latter includes StoqMA (Bravyi Proof. It follows directly from the stoquastic property that the corresponding Gibbs density matrix ρ = exp( βH)/Tr[exp( βH)] has non-negative matrix elements− in the computational− basis for any β > 0. In 27 We are indebted to Elizabeth Crosson for her help in formulating the StoqAQCEval problem, the BStoqP class, and working out particular, if H is stoquastic then for sufficiently small the relations of BStoqP to other complexity classes. β, 11 βH has only non-negative matrix elements. The 28 − As far as we know the related term StoqP was informally intro- largest eigenvalue of 11 βH corresponds to the ground duced by Stephen Jordan in a talk presented at the AQC 2016 state energy of H. Thus,− by the Perron-Frobenius theo- conference (Jordan, 2016a), showing that StoqP is not equal to BQP unless BQP is in the third level of the polynomial hierarchy. rem (see Sec. III.E.1) the ground state of H can be chosen 29 Like MA, the class AM (Arthur Merlin) is a probabilistic gen- to have non-negative amplitudes. eralization of NP. See https://complexityzoo.uwaterloo.ca/ Complexity_Zoo for definitions of the complexity classes men- Consequently, if the Hamiltonian is stoquastic, a clas- tioned here. sical probability distribution can be associated with the 31

ground state. This raises the question as to whether Sto- With this evidence for the potential limitations of sto- qAQC is a model that is capable of quantum speedup quastic Hamiltonians, the question arises if they are wor- over classical algorithms. Following is the evidence re- thy of pursuit, either theoretically or experimentally. garding this question. However, it is important to remember that the weak- ness of stoquastic Hamiltonians arises when one assumes 1. The ground state energy of the fully ferromagnetic that they generate an evolution that occurs in the ground transverse field Ising model can be found to a given state. Indeed, we will see in Sec. VI.C that excited additive error in polynomial time with a classical state stoquastic evolution can be as powerful as universal algorithm on any graph, with or without a trans- AQC. Moreover, in the next subsection we briefly review verse magnetic field (Bravyi and Gosset, 2016). counterexamples to the claim that stoquastic Hamiltoni- 2. In (Bravyi et al., 2008b) it was shown that for ans are necessarily easy to simulate using heuristic clas- any fixed k, stoquastic k-local Hamiltonian is con- sical algorithms. tained in the complexity class AM. Thus, unless QMA AM (which is believed to be unlikely), stoquastic⊆ k-local Hamiltonian is not QMA-complete. B. Why it might be hard to simulate stoquastic Hamiltonians 3. It was also shown in (Bravyi et al., 2008b) that gapped StoqAQC can be simulated in PostBPP, There does not exist a general theorem that rules out the complexity class described by a polynomial- a quantum speedup of StoqAQC over all possible classi- time classical randomized computer with the ability cal algorithms. However, it is often stated that Monte to post-select on some subset of the bits after the Carlo simulations of StoqAQC do not suffer from the algorithm is run.30 I.e., it suffices to call an oracle sign problem and will therefore simulate StoqAQC with- for problems in PostBPP a polynomial number of out a slowdown. Specifically, the conjecture is that if the times to efficiently sample from the ground state of Monte Carlo simulation starts at s = 0 in the equilibrium a gapped stoquastic Hamiltonian.31 state, and if s changes by a small amount from one step Suppose that StoqAQC could be used to perform to the next, where is polynomially small in the system universal quantum computation. Since gapped Sto- size n, the inverse temperature β and the spectral gap qAQC can be simulated in PostBPP, this would ∆, then the Monte Carlo simulation stays close to the imply that SampBQP SampPostBPP.32 In other equilibrium state along the path. For sufficiently large words, this would imply⊆ that polynomial time β, this would correspond to following the instantaneous quantum algorithms can be simulated classically in ground state. In this subsection we briefly review theo- polynomial time using post-selection. This would retical evidence that such a conjecture is not always true. then imply that PostBPP=PostBQP which in turn We focus on two of the most direct classical competitors would collapse the polynomial hierarchy. Thus it is to StoqAQC: path integral quantum Monte Carlo (PI- unlikely that StoqAQC is universal for AQC. QMC), and diffusion quantum Monte Carlo (D-QMC). 4. In (Bravyi and Terhal, 2009) it was shown that adiabatic evolution along a path composed entirely 1. Topological obstructions of stoquastic frustration-free Hamiltonians (recall Definition4) may be simulated by a sequence of In (Hastings and Freedman, 2013) examples were given classical random walks, i.e., is contained in BPP. of StoqAQC with a polynomially small eigenvalue gap, but where PI-QMC take exponential time to converge. Loosely, the failure of convergence was due to topological 30 See also (Farhi and Harrow, 2016), where gapped StoqAQC was obstructions around which the worldlines (trajectories in called stoquastic gapped adiabatic evolution, QADI-SG. 31 imaginary time) can get tangled. PostBPP, also known as BPPpath, contains NP. For example, consider the Grover problem with two registers, a bit-string x The simplest of the examples can be understood infor the input and a second register where f(x) is stored. Now if tuitively as follows. A sombrero-like potential is con- we pick x at random and post-select on the second register being structed for a single particle with a deep circular min- 1, we find a marked item. PostBPP is known to be contained in imum of radius r. The worldline of the particle in PI- the third level of the polynomial hierarchy (Han et al., 1993). 32 In sampling problems we are given an input x ∈ {0, 1}n, and the QMC with closed boundary conditions is some closed goal is to sample (exactly or approximately) from some probabil- path that follows this circle in imaginary time. Because ity distribution over poly(n)-bit strings. SampBQP and Samp- of the depth of the potential at the minimum the distribu- PostBPP are the classes of sampling problems solvable on quantum computers and probabilistic classical computers with post- tion of trajectories has very small probability to include selection, respectively, to within error in total variation (or any point with radius larger than r and it takes an ex- trace-norm) distance, in time polynomial in n and 1/ (Aaron- ponential time in the winding number to transition from son, 2010). one winding number sector to another. Therefore, if an 32 appropriate dimensionless combination of the radius r or D-QMC algorithms perform random walks designed to the mass of the particle is changed sufficiently fast then ensure that a population of random walkers converges to (1) PI-QMC fails to equilibrate. At the same time it can be ps (x). However, in exponentially large Hilbert spaces shown that for this example the gap closes polynomially there can be vertices such that the distribution asso- (2) and so one expects that adiabatic evolution requires only ciated with the L2-normalized wavefunction ps (x) is polynomial time to find the ground state. polynomial, but the distribution associated with the L1- (1) While this example uses winding numbers to construct normalized wavefunction ps (x) is exponentially small. a protocol for which PI-QMC takes exponential time The idea behind the examples in (Jarret et al., 2016) to equilibrate, PI-QMC can still find the ground state. is to exploit this discrepancy to design polynomial-time To observe a more dramatic effect where not only equi- stoquastic adiabatic processes that the corresponding D- libration is hampered but also the probability of find- QMC simulations will fail to efficiently simulate. ing the ground state is low, one can introduce stronger The main example given in (Jarret et al., 2016) is the topological effects and additionally exploit the discrep- stoquastic Hamiltonian H(s) = 1 [L + b(s)W ] c(s)P , n − ancy between L1 and L2-normalized wavefunctions. This where L is the graph Laplacian of the n-bit hypercube, was first done in the “bouquet of circles” example intro- W is the Hamming weight operator (i.e., W x = x x duced in (Hastings and Freedman, 2013), which shows where x is the Hamming weight of the bit-string| i | ||x),i that PI-QMC can fail to converge even when using open P = 0| | 0 0 0 . In terms of Pauli matrices this boundary conditions. The example was designed so Hamiltonian| ··· ih can··· be| written, up to an overall constant, that the majority of the amplitude ψ lies within an ex- as pander graph, although the majority of the probability n ψ 2 does not. Because the endpoints of the wordlines 1 1 H(s) = Xj + b(s)Zj c(s)P. (102) are| | distributed according to ψ and not ψ 2, this effec- −n 2 − j=1 tively “pins” them to the expander graph.| | This pinning X means that even though the worldline is in principle open, The schedules b(s) and c(s) are: the worldline is nevertheless prevented from changing its 2sb 0 s [0, 1/2) topological sector within the bouquet of circles. This b(s) = c(s) = . b (2s 1)c s ∈ [1/2, 1] then causes failure of convergence. − ∈ A more general method using perturbative gadgets is (103) explained in (Hastings and Freedman, 2013), that allows For s [0, 1/2) this is a Hamiltonian of n non-interacting one to map between continuous variables and spins and qubits∈ whose gap is easily seen to be 2 1 + (sb/2)2, applies to all the examples given there. n minimized at s = 0 where it equals 2/n. The ground n p state is given by ψ(θ) ⊗ , where ψ(θ) = cos(θ/2) 0 + | i 1 | i | i sin(θ/2) 1 and θ = tan− [2/(sb)]. For s [1/2, 1] it can 2. Non-topological obstructions be shown| i that the minimum gap is attained∈ at s = 1/2, 3/2 where it equals 1/√2n+O(n− ). Thus the overall min- Diffusion Monte Carlo algorithms should not be af- imum gap is polynomial (2/n) and the StoqAQC pro- fected by topological obstructions that depend on closed cess converges to the ground state 0 0 in polynomial boundary conditions, since they do not exhibit period- time. By choosing b so that at s =| 1··· we havei cos(θ/2) = icity in the imaginary time direction. Rather than use 1 1/(4n), it is easy to show from the analysis of the non- topological obstructions, it is possible to rely entirely on interacting− problem that the probability of ending up in the discrepancy between L1 and L2-normalization to de- (2) 2n 1/2 the ground state is ps=1(0 0) = cos (θs=1/2) e− sign examples where Monte Carlo methods have differ- in the limit n . ··· → ing convergence from AQC. This discrepancy was used On the other→ hand,∞ for the non-interacting problem et al. in (Jarret , 2016) to ensure that the walkers in a (when s [0, 1/2)) the D-QMC process33 samples from DQMC algorithm never “learn” about a potential well ∈ (1) x n x the distribution ps (x) = sin(θ/2)| | cos(θ/2) −| |/Zs, that contains the solution, causing DQMC to take expo- x n x where Z = n sin(θ/2) cos(θ/2) = nential time to converge. Since the gap for the adiabatic s x 0,1 | | −| | [sin(θ/2) + cos(θ/2)]n∈{, so} that for large n we have Z process is large, QA takes only polynomial time. P 1 ≈ (1 + 1/√2n)n e√n/2 for the same choice of b. Thus, Let H(s) be some stoquastic Hamiltonian acting on a → Hilbert space whose basis states can be equated with the for D-QMC the probability of being in the ground state (1) n vertices V of some graph. Let ψ (x): V denote the at s = 1/2 is p (0 0) = cos (θs=1/2)/Zs=1 s C s=1/2 ··· → ground state of H(s). Define probability7→ distributions (1) ψs(x) (2) 2 ps (x) = P and ps (x) = ψs (x). The stoquas- y V ψs(y) ∈ (1) 33 ticity of H(s) ensures that ψs(x) 0, so that ps (x) is Here D-QMC refers to the “Substochastic Monte Carlo (SSMC)” a valid probability distribution. ≥ algorithm introduced in (Jarret et al., 2016). 33

1/4 √n/2 ˜ e− e− . Since at s = 1/2 the random walkers that interpret Tk as being 2-local or 3-local acting on n + 1 diffuse in the D-QMC process have a probability to be qubits. Note furthermore that Tk is such that it only at the all-zeros string that is of order e √n/2, with high has one non-zero entry per row and column, hence with − ˜ likelihood, no walkers will land on the all-zeros string un- the substitution in Eq. (106), the Tk’s are permutation til the number of time-steps times the number of walkers matrices. We can write the following Hamiltonian acting on n + 1 qubits: approaches e√n/2. Until this happens it is impossible for the distribution of walkers to be affected by the change in H˜ = α T˜ (107) the potential at the all-zeros string that is occurring from ZZXX − k k k s = 1/2 to s = 1; no walkers have landed there, and the X D-QMC algorithm has therefore never queried the value which is a linear combination of permutation matrices of the potential at that site. Only after allowing for this with negative coefficients. This makes H˜ZZXX a (3-local) exponential cost, and by appropriately choosing c, does stoquastic Hamiltonian. We can write it as: the D-QMC algorithm find the ground state with high 34 H˜ = H + H¯ + + , (108) probability. ZZXX ZZXX ⊗ |−ih−| ZZXX ⊗ | ih | where H¯ = α T and T is the entrywise ZZXX − k k| k| | k| C. QMA-complete problems and universal AQC using absolute value of Tk. To see why this is the case, first P stoquastic Hamiltonians with excited states consider a positive element (Tk)ij. Then:

Our definition of StoqAQC (Definition6) stipulates αk(Tk)ij αk(Tk)ij + + = αk(Tk)ij 11 − ⊗ |−ih−| − ⊗ | ih | − ⊗ that the computation must proceed in the ground state. It turns out that if this condition is relaxed, computation corresponding to the first replacement in Eq. (106). For with stoquastic Hamiltonians is as powerful as AQC, i.e., a negative element, we have: it is universal. Here we review a construction by (Jordan α (T ) +α (T ) + + = α (T ) σx et al., 2010) of a 3-local stoquastic Hamiltonian that, − k k ij ⊗|−ih−| k k ij ⊗| ih | − k k ij ⊗ by allowing for excited state evolution, is both QMA- corresponding to the second replacement in Eq. (106). complete and universal for AQC. The spectrum of H˜ZZXX separates into two sectors . We start with the QMA-complete Hamiltonian intro- L± The sector is spanned by εj , where εj are duced in Sec. V.D, that can be written as: L− | i ⊗ |−i | i the eigenstates of HZZXX , while the sector + is spanned L ¯ x z by ε¯j + , where ε¯j are the eigenstates of HZZXX . HZZXX = diXi + hiZi + JijXiXj + JijZiZj , Because| i ⊗ the | i Hamiltonian| i does not couple the two sectors i i j X X≤ (there are no interactions that take the ancilla qubit from (104) to ), a closed system evolution initialized in the where d , h , J x and J z are arbitrary real coefficients. i i ij ij |±i sector|∓i will remain in that sector. The key idea is to eliminate the negative matrix elements − L Because the spectrum in the sector is identical to in each term. Toward this end the Hamiltonian is written − that of H , which is capableL of universal adiabatic as: ZZXX quantum computation, then universal adiabatic quantum computation can be performed in the sector. How- HZZXX = αkTk , (105) − ever, the lowest energy state in may notL− necessarily be Xk L− the ground state of H˜ZZXX . Therefore, this establishes where Tk Xi, Zi, XiXj, ZiZj and such that universal AQC using a stoquastic Hamiltonian only if we ∈ {± ± ± ± } αk > 0. For an n-qubit system, the operators Tk are do not restrict ourselves to the ground state of the Hamil- represented by 2n 2n symmetric matrices with entries tonian. Attempting to make the lowest energy state in × taking value +1, 1, 0. We use the regular representation be the ground state requires introducing a sufficiently − L− of the Z2 group to make the replacement large term proportional to 11 + + to the Hamiltonian ⊗| ih | H˜ZZXX , but such a term would make the new Hamil- 1 0 0 1 0 0 1 , 1 , 0 (106) tonian non-stoquastic since it would introduce positive → 0 1 − → 1 0 → 0 0 off-diagonal elements. Therefore this method does not establish universal adiabatic quantum computation us- in T to define a new operator T˜ . The matrix represen- k k ing the ground state of a stoquastic Hamiltonian. tation of T˜ is of size 2n+1 2n+1, and since the original k × Tk was either 1-local or 2-local acting on n qubits, we can D. Examples of slowdown by StoqAQC

It should not come as a surprise that AQC with an 34 For c = 2 one finds that p(1) (0 ··· 0) = 1/2 + O(n 1/2). s=1 − arbitrary final Hamiltonian, which is essentially a black 34 box approach, does not guarantee quantum speedups. It Let us write the corresponding StoqAQC Hamiltonian can be vulnerable to the same sorts of locality traps con- to make the perturbation explicit: fronted by heuristic classical algorithms such as simu- H(s) = H (s) s(n + 1) 1n 1n , (111) lated annealing. HW − | ih | A slowdown, or failure of AQC to provide a speedup, where 1n is the all-one state. Denote the instanta- is a scenario wherein a more efficient classical algorithm | i neous eigenstates of HHW(s) by vi(s) (v0 denotes the is known. All the known examples that fall into this cat- ground state). Note that the overlap{| ofi} the all-one-state egory arise when the gap closes “too fast” in the problem with the instantaneous ground state of the plain Ham- size. However, it is important to note that adiabatic the- ming Weight algorithm is always exponentially small: orems provide only upper bounds on run time, not lower bounds. Thus, e.g., an exponentially small gap does not n 1 1 v0(s) . (112) strictly imply an exponentially long run time. The in- h | i ≤ √2n verse gap is often treated as a proxy for run time, but a We will show that this fact causes the adiabatic algorithm claim such as an equal scaling of the inverse gap and the as defined in Eq. (111) to take exponential time because run time does not hold as a general theorem. it leads to an exponentially small gap. With this caveat in mind, in this subsection we re- Define a matrix A(s) with elements: view such “small-gap” examples in increasing order of generality or difficulty of analysis, which all arise in the Aij = vi(s) H(s) vj(s) . (113) StoqAQC context. However, we must first note another h | | i important caveat. Namely, in some of the examples we Note that A(0) is diagonal and A00(0) = 0, equal to present numerical evidence that is based, in necessity, on the ground state eigenvalue. Also A(1) is diagonal, but finite size calculations. One is then often tempted to ex- now A2n 1,2n 1(1) = 1 is equal to the ground state − − − trapolate such evidence to the asymptotic scaling. Of eigenvalue. Define a matrix B in the same basis as: course, any such extrapolations based purely on numerics are conjectures. For example, a claim of exponential A00(s) i = j = 0 0 i = 0, j > 0 scaling can never be proven based on numerics alone, as Bij(s) =  (114)  0 i > 0, j = 0 any finite set of data points can always be perfectly fit  by a polynomial of sufficiently high degree. Neverthe- Aij(s) otherwise less, numerics-driven conjectures about scaling can be  The matrix B always has A00 as an eigenvalue. By con- quite useful, especially if supported by other, analytical struction, we know that at s = 1, the matrix B has 1 arguments. n 1 n−1 as its ground state eigenvalue (located in the 2 − 2 − Subsequently, we will see in Sec.VII that there are sub-matrix). Because the matrix transforms continu-× various methods for circumventing slowdowns, e.g., via ously between these two extremes, there cannot be a the introduction of non-stoquastic terms. jump in the ground state eigenvalue, so there must be a critical value of s, which we denote by sc, where B has a vanishing gap. 1. Perturbed Hamming Weight Problems with Exponentially Small Overlaps The optimal matching distance between A and B ex- presses how close their eigenvalue spectra are: The plain Hamming weight problem is described by d(A, B) = min max λj µπ(j) , (115) π 1 j 2n | − | 1 ≤ ≤ H (s) = (1 s) (1 σx) + s x x x . (109) HW − 2 − i | || ih | where π denotes a permutation. Since A and B are i x X X Hermitian, this is upper-bounded by A B 2 (Bhatia, Its cost function is simply the Hamming weight x of R., 1997). The matrix A B only hask non-zero− k entries | | − the binary bit-string x, which is trivially minimized at (A B)0,j>0 = A0,j>0 and (A B)j>0,0 = Aj>0,0 = n − − n n x = 0 . Consider the following perturbation of the plain A0∗,j>0, with A0,j>0 = s(n + 1) v1(s) 1 1 vj(s) . Hamming weight problem (van Dam et al., 2001): Therefore: − h | ih | i x if x < n h(x) = . (110) A B 2 = (116) | 1| if |x| = n k − k − | | 2n 1 − This is a toy problem that is designed to be hard for s(n + 1) v (s) 1n 1n v (s) v (s) 1n |h 1 | i|v h | j ih j | i classical algorithms based on local search: its global op- u j=1 u X timum lies in a narrow basin, while there is a local op- = s(n + 1) v (s) 1tn 1 1n v (s) 2 timum with a much larger basin. An algorithm such as |h 1 | i| − |h | 0 i| n p s(n + 1) simulated annealing with single spin updates would re- s(n + 1) v1(s) 1 . quire exponential time to find the global minimum. ≤ |h | i| ≤ √2n 35

Thus, the gap of A [and hence of H(s)] is always upper- Under this unitary transformation we have: bounded by the gap of B plus twice A B 2. Since k − k 1 at s = sc the gap of B is zero, it follows that the z z H0 = UH1U † = 11i σ σ , (119) √ n 2 1 2 − i i+1 gap of H(sc) is sc(n + 1)/ 2 − (van Dam et al., i 2001). Therefore,≤ the exponentially small overlap be- X tween the unperturbed instantaneous ground state and i.e., the new final Hamiltonian is a sum of just agree the perturbed final ground state results in the adiabatic clauses. Note that this unitary transformation requires algorithm requiring exponential time to reach the final us to know the ground state, but H10 and H1 are isospec- ground state. Informally, this can also be viewed as the tral, so we can use it for convenience in our gap analysis. inability of local quantum search (fluctuations induced The adiabatic computation procedure will be governed by the local initial Hamiltonian) to explore the entire by the following time-dependent Hamiltonian: (non-local) energy landscape effectively. H(s) = (1 s)H + sH0 , 0 s 1 , (120) − 0 1 ≤ ≤ 2. 2-SAT on a Ring x with the initial Hamiltonian H0 = i 11i σi . We wish to diagonalize H(s) in order to find its ground− state gap. In this subsection we review the “2-SAT on a Ring” P n x First, define the negation operator G = i=1 σi such problem introduced in the seminal work (Farhi et al., that: 2000), which launched the field of AQC. This example is Q n n instructive because of its use of the Jordan-Wigner and G ( i=1 zi ) = i=1 zi , (121) Fourier transformation techniques, and is also of histori- ⊗ | i ⊗ |¬ i cal interest. It also serves to illustrate that even a polyno- which clearly commutes with H(s). The uniform su- mially small gap does not guarantee a quantum speedup. perposition state, which is the ground state of H(0), is We thus review it in detail. invariant under G, i.e., it has eigenvalue +1 under G. Consider an n-bit SAT problem with n clauses. Each Therefore, the unitary dynamics will keep the state in clause only acts on adjacent bits, i.e., the clause Cj only the sector with G = +1 if it starts in the ground state acts on bits j and j + 1, where we identify bit n + 1 of H(0). Let us then write H(s) purely in the G = +1 with bit 1. Let each clause be of only two forms: “agree” sector. Second, define the Jordan-Wigner transformation clauses where 00 and 11 are satisfying assignments, and “disagree” clauses where 01 and 10 are satisfying assign- x x x ments. Since an odd number of satisfied disagree clauses bj = σ1 σ2 . . . σj 1σj− (122a) − means that the first bit of the first disagree clause is the x x x + bj† = σ1 σ2 . . . σj 1σj (122b) opposite of the second bit of the last disagree clause, yet − bits 1 and n+1 must agree, there must be an even number 1 z y where σ± = σ iσ . These are fermionic operators of disagree clauses in order for a satisfying assignment to j 2 j j that satisfy: ± exist. The classical computational cost of finding a satisfying assignment is at most n: given the list of clauses, a b , b = 0 (amounts to σ−, σ− = 0) (123a) satisfying assignment is found (assuming an even number { j k} j k + of disagree clauses) simply by going around the ring and bj, b† = δjk (amounts to σj−, σ = δjk) . (123b) n { k} k satisfying each clause one at a time. Note that if wi i=1 n { } is a satisfying assignment then so is w . Note that {¬ i}i=1 Let us now define the final Hamiltonian H1 = n H associated with the SAT problem, where each 1 x i=1 Ci bj†bj = 11j σj , j = 1, . . . , n (124a) clause is represented by: 2 − P z z b† b b† +b = σ σ , j = 1, . . . , n 1 j − j j+1 j+1 j j+1 − 1 xi z z HCi = 1 ( 1) σi σi+1 (117) (124b) 2 − − z z xi = 0 (1) if Ci is an agree (disagree) clause . b† b b† + b = Gσ σ . (124c) n − n 1 1 − n 1 n The ground states of HP are then given by 0 1 i=2 wi i n i 1 | i ⊗ | i In order to make Eqs. (124b) and (124c) consistent in the and wi i, where wi = − xj (i 2 and addition i=1 j=1 G = +1 sector, we take bn+1 b1. Using this, we have: modulo⊗ 2).|¬ Iti is possible to gauge away≥ all the disagree ≡ − clauses. To see this, let U beL the unitary transformation n defined such that H(s) = 2(1 s)b†bj (125) |G=+1 − j j=1 X h zi , if wi = 1 s U zi = |¬ i . (118) + 11 b† b b† + b | i zi , if wj = 0 j j j j+1 j+1 | i 2 − − i 36

Third, since this Hamiltonian is invariant under transla- The instantaneous ground state energy of H(s) is thus tions j j + 1, define Fourier operators βp: given by p=1,3,... Ep−(s). The first excited state energy 7→ + is given by E (s)+ E−(s). The energy gap ∆(s) 1 n P 1 p=3,... p β = eiπpj/nb , p = 1, 3,..., (n 1) , is therefore given by: p √n j ± ± ± − P j=1 + X ∆(s) = E (s) E−(s) (132) (126) 1 − 1 πp 1/2 where for simplicity it is assumed that n is even. Equiv- = 2 (2 3s)2 + 4s(1 s)(1 cos . alently: − − − n h i 2 2+cos π 1 iπpj/n ( n ) bj = e− βp , (127) The minimum occurs at s∗ = 5+4 cos π 2/3 as n √n n → → p= 1,... . Therefore, the minimum gap is given by: X± ∞ iπp(j j0)/n where we used the fact that p= 1,... e − = π 1 4π ± ∆(s∗) = 4 sin π , (133) nδj,j0 . Furthermore, note that: n 5 + 4 cos → 3n P n 1 iπ((2a 1)j+(2b 1)j0)/n which implies a polynomial p run time for the adiabatic al- β2a 1, β2b 1 = e − − bj, bj0 { − − } n { } gorithm. As mentioned, the classical computational cost Xj,j0 = 0 (128a) of finding a satisfying assignment is at most n. There- fore, despite the polynomially small gap in this example, 1 iπ((2a 1)j (2b 1)j0)/n β2a 1, β2†b 1 = e − − − bj, bj† there is no quantum speedup. This illustrates that a Sto- − − n 0 j,j qAQC slowdown need not necessarily be associated with n o X0 n o n an exponentially small gap. 1 2πi(a b)/n = e − = δ (128b) n a,b j=1 X 3. Weighted 2-SAT on a chain with periodicity so the set β comprises valid fermionic operators. Writ- { p} ing the Hamiltonian in terms of this set, we have: We now discuss another problem, proposed in (Re- ichardt, 2004), that combines 2-SAT with an exponential H(s) = 2(1 s) βp†βp + β† pβ p − − − slowdown of StoqAQC. It can thus be viewed as exhibit- p=1,3,... X h πp ing aspects of the two previous problems we discussed. + s 11 cos βp†βp β pβ† p Consider a weighted 2-SAT problem on a chain with − n − − − “agree” clauses between bits i, i + 1 for i = 1,...,N 1 πp − +i sin β† pβp† βpβ p (129a) with weights: n − − − i Ap(s) (129b) i ≡ w if n is odd , p=1,3,... Ji = d e (134) X 1 if i is even d n e Now that H(s) has finally been written as sum of com- where n is the period and w > 1. As for the previous muting operators ([Ap,Ap0 ] = 0 for p = p0), we can diagonalize each summand separately.6 For a given p, 2-SAT problem, we can map this to a spin-chain with ferromagnetic couplings with strength given by Ji. The let us denote by Ωp the state that is annihilated by | i adiabatic Hamiltonian is given by: βp and β p, i.e., βp Ωp = β p Ωp = 0. Note that − | i − | i Ap(s = 0) Ωp = 0, so Ωp is the ground state of Ap N N 1 | i | i − at s = 0 (recall that we already knew that the ground H(s) = (1 s) σx s J σzσz . (135) − − i − i i i+1 state energy at s = 0 was zero). Let Σp = β† pβp† Ωp . i=1 i=1 A (s) keeps states in the subspace spanned| i by− Ω | andi X X p | pi Σp in the same subspace; the initial state is in this sub- This chain has coefficients that alternate between w and space,| i so we can restrict our attention to it. Let us write 1 in sectors of size n each, with the b + 1 odd-numbered A (s) in the Ω , Σ basis: sectors being “heavy” (Ji = w > 1), b even-numbered p {| pi | pi} sectors being “light” (Ji = 1), and where the total num- s + s cos πp is sin πp ber of sectors is (N 1)/n = 2b + 1. Since the chain is A (s) = n n . (130) p is sin πp 4 3s s cos πp ferromagnetic, the ground− state of H(1) is trivially the − n − − n all-0 or all-1 computational-basis state. The problem is Diagonalizing this, we find for the energies: thus classically easy and can be solved by inspection or 1/2 in time O(N) by a heuristic classical algorithm such as 2 πp E±(s) = 2 s (2 3s) + 4s(1 s)(1 cos . p − ± − − − n simulated annealing, by simply traversing the chain and h i(131) updating one spin at a time. 37

Note that at s = 0 there is a unique ground state, have not ordered yet. Since the initial Hamiltonian gen- while as we just noted, at s = 1 the ground state is dou- erates only local spin flips, the algorithm is likely to get bly degenerate. Therefore, the relevant quantum ground stuck in a local minimum with a domain wall in one or state gap ∆ is not the gap to the first excited state (since more disordered sectors, if run for less than exponential at the end of the evolution, this merges with the ground time in n. This mechanism in which large local regions state), but to the second excited state. order before the whole is well-known in disordered, ge- It turns out this gap is exponentially small in the sec- ometrically local optimization problems, giving rise to a tor size n across a constant range s (1/(1 + w), 1/2). Griffiths phase (Fisher, 1995). Moreover, there are exponentially many∈ (in √N) exponentially small excitations above the ground state for n √N. 4. Topological slowdown in a dimer model or local Ising ladder ∼ More precisely, let µw = sw/(1 s). Theorem 4 in (Reichardt, 2004) states that: − Another interesting example of a local spin model that leads StoqAQC astray was given in (Laumann et al., 1. For any fixed s > 1/(1 + w), i.e., µw < 1, H(s) has 2012). They showed that a translation invariant quasi-1D n one eigenvalue only O(µw) above the ground state transverse field Ising model with nearest-neighbor inter- energy. This means that the gap ∆ is exponentially actions only, the ground state of which is readily found decreasing with the sector size n. by inspection, results in exponentially long run times for StoqAQC. The model can be understood as either 2. For s (1/(1+w), 1/(1+√w)] (i.e., again µw < 1), a dimer model on a two-leg ladder of even length L, or, H(s)∈ has 2b+1 1 eigenvalues only O(bµn ) above − w using a duality transformation, a two-leg frustrated Ising the ground state energy. This means that there are ladder of the same length in a uniform magnetic field, the exponentially many (in the number of odd sectors ground states of which map onto the dimer states. The b + 1) excited states, that likewise have an expo- frustrated Ising ladder Hamiltonian is: nentially small (in n) gap from the ground state. Note that b = [(N 1)/n + 1]/2, so b √N when 1 − ∼ H = J σzσz K σz + U σz , n √N. 1 − ij i j − i 2 i ∼ i,j i upper i lower hXi ∈X ∈X (136) 3. For s [1/(1+√w), 1/2), where µ1 = s/(1 s) > 1, ∈ b+1 n − where upper and lower refer to the legs of the ladder, H(s) has 2 1 eigenvalues O(bµ1− ) above the − J = K for the upper-leg couplings and J = K for ground state energy. This again means an expo- ij − ij nentially large number (in b) of excited states with all other (lower-leg and rungs) couplings. an exponentially small (in n) gap. Quantum dynamics is introduced via a standard initial Hamiltonian Γ(t) σx, where Γ(0) H and − i i k 1k The proof uses a Jordan-Wigner transformation to diag- Γ(tf ) = 0. The dimer model exhibits a first order quan- onalize H(s), similarly to the technique in Sec. VI.D.2. tum phase transition withP an exponentially small gap The spectral gaps of the 2N 2N matrix H(s) are the when K U, which is inherited by the frustrated Ising square roots of the eigenvalues× of an N N symmetric, ladder model. Namely, the system prefers the sector with tridiagonal matrix. The Sturm sequence× of the principal exponentially many ground states, while any degeneracy- leading minors of this matrix is then analyzed to bound lifting interaction favors another containing only O(1) the eigenvalue gaps of H(s). states. StoqAQC selects the wrong sector, tunneling out Why might we expect this problem to be hard for the of which becomes exponentially slow as Γ is reduced. adiabatic algorithm? Within any given light or heavy More specifically, for K U, the Hilbert space is sector, the problem (at fixed s) is that of a uniform trans- spanned by an orthonormal basis of hardcore dimer cov- verse field Ising chain. Consider the thermodynamic limit erings (“perfect matchings”) of the ladder. These fall N 1, and also let n 1. In this limit the transverse into three sectors which are topological in that they are field Ising chain encounters a phase transition separat- not connected by any local rearrangement of the dimers. ing the disordered phase and the ordered phase when The sectors are labeled by a winding number w, the dif- 1 s = sJi (Sachdev, 2001), i.e., at s = 1/(1 + Ji). (The ference between the number of dimers on the top and boundary− of the chain only adds O(1) energy, so it does bottom rows (on any fixed plaquette). The model as- not impact this intuitive argument in the thermodynamic signs extensive energy L to every state in the w = 0 limit.) This means that the heavy sectors encounter the sector while leaving the∝ two staggered states w = 1 as 1 ± phase transition at s = 1+w whereas the light sectors ground states with energy 0. On the other hand, at large encounter the phase transition at s = 1/2, i.e., the light Γ (strong transverse field) the w = 0 sector is favored. 1 sectors order after the heavy ones. At s = 1+w each Intuitively, slowly turning the transverse field off by re- heavy sector orders in either the all-0 or all-1 state, and ducing Γ does not help change the topological sector since different heavy sectors are separated by light sectors that any off-diagonal term in the dimer Hilbert space involv- 38 ing only a finite number of rungs in the ladder leaves the energy density given by: winding number w invariant. This is depicted in Fig.5. p 1 2 2 2p 2 Numerical analysis of the Ising ladder confirms this pic- F = (p 1)m log 2 cosh β Γ + p m − . ture by revealing that the gap is exponentially small in − − β h p i(139) L when K > U. The critical point is found to be at The dominant contribution to F comes from the saddle- Γ U/b + U 2/(4Kb3), where b 0.6 (from exact diag- c point of the partition function Z, which provide consis- onalization≈ numerics). ≈ tency equations for the field m:

2 2 2(p 1) tanh β Γ + p m − E p 1 m = pm − . (140) 2 2 2(p 1) Γp+ p m −

L Solving this equationp numerically for p > 2 reveals a

U L b

2 discontinuity in the value of m that minimizes the free energy as Γ is tuned through the phase transition point. At this critical point, the free energy exhibits a degenerate double-well potential, and an instantonic calculation on this potential gives an exponentially small energy gap 0 with system size (J¨org et al., 2010a). !c !

6. 3-Regular 3-XORSAT FIG. 5 Energy spectrum of the dimer model on an even length periodic ladder, with the dimer configurations illustrated. The w = ±1 states are at energy E = 0, while the All the problems we discussed so far were amenable w = 0 sector splits into a band for Γ > 0. For sufficiently large to a classical solution “by inspection” (i.e., the solution Γ, the w = 0 sector contains the ground state of the system. is obvious from the form of the cost function). Some An unavoided level crossing (first order quantum phase tran- problems were even easy for classical heuristic algorithms sition) occurs at Γ = Γc, which is responsible for the quantum performing local search. We now discuss a problem that slowdown. From (Laumann et al., 2012). is non-trivial in this respect, i.e., classically only yields in polynomial time to a tailored approach. In 3-XORSAT, each clause involves three bits, and there are M clauses and n bits in total. A clause is sat- 5. Ferromagnetic Mean-field Models isfied if the sum of the three bits (mod 2) is a specified value; it can be 0 or 1 depending on the clause. For 3- The quantum ferromagnetic p-spin model is given by: regular 3-XORSAT, every bit is in exactly three clauses and M = n. This problem is associated with a spin glass p phase but is “glassy without being hard to solve” (Franz 1 n n H = σz Γ σx . (137) et al., 2001; Ricci-Tersenghi, 2010): the problem of find- −np 1 i − i − i=1 ! i=1 ing a satisfying assignment can be solved in polynomial X X time using Gaussian elimination because the problem in- By inspection it is clear that when p is even, the ground volves only linear constraints (mod 2) (Haanpaa et al., state at Γ = 0 is either of the two fully-aligned ferro- 2006). magnetic states, while when p is odd, the unique ground A final Hamiltonian involving n spins can be be written state at Γ = 0 is the fully-aligned spin-down state. As such that each satisfied clause gives energy 0 and each Γ is tuned from a large value towards zero, the system unsatisfied clause gives energy 1: encounters a first-order phase transition for p > 2. This n z z z 11 Jcσ σ σ can be readily shown by employing the Suzuki-Trotter de- H = − i1,c i2,c i3,c . (141) composition and the static approximation (Chayes et al., 1 2 c=1 2008; Jörg et al., 2010a; Krzakala et al., 2008; Suzuki X et al., 2013) to calculate the partition function Z in the Here the index (ik, c) denotes the three bits associated large n limit, where with clause c, and Jc 1 depending on whether the clause is satisfied if the∈ {± sum} of its bits (mod 2) is 0 or βnF (β,Γ,m) 1. The StoqAQC Hamiltonian is then given, as usual, by Z = dm e− . (138) n x H(s) = (1 s) i=1 σi + sH1. The median minimum Z gap for− random− 3-regular 3-XORSAT has numerically Here β is the inverse temperature, m is the Hubbard- been shown to beP exponentially small in the system size Stratonovich field (Hubbard, 1959), and F is the free up to n = 24 (Jörg et al., 2010b) and n = 40 (Farhi et al., 39

(µ) 2012) [both using the quantum cavity method (Krzakala (Hebb rule), where ξi are zero-mean i.i.d. random vari- et al., 2008; Laumann et al., 2008) and QMC simula- ables of unit variance. By focusing on the analytically tions], with a first order quantum phase transition at more tractable Hopfield model, (Knysh, 2016) rigorously s = 1/2. Thus, the numerical evidence suggests that analyzed for r = 2 (the two-pattern case) and p = 2 StoqAQC takes exponential time to solve this problem. (two-local interactions) the properties of local minima The same is true for classical heuristic local search algo- away from the global minimum. rithms such as WalkSAT (Guidetti and Young, 2011). We note that since the Hamiltonian gap is not a ther- The main insight gained from the theoretical analysis modynamic quantity, one must be careful not to auto- of (Knysh, 2016) is that the complexity of the model is matically associate a first order quantum phase transition not determined by the phase transition, but rather by the with an exponentially small gap. While the examples pre- existence of small-gap bottlenecks in the spin glass phase. sented in this review agree with this rule [for additional Namely, after the occurrence of the polynomially closing examples see (Bapst and Semerjian, 2012; Dusuel and Vi- gap associated with the second order phase transition dal, 2005; Jörg et al., 2008, 2010a; Knysh and Smelyan- separating the paramagnetic and glass phases, there are skiy, 2006)], counterexamples wherein a first order quan- O(log n) additional gap minima in the spin glass phase tum phase transition is associated with a polynomially appearing in an approximate geometric progression, a small gap are known (Cabrera and Jullien, 1987; Lau- phenomenon that can be attributed to the self-similar mann et al., 2012, 2015; Tsuda et al., 2013). properties of the free energy landscape in a Γ interval bounded by the appearance of the spin-glass phase. At these bottlenecks, the gaps scale as a stretched exponen- 7. Sherrington-Kirkpatrick and Two-Pattern Gaussian Hopfield 3/4 3/4 tial e cΓm n , where Γ is the location of the m-th Models − m minimum. This is illustrated in Fig.6. Nevertheless this means that StoqAQC suffers a (stretched) exponen- The Sherrington-Kirkpatrick (SK) model, the proto- tial slowdown, since the two pattern Gaussian Hopfield typical spin glass model, is NP-hard, yet its quantum model admits an efficient classical solution based on angle transverse field Ising model version (Das et al., 2005; Ishii sorting and exhaustive search, that scales as O[n log(n)] and Yamamoto, 1985; Ray et al., 1989; Usadel, 1986) (Knysh, 2016). Thus, this is another case where a Sto- exhibits a second order phase transition separating the qAQC algorithm is too generic to exploit problem struc- paramagnetic phase from the spin glass phase (Miller and ture, and consequently a tailored classical algorithm has Huse, 1993; Ye et al., 1993). The model is defined via the exponentially better scaling. final Hamiltonian

z z H1 = Ji1i2 σi1 σi2 (142) i ip = ξi ξi (144) ··· np 1 1 ··· p − µ=1 X 40

E. StoqAQC algorithms with a scaling advantage over of n non-interacting spins in a global (longitudinal) field, simulated annealing which is of course a trivial problem that can be solved in time O(1), e.g., by parallelized SA running with a a single A substantial effort is underway to develop problems thread for each spin. The scaling of the time needed by that may exhibit any form of a quantum speedup (re- the quantum algorithm is O(n1/2), and the full cost of the call the classification given in Sec.III). One approach quantum algorithm is O(n3/2) according to Eq. (1), since has been to develop “tunneling gadgets”, i.e., small it requires O(n) single-qubit terms in the Hamiltonian. A toy Hamiltonians that exhibit tunneling (Boixo et al., fairer comparison is to an SA algorithm that is ignorant 2016), and use these gadgets to construct larger prob- of the structure of the problem. In this case one can show lems (Denchev et al., 2016). An alternative approach that the cost for single-spin update SA with random spin has been to develop instances that are believed to exhibit selection is lower bounded by (n log n)(Muthukrishnan “small-and-thin” energy barriers in their classical energy et al., 2016). O landscape (Katzgraber et al., 2015) in the hope that such Next we consider a more interesting problem, referred barriers persist in the quantum energy landscape where to as the “spike”, first studied in (Farhi et al., 2002a). tunneling occurs. These approaches have been used pri- The cost function is given by: marily to assess the performance of the D-Wave devices and are based on numerical analysis, which makes extrap- n if x = n/4 f(x) = (147) olation and conclusions about asymptotic scaling rather x otherwise| | challenging (Brady and Dam, 2016; Mandrà et al., 2016). | | In this subsection we consider several examples of Sto- Since the barrier scales with n, we can expect that single- qAQC with a demonstrable quantum scaling advantage spin-update SA will take exp(n) time to traverse the over simulated annealing (SA). While none of the ex- barrier. However, it can be shown that the quantum 1/2 amples are demonstrations of an unqualified quantum gap scales as Ω(n− )(Farhi et al., 2002a; Kong and speedup, these examples are illustrative in that they re- Crosson, 2015), so the adiabatic algorithm only takes veal important qualitative differences between SA, where polynomial time. thermal fluctuations are used to explore the energy land- This type of “perturbed” Hamming weight problem scape, and StoqAQC, where quantum fluctuations are can be generalized, while still retaining an advantage over used to explore a different energy landscape. Still, these single-spin-update SA. For cost functions of the form results are based on a comparison with SA that uses only x + h(n) if l(n) < x < u(n) single-spin updates. SA-like algorithms with cluster-spin f(x) = (148) |x| otherwise| | updates can be significantly more efficient (Houdayer, J., | | 2001; Mandrà et al., 2016; Swendsen and Wang, 1987; satisfying h[(u l)/√l] = o(1), the minimum gap of the Wolff, 1989; Zhu et al., 2015; Zintchenko et al., 2015), − and their performance relative to StoqAQC is largely an quantum algorithm is lower bounded by a constant (Re- open question. The same is true for parallel temper- ichardt, 2004) [see Appendix A of Ref. (Muthukrishnan ing (aka exchange Monte Carlo) (Earl and Deem, 2005; et al., 2016) for a pedagogical review of the proof]. The Hukushima and Nemoto, 1996; Katzgraber et al., 2006; SA run time, on the other hand, scales exponentially in Marinari and Parisi, 1992; Swendsen and Wang, 1986). maxn h(n). Similarly, consider barriers with width proportional to nα and height proportional to nβ, i.e. 1. Spike-like Perturbed Hamming Weight Problems x + nα if n 1 nβ < x < n + 1 nβ f(x) = 4 2 4 2 . |x| otherwise− | | We start with a problem for which there is no (limited) | | quantum speedup, in order to set up the more interesting (149) problems that follow. Consider a cost function f(x) to When α and β satisfy α + β 1/2, α < 1/2, and 2α + n ≥ 1/2 α β be minimized with x 0, 1 an n-bit string. The final β 1, the minimum gap scales polynomially as n − − ≤ Hamiltonian can generically∈ { } be written as: (Brady and van Dam, 2016a,b), while the SA run time scales exponentially in nα. H = f(x) x x . (145) 1 | ih | x X 2. Large plateaus We first consider the cost function of the “plain” Ham- ming weight problem: The above examples have relied on energy barriers in f(x) = x (146) the classical cost that scale with problem size to foil | | single-spin-update SA. This agrees with the intuition where x denotes the Hamming weight of the n-bit string that a StoqAQC advantage over SA is associated with x [as in| Eq.| (109)]. This problem is equivalent to a system tall and thin barriers (Das and Chakrabarti, 2008; Ray 41 et al., 1989). However, somewhat counterintuitively, it is to finding the minimum in a set of 2n numbers (Mertens, also possible to foil SA by having very large plateaus in 2001). To translate it into Ising spin variables let sj = 1 the classical cost function. Specifically, consider: when j and s = 1 otherwise, so that ∈ P j − u 1 if l < x < u f(x) = − | | (150) n x otherwise E = a s , (152) | | j j j=1 where l, u = (1) [a special case of Eq. (148)]. SA with X single-spin-updatesO and random spin selection has run which can then be turned into a Mattis-like Ising Hamil- time O(nu l 1), where u l 1 is the plateau width − − tonian whose ground state is the minimizing partition: (Muthukrishnan et al., 2016−).− This polynomial scaling arises because the energy landscape provides no preferred n direction and SA then behaves as a random walker on H1 = aiajsisj . (153) the plateau. Numerical diagonalization shows that the i,j=1 X quantum minimum gap is constant and the adiabatic run time is only O(n1/2), where the scaling with n arises from The energy landscape of this final Hamiltonian is known the numerator of the adiabatic condition (Muthukrishnan to be extremely rugged in the hard phase (Smelyanskiy et al., 2016). Thus StoqAQC has a polynomial scaling et al., 2002; Stadler et al., 2003), and the asymptotic be- advantage over SA in this case. havior can already be seen for small sizes n. While SA A natural question is whether these potential quan- effectively requires the searching of all possible bit configurations with a run time 20.98n (Stadler et al., 2003), tum speedup results for StoqAQC relative to SA survive ∝ when other algorithms are considered. The answer is numerical simulations of StoqAQC exhibit a slightly bet- 0.8n negative. (Muthukrishnan et al., 2016) showed that a di- ter run time 2 (Denchev et al., 2016). State-of-the- ∝ 0.291n abatic evolution is more efficient than the adiabatic evo- art classical algorithms have scalings as low as 2 lution to solve these problems, and a similar efficiency is (Becker et al., 2011). achieved using classical spin-vector dynamics. There is It should be noted that number partitioning is known also a growing body of numerical (Brady and van Dam, as the “easiest hard problem” (Hayes, 2002) due to the 2016a; Crosson and Deng, 2014; Denchev et al., 2016; existence of efficient approximation algorithms that ap- Muthukrishnan et al., 2016) and analytical (Crosson and ply in most (though of course not all) cases, e.g., a poly- Harrow, 2016; Isakov et al., 2016) research that shows nomial time approximation algorithm known as the dif- that quantum Monte Carlo methods exhibit similar or ferencing method (Karmarkar and Karp, 1982). It should even identical advantages over SA for many spike-like further be noted that if all the aj’s are bounded by a perturbed Hamming weight problems. polynomial in n, then integer partitioning can be solved in polynomial time by dynamic programming (Mertens, 2003). The NP-hardness of the number partitioning F. StoqAQC algorithms with undetermined speedup problem requires input numbers of size exponentially large in n or, after division by the maximal input number, In this subsection we focus on examples where it is of exponentially high precision. This is problematic since currently unknown whether there is a quantum speedup the aj are used as coupling coefficients in the adiabatic or slowdown for StoqAQC. Hamiltonian{ } (153), and suggests that a different encoding will be needed in order to allow AQC to meaningfully address number partitioning. 1. Number partitioning

The number partitioning problem is a canonical NP- 2. Exact Cover and its generalizations complete problem (M.R. Garey and D.S. Johnson, 1979) that is defined as follows: given a set of n positive num- We briefly review the adiabatic algorithm for Exact bers a n , the objective is to find a partition of this { i}i=1 P Cover, which initiated and sparked the tremendous in- set that minimizes the partition residue E defined as: terest in the power of AQC when it was first studied in (Farhi et al., 2001). While the optimistic claim made in that paper, that “the quantum adiabatic algorithm E = aj aj . (151) − worked well, providing evidence that quantum comput- j j / X∈P X∈P ers (if large ones can be built) may be able to outper-

The problem exhibits an easy-hard phase transition at form ordinary computers on hard sets of instances of the critical value b/n = 1, where b is the number of NP-complete problems” turned out to be premature, the bits used to represent the set ai (Borgs et al., 2001; historical impact of this study was large, and it led to Mertens, 1998). In the hard phase{ it} roughly corresponds the avalanche of work that forms the core of this review. 42

The Exact Cover 3 (EC3) problem is an NP-complete worst-case instances. Using StoqAQC has, in all cases problem that is a particular formulation of 3-SAT [re- that have been studied to date, resulted in exponentially call Sec. V.A] whereby each clause C (composed of three small gaps. Thus, whether these problems can be sped up bits xC1 , xC2 , xC3 that are taken from the set of variables (even polynomially) is at this time still an open problem. n xi 0, 1 i=1) is satisfied if xC1 +xC2 +xC3 = 1. There are{ ∈ only { three}} satisfying assignments: (1,0,0), (0,1,0), 3. 3-Regular MAXCUT and (0,0,1). A 3-local Hamiltonian HC can be associated with each clause, that assigns an energy penalty to the unsatisfying assignments (Farhi et al., 2001; Latorre For 3-regular MAXCUT, the problem is to find the and Orus, 2004): assignment that gives the maximum number of satisfied clauses, where each bit appears in exactly three clauses. 1 z z z Each clause involves only two bits and is satisfied if and HC = 1 + σC 1 + σC 1 + σC 8 1 2 3 only if the sum of the two bits (mod 2) is 1. The number z z z +1 σC 1 σC 1 σC of clauses is M = 3n/2. The final Hamiltonian can be − 1 − 2 − 3 + 1 σz 1 σz 1 + σz written as: − C1 − C2 C3 z z z + 1 σ 1 + σ 1 σ 3n/2 z z C1 C2 C3 1 − − 1 + σi1,cσi2,c + 1 + σz 1 σz 1 σz . (154) H1 = , (156) C1 C2 C3 2 − − c=1 A 2-local alternative is (Young et al. , 2010): X where the index (ik, c) denotes the two bits associated 1 z z z 2 with clause c. This model can also be viewed as an anti- HC = σC + σC + σC 1 . (155) 4 1 2 3 − ferromagnet on a 3-regular random graph. Because the random graph in general has loops of odd length, it is The final Hamiltonian is then given by H1 = C HC. If the ground state energy is 0, then an assignment ex- not possible to satisfy all of the clauses. This problem is ists that satisfies all clauses. The adiabatic algorithmP NP hard. For random instances of this problem, where there is given as usual by H(s) = (1 s)H0 + sH1, with H = n 1 (1 σx). − is a doubly degenerate ground state (the smallest pos- 0 i=1 2 − i For instances with a unique satisfying assignment, sible because of the Z2 symmetry) and with a speci- whileP the initial (small n) scaling of the typical mini- fied energy of n/8, the standard adiabatic Hamiltonian n x mum gap (median) is consistent with polynomial (Farhi H(s) = (1 s) i=1 σi + sH1 exhibits, for sufficiently large sizes− of− up to n = 160, two minima in the energy et al., 2001; Latorre and Orus, 2004), the true (large n) P scaling is exponential and can be associated with a first gap (Farhi et al., 2012) (see Fig.7 for an example). The order phase transition (Young et al., 2008, 2010) occur- first minimum, at s 0.36, is associated with a second- order phase transition≈ from paramagnetic to glassy, and ring at intermediate s = sc > 0. The fraction of instances with this behavior increases with increasing problem size the gap closes polynomially with system size. The sec- (Young et al., 2010). This illustrates the perils of ex- trapolating the asymptotic scaling from studies based on 0.35 small problem sizes. A natural generalization of the Exact Cover problem 0.3 is to have the sum of K variables sum to 1 for a clause 0.25 to be satisfied, which defines the problem known as “1- in-K SAT”. Another is to have the clause satisfied un- 0.2 1 less all the variables are equal, which defines the problem E ∆ “K-Not-All-Equal-SAT”. Both of these are NP-complete 0.15 and have been shown analytically to exhibit a first order phase transition for sufficiently large K (Smelyanskiy 0.1 et al., 2004). Numerical results for locked 1-in-3 SAT and 0.05 locked 1-in-4 SAT — where “locked’ is the additional requirement that every variable is in at least two clauses 0 and that one cannot get from one satisfying assignment 0.35 0.4 0.45 0.5 to another by flipping a single variable (Zdeborováand s Mézard, 2008a,b) — have been shown to exhibit an ex- FIG. 7 The gap to the first even excited state for an instance ponentially small gap at the satisfiability transition (Hen of size n = 128, exhibiting two minima. The lower minimum and Young, 2011). occurs well within the spin-glass phase, while the higher min- Since all these problems are NP-complete, there is imum is associated with the second order phase transition. no polynomial-time classical algorithm known for their From (Farhi et al., 2012). 43

ond minimum occurs inside the spin-glass phase, with a described above. It is unknown whether its StoqAQC N(N 1)/2 gap that closes exponentially (or possibly a stretched ex- version improves upon the classical brute force 2 − ponential). Therefore, while the first minimum does not scaling. The adiabatic quantum algorithm was simulated pose a problem for the adiabatic algorithm (although it in (Gaitan and Clark, 2012) and shown to correctly de- has been shown that the quantum algorithm with a linear termines the Ramsey numbers R(3, 3) and R(2, s) for interpolating schedule does not pass through the associ- 5 s 7. It was also shown there that Ramsey number ated glass phase transition faster than SA (Liu et al., computation≤ ≤ is in QMA. 2015)), the second minimum implies an exponential run An adiabatic algorithm for generalized Ramsey num- time. bers (where the induced subgraphs are trees rather than complete graphs) was presented in (Ranjbar et al., 2016). Whether this results in a quantum speedup is also un- 4. Ramsey numbers known. We also remark that Ising formulations for many NP-complete and NP-hard problems, including all An adiabatic algorithm for the calculation of the Ram- of Karp’s 21 NP-complete problems (Karp, 1972), are sey numbers R(k, l) was proposed in (Gaitan and Clark, known (Lucas, 2014), but it is unknown whether they 2012). R(k, l) is the smallest integer r such that every are amenable to a quantum speedup. graph on r or more vertices contains either a k-clique or an l-independent set.35 Computing them by brute force is doubly exponential in N = max k, l [note that 5. Finding largest cliques in random graphs R(k, l) = R(l, k)] using graph coloring techniques,{ } as fol- N(N 1)/2 lows: Try every one of the 2 − colorings of the The fastest algorithm known to date for the NP-hard edges of the complete graph KN with the colors blue and problem of finding a largest clique in a graph runs in time red. For every coloring, check whether or not there is an O(20.249n) for a graph with n vertices (Robson, 2001).36 induced subgraph on k vertices with only blue edges, or For random graphs, a super-polynomial time is required an induced subgraph on l vertices with only red edges. to find cliques larger than log n using the Metropolis al- If every coloring contains at least one of the desired sub- gorithm, while the maximum clique is likely to be of graphs, we are done. Otherwise, increment N by 1 and size very close to 2 log n (Jerrum, 1992). One of the repeat. Except for certain special values of k and l, no earliest papers on the quantum adiabatic algorithm was better algorithm is currently known. concerned with the largest clique problem for random The idea in (Gaitan and Clark, 2012) is to construct a graphs (Childs et al., 2002), though the algorithm pre- cost function h(G) for a graph G where sented there works for general graphs. The results were numerical and showed, by fixing the desired success prob- h(G) = (G) + (G) (157) C I ability, that the median time required by the adiabatic where (G) counts the number of m-cliques in the graph algorithm to find the largest clique in a random graph G andC (G) counts the number of l-independent sets are consistent with quadratic growth for graphs of up in the graphI G. The cost h(G) equals zero only if to 18 vertices. These results on small graphs probably there does not exist an k-clique or an l-independent do not capture the asymptotic behavior of the algorithm set. This will only occur if R(k, l) > N. The algo- (the coefficients grow rapidly and have alternating sign), rithm then proceeds as follows. By mapping h(G) over which is likely to be dominated by exponentially small gaps [however, to the extent that these are due to per- KN to a final Hamiltonian H1, the adiabatic algorithm n x turbative crossings, they can be avoided by techniques H(s) = (1 s) i=1 σi +sH1 is performed and the final energy of− the− state is measured. If h(G) = 0, then N is we discuss in Sec. VII.G, in particular as related to the incremented byP 1 and the experiment is repeated. This maximum independent set problem (Choi, 2011)]. process continues until the first occurrence of h(G) > 0, in which case N = R(k, l). Thus the algorithm is essentially an adiabatic version of the graph coloring method 6. Graph isomorphism In the graph isomorphism problem, two N-vertex graphs G and G0 are given, and the task is to determine 35 A k-clique is a subset of k vertices such that every two distinct whether there exists a permutation of the vertices of G vertices are adjacent. Equivalently, the subgraph induced by the clique is a complete graph. An independent set is a subset of the vertices no two of which are adjacent. R(k, l) can be phrased as the “party problem”: What is the smallest number of guests one can invite to a party such that there is always either a group of 36 As stated, this is actually an algorithm for the complementary k guests that all know each other, or a group of l guests, none of maximum independent set (MIS) problem, but this is sufficient whom know each other? Such a threshold number always exists since MIS(G) = max-clique(G¯) for any graph G and its comple- (Ramsey, 1930). ment G¯, and the algorithm applies for arbitrary G. 44 such that it preserves the adjacency and transforms G to Another idea is to learn the weights of a Boltzmann G0, in which case the graphs are said to be isomorphic. machine or, after the introduction of a hidden layer, a If and only if the graphs are isomorphic does there exist reduced Boltzmann machine (Hinton et al., 2006). The a permutation matrix σ that satisfies latter forms the basis for various modern methods of deep learning. StoqAQC approaches for this problem were T A0 = σAσ , (158) developed in (Adachi and Henderson, 2015; Amin et al., 2016; Benedetti et al., 2016) where A and A0 are the adjacency matrices of G and Neither the classical nor the quantum computational G0 respectively. An adiabatic algorithm to determine complexity is known in this case, but scaling of the so- whether a pair of graphs are isomorphic was first pro- lution time with problem size is not the only relevant posed in (Hen and Young, 2012), mostly for strongly criterion: classification accuracy on the test data set is regular graphs, and generalized to arbitrary graphs in clearly another crucial metric. It is possible, though at (Gaitan and Clark, 2014), which also showed how to de- this point entirely speculative, that the quantum method termine the permutation(s) that connect a pair of iso- will lead to better classification performance. This can morphic graphs, and the automorphism group of a given come about in the case of ground state degeneracy, if graph. The final Hamiltonian formulated in (Gaitan and the weights are reconstructed via ground state solutions Clark, 2014) is such that when the ground state energy and if quantum and classical heuristics for solving the vanishes the graphs are isomorphic and the bit-string QUBO problem find different ground states (Azinović s = (s0, . . . , sN 1) associated with the ground state gives − et al., 2016; Mandrà et al., 2017; Matsuda et al., 2009; an N N permutation matrix σ(s) to perform the trans- Zhang et al., 2017). formation:×

0 if s > N 1 σ(s) = j . (159) ij δ if 0 s N − 1 G. Speedup mechanisms? i,sj ≤ j ≤ − The computational complexity of these adiabatic algo- While the universality of AQC suggests that similar rithm is currently unknown. However, a recent break- speedup mechanisms are at play as in the circuit model through gave a quasipolynomial (exp[(log n)O(1)]) time of quantum computing, the situation is less clear regard- classical algorithm for graph isomorphism (Babai, 2015). ing StoqAQC. Here we discuss two potential mechanisms, It seems unlikely that this can be improved upon by using tunneling and entanglement, that might be thought to StoqAQC without deeply exploiting problem structure. endow StoqAQC with an advantage over classical algorithms.

7. Machine learning 1. The role of tunneling Quantum machine learning is currently an exciting and rapidly moving frontier in the context of the cir- It is often stated that an advantage of StoqAQC cuit model (Lloyd et al., 2014; Rebentrost et al., 2014; over classical heuristic local-search algorithms is that the Wiebe et al., 2014), though it must be evaluated care- quantum system has the ability to tunnel through en- fully (Aaronson, 2015). One StoqAQC approach is to find ergy barriers, which can provide an advantage over clas- a quantum version of the classical method of boosting, sical algorithms such as simulated annealing that only al- wherein multiple weak classifiers (or features) are com- low probabilistic hopping over the same barriers. Indeed, bined to create a single strong classifier (Freund et al., such a qualitative picture motivated some of the early re- 1999; Meir, 2003). The task is to find the optimal set search on quantum annealing [e.g., (Finnila et al., 1994)]. of weights of the weak classifiers so as to minimize the However, this statement requires a careful interpretation training error of the strong classifier on a training data as it has the potential to be misleading. Whereas only set. After this training step, the strong classifier is then the final cost function — which generates the energy applied to a test data set. This optimization problem landscape that the classical random walker explores — can be mapped to a quadratic unconstrained binary op- matters for the classical algorithm, this energy landscape timization (QUBO) problem, which can then be trivially does not become relevant for the quantum evolution un- turned into an Ising spin Hamiltonian suitable for adia- til the end. Therefore, tunneling does not occur on the batic quantum optimization, where the binary variables energy landscape defined by the final cost function alone, represent the weights. This idea was implemented in if it occurs at all. A different notion of tunneling is at (Babbush et al., 2014a; Denchev et al., 2012; Neven et al., work, which we now explain. 2008a, 2009, 2008b; Pudenz and Lidar, 2013), where the The standard notion of tunneling from single-particle ground states found by the adiabatic algorithm encode quantum mechanics involves a semiclassical potential the solution for the weights. where classically allowed and classically forbidden regions 45

1 1 0.5

0.8 0.9 0.4 ) 1 2 ) = , s 0.6 0.8 0.3 0 /n , , s i θ 0 ( , θ SC 0.4 HW s = 0 ( 0.7 0.2 h V

s = 1/4 SC V 0.2 s = 1/2 0.6 n = 10 0.1 n = 10 s = 3/4 n = 20 n = 20 s = 1 n = 100 n = 100 0 0.5 0 0 π/4 π/2 3π/4 π 0 π/4 π/2 3π/4 π 0.45 0.5 0.55 θ θ s (a) (b) (c)

FIG. 8 Analysis of tunneling in the Grover problem. (a) The semiclassical potential for n = 20 at different dimensionless times s. The arrows indicate the behavior of the local minima as s increases. There is a discrete jump in the position of the global minimum at s = 1/2, where it changes from being at θ ≈ π/2 to θ ≈ 0, corresponding to a first order quantum phase transition. (b) The behavior of the potential when the two minima are degenerate at s = 1/2. As n grows, both the barrier height grows (and saturates at 1) and the curvature of the local minima grows. (c) The expectation value of the Hamming Weight operator [defined in Eq. 167] of the instantaneous ground state as n grows. This is to be interpreted as the system requiring O(n) spins to tunnel in order to follow the instantaneous ground state as the system crosses the minimum gap at s = 1/2.

can be defined. Starting from a many-body Hamiltonian, We illustrate this approach with the Grover Hamil- there is no unique way to take the semiclassical limit. tonian [Eq. (16)]. Recall that the final Hamiltonian is Consider one such limit, based on the spin-coherent path H1 = 11 m m , where m is the marked state associ- integral formalism (Klauder, 1979): ated with− the | ih marked| item.| Asi a cost function, this is the antithesis of the “tall and narrow” potential that is often tf i S[Ω(t)] associated with a classical speedup: x H1 x = 1 δx,m, Ω(tf ) Texp[ i dτH(τ)] Ω(0) = Ω(t)e ~ , h | − | i D i.e., the potential is flat everywhere,h except| | i for a− well of Z0 Z (160) constant depth at the marked state. Nevertheless, we where the action S[Ω(t)] is given by: now show that following the instantaneous ground state will involve the tunneling of O(n) qubits. tf S[Ω(t)] = dt (i~ Ω(t) ∂t Ω(t) Ω(t) H(t) Ω(t) ) , Without loss of generality we may assume that the h | | i − h | | i 0 “marked” state is the all-zero bit string. Setting θj θ Z (161) ≡ and ϕj ϕ j in Eq. (162), the Hamiltonian can be and written succinctly≡ ∀ as: Ω θ, ϕ (162) H(s) = (1 s)(11 Ω(π/2, 0) Ω(π/2, 0) ) | i ≡ | i − − | ih | n cos(θ /2) 0 + eiϕj sin(θ /2) 1 . +s (11 Ω(0, 0) Ω(0, 0) ) . (164) ≡ ⊗j=1 j | ij j | ij − | ih | is the spin-coherent state (Arecchi et al., 1972). The semiclassical potential for the Grover problem is Despite the absence of a true kinetic term, we can iden- then: tify the semiclassical potential as: 1 V (θ, 0) = (1 s) 1 (1 + sin θ)n SC − − 2n V ( θ , ϕ , t) = Ω H(t) Ω (163) SC { j} { j} h | | i 1 +s 1 (1 + cos θ)n . (165) This form for V has been used (Boixo et al., 2016; − 2n SC et al. et al. Farhi , 2002a; Muthukrishnan , 2016; Schaller The locations of the two degenerate minima at s = 1/2 and Schützhold, 2010) to capture many of the rele- are given by the pair of transcendental equation: vant features of StoqAQC problems endowed with qubit- permutation symmetry; this symmetry often allows for 1 cos θ + sin θ 1 + sin θ n − = , (166a) analytical and numerical progress.37 1 + cos θ sin θ 1 + cos θ − 1 + cos θ sin θ 1 + cos θ n − = , (166b) 1 cos θ + sin θ 1 + sin θ − 37 Note that by using a product-state ansatz via the symmetric spin-coherent state, the semiclassical approach implicitly takes which in the limit of n have solutions 0 and π/2 respectively. This equation→ ∞is invariant under θ π/2 θ, advantage of the bit-symmetry of the problem. This is inacces- → − sible to an algorithm that has only black-box access to f, thus which corresponds to the two minima. Since the semi- limiting the generality of this approach. classical potential in Eq. (165) at s = 1/2 is also invariant 46 under θ π/2 θ, the local minima have identical struc- if entanglement is present, its computational role in AQC ture. Using→ the− Hamming Weight operator defined as: is unclear. The area law asserts that for any subset S of particles, n 1 the entanglement entropy between S and its complement HW = (11 σz) (167) 2 − i is bounded by the surface area of S rather than the trivial i=1 X bound of the volume of S. While generic quantum states this potential suggests that in the large n limit, we can do not obey an area law (Hayden et al., 2006), and there expect that n/2 spins need to be flipped in order to move are 1D systems for which there is exponentially more en- from the θ π/2 minimum to the θ 0 minimum, i.e., tanglement than suggested by the area law (Movassagh ≈ ≈ and Shor, 2016), a sweeping conjecture in condensed mat- Ω(π/2, 0) HW Ω(π/2, 0) Ω(0, 0) HW Ω(0, 0) = n/2 . ter physics is that in a gapped system the entanglement h | | i−h | | i (168) spreads only over a finite length, which leads to area laws The instantaneous ground state, as it passes through the for the entanglement entropy (Eisert et al., 2010).38 E.g., minimum gap at s = 1/2, indeed exhibits this behavior, the area law for gapped 1D systems, proved in (Hastings, as shown in Fig.8. 2007), states that for the ground state, the entanglement However, the more general role of tunneling in proof any interval is upper bounded by a constant indepen- viding quantum speedups is not by any means evident. dent of the size of the interval. While this leaves open the This topic was studied in detail in (Muthukrishnan et al., question of the general dependence of the upper bound 2016), which showed that tunneling is neither necessary on the spectral gap ∆, this means that the ground state nor sufficient for speedups in the class of perturbed Ham- of such systems is accurately described by polynomial- ming weight optimization problems with qubit permuta- size matrix product states (MPSs) (Ostlund¨ and Rom- tion symmetry. mer, 1995; White, 1992; White and Noack, 1992). In Our discussion here has been restricted to coherent (Gottesman and Hastings, 2010) it was shown that for tunneling, and compelling arguments have been pre- certain 1D system the entanglement entropy in some re- sented in (Andriyash and Amin, 2017; Boixo et al., 2016; gions can be as high as poly(1/∆). This demonstrates Denchev et al., 2016) that incoherent, thermally assisted that the entanglement entropy can become large as the tunneling plays a computational role in quantum anneal- gap becomes small. Two other important recent results ing. However, this mechanism is in the open-system set- are the existence of a polynomial time algorithm for the ting, which is outside the scope of this review. Moreover, ground state of 1D gapped local Hamiltonians with con- its role in (Boixo et al., 2016; Denchev et al., 2016) is lim- stant ground-state energy (Huang, 2014; Landau et al., ited to a prefactor, and does not translate into a scaling 2015), and the fact that 1D quantum many-body states advantage, i.e., it does not qualify as a speedup according satisfying exponential decay of correlations always fulfill to the classification of (Rønnow et al., 2014). an area law (Brandao and Horodecki, 2013). However, the connection between entanglement entropy and gaps is not nearly as clear in higher dimen- 2. The role of entanglement sional systems, even though entanglement close to quantum phase transitions is a well developed subject (Amico The role that entanglement plays in quantum compu- et al., 2008; Osborne and Nielsen, 2002; Osterloh et al., tation with pure states in the circuit model depends on 2002; Vidal et al., 2003; Wu et al., 2004). the entanglement measure used. On the one hand, it is It is not surprising that entanglement is necessary for well known that for any circuit-model quantum algorithm the computation to succeed if the intermediate ground operating on pure states, the presence of multi-partite states that the system must follow are entangled. This entanglement quantified via the Schmidt-rank (with a was verified explicitly in (Bauer et al., 2015), where number of parties that increases unboundedly with in- the quantum state was represented by an MPS and put size), is necessary if the quantum algorithm is to projected entangled-pair states (PEPS) (Verstraete and offer an exponential speedup over classical computation Cirac, 2004; Verstraete et al., 2008). This work showed (Jozsa and Linden, 2003). On the other hand, universal that the probability of finding the ground state of an quantum computation can be achieved in the standard Ising spin glass on either a planar or non-planar two- pure-state circuit model while the entanglement entropy dimensional graph increases with the amount of entan- (or any other suitably continuous entanglement measure) glement in the MPS state or PEPS state. Furthermore, of every bipartition is small in each step of the compu- even a small amount of entanglement gives improved suc- tation (Van den Nest, 2013). The corresponding role of cess probability over a mean-field model. However, this entanglement in the computational efficiency of AQC remains an open question. Partly this is because the connection between entanglement and spectral gaps is not 38 Here “gapped” means O(1), whereas in AQC “gapped” usually yet very well understood, and partly this is because even means O[1/poly(n)]. 47 does not resolve the role entanglement plays in generat- A. Avoiding poor choices for the initial and final ing a speedup. Hamiltonians In an attempt to address this, the entanglement entropy for the adiabatic Grover algorithm was studied, We first show that if one chooses the initial Hamilto- and it was found to be bounded ( 1) throughout the nian to be the one-dimensional projector onto the uni- ≤ form superposition state φ , and uses a linear inter- evolution (Orúsand Latorre, 2004). This was also ob- | i served numerically for systems with 10 qubits (Wen and polation, then an improvement beyond a Grover-like Qiu, 2008). In an effort to check whether more entangle- quadratic speedup is impossible as long as the final ment may help the Grover speedup, (Wen et al., 2009) Hamiltonian H1 is diagonal in the computational basis. considered adding an additional term to the Hamiltonian Specifically, for an adiabatic algorithm of the form to make the ground state more entangled, to reach an t t O(1) scaling in a Grover search task. However, since it H(t) = 1 E (11 φ φ ) + H1 , (170) − t − | ih | t is impossible to achieve a better-than-quadratic speedup f f in the Grover search problem without introducing an ex- the run time tf for measuring the ground state of H1 with plicit dependence on the marked state (Bennett et al., probability p is lower bounded by [Theorem 1 of (Farhi 1997), this result is not conclusive in linking entangle- et al., 2008); see also (Znidariˇcandˇ Horvat, 2006)]: ment with enhanced computational efficiency. Further- more, a two-dimensional path for the Grover problem 2 N √p t 1 1 p 2 , (171) using the quantum adiabatic brachistochrone approach f ≥ E − − k − E r [see Sec. VII.B] that gives a higher success probability n p for the same evolution time relative to the standard one- where N = 2 and k is the degeneracy of the ground dimensional path for the Grover problem, in fact has less state of H1. To see this, define an operator Vx for x = 0,...,N 1 that is diagonal in the computational basis: entanglement (negativity) (Rezakhani et al., 2009). − The entanglement entropy in the adiabatic algorithm 2πizx/N z Vx z = e , (172) for the Exact Cover problem, where no speedup is known h | | i 1 N 1 2πizx/N (recall Sec. VI.F.2), scales linearly with problem size for and let x = V φ = − e z . Now define x √N z=0 n 20 (Latorre and Orus, 2004; Orúsand Latorre, 2004). | i | i | i ≤ the modified adiabatic algorithm: Further studies have also shown this lack of correlation P between performance and the amount of entanglement t t Hx(t) = 1 E (11 x x ) + H1 . (173) et al. − t − | ih | t entropy. In (Hauke , 2015) simulations of adiabatic f f quantum optimization were performed of a trapped ion Hamiltonian with n = 16 of the form: Note that x = 0 = φ implies that H0(t) = H(t). For each x, the| finali state| i is given by ψ = U (t , 0) x , | xi x f | i n z z n with success probability px = ψx P ψx , where P is σi σj z z z z h | | i H1 = J + h σ + V σ σ , (169) the projector onto the ground subspace of H . Using i j i i i j 1 i=j i i=j H (t) = V H V , we have U (t, 0) = V U (t, 0)V , and X6 | − | X X6 x x 0 x† x x 0 x† hence px = p, x since Vx commutes with P . We should z ∀ with 100 disorder realizations of hi . It was found that a already see a potential problem for having tf scale better large entanglement entropy has little significance for the than √N, since if we were to run the algorithm backward, success probability of the optimization task. we would find the state x , which would be solving the Overall, these results indicate that the connection be- Grover problem (note that| i the initial Hamiltonian (173) tween entanglement and algorithmic efficiency in AQC is is the Grover Hamiltonian in a rotated basis). currently wide open and deserves further study. Now define an evolution according to an x-independent Hamiltonian: t t VII. CIRCUMVENTING SLOWDOWN MECHANISMS HR(t) = 1 E11 + H1 , (174) − t t FOR AQC f f 1 and let gx = P ψx . Consider the difference in the In this section we collect several insights into mech- | i √p | i reverse-evolutions associated with H (t) and H (t) from anisms that explain slowdowns in the performance of R x g : adiabatic algorithms. We also discuss mechanisms for | xi circumventing such slowdowns. Several important ideas 2 S(t) = U †(t , t) U † (t , t) g . (175) will be reviewed: avoiding the use certain initial and final k x f − R f | xik x Hamiltonians, modifying the adiabatic schedule, avoid- X ing quantum phase transitions, and avoiding perturba- We can write gx = √p ψx + √1 p ψ⊥ , where ψ⊥ | i | i − | x i | x i tive energy level crossings. is orthogonal to ψ . Using U †(t , 0) ψ = x and | xi x f | xi | i 48

defining Rx = UR† (tf , 0) gx , we have: as √n (Brady and van Dam, 2017; Muthukrishnan et al., | i | i 2016). If, however, we were to choose instead the pro- 2 S(0) = √p x + 1 p x⊥ R (176a) jector initial Hamiltonian, the result above shows that k | i − | i − | xik x we would find a dramatically poor scaling despite the X p simplicity of the final Hamiltonian. = 2N √p x R + 1 p x⊥ R + c.c. − h | xi − h | xi x A similar result is found if all structure is removed from X h p i (176b) the final Hamiltonian. Namely, if H1 = z h(z) z z , we can define a permutation π over the N computational| ih | 2N 2√p x Rx 2N 1 p . (176c) [π] 1 P ≥ − |h | i| − − basis states such that h (z) = h(π− (z)). Assume that x X p the initial Hamiltonian is π-independent and that c(t) Since H commutes with H , the state R is an element satisfies c(t) 1. Then, for the permuted Hamiltonian R 1 | xi | | ≤ of the k-dimensional ground subspace of H1. Choosing H1,π = z h(z) π(z) π(z) , one can show that if the k adiabatic algorithm| ih | a basis Gi i=1 for this subspace, and writing Rx = k {| i} | i P i=1 αx,i Gi , we have: | i Hπ(t) = H0 + c(t)H1,π (182) P x Rx αx,i x Gi (177) |h | i| ≤ | | · |h | i| succeeds with probability p for a set of N! permutations, x x,i X X then [Theorem 2 of (Farhi et al., 2008)]: √ αx,i x0 Gi0 = Nk . ≤ | | |h | i| 2 s x,i x0,i0 p /2 X X tf √N 1 , (183) ≥ 16h∗ − − 4h∗ Therefore, we have: p 2 where h∗ = z h(z) /N 1. This result means that S(0) 2N 1 1 p 2 Nkp . (178) no algorithm of the form of− Eq. (182) can find the mini- ≥ − − − p mum of H withP a constant probability for even a frac- p p 1,π In order to upper-bound S(0), we use S(tf ) S(0) tion of all permutations if tf is o(√N). t − ≤ f d S(t) dt with S(t ) = 0. The derivative can be The lesson from this analysis is what not to do when 0 | dt | f computed using the Schrödingerequation: designing quantum adiabatic algorithms: avoid choosing R the initial Hamiltonian to be the one-dimensional pro- d S(t) = i g U (t , t)[H (t) H (t)] U † (t , t) g jector onto the uniform superposition state if a better- dt − h x| x f x − R R f | xi x than-quadratic speedup is hoped for, and avoid removing X + c.c. structure from the final Hamiltonian. t = 2 1 E g U (t , t) x − = − t h x| x f | i× x f B. Quantum Adiabatic Brachistochrone X x UR† (tf , t) gx . (179) h | | i Modifying the adiabatic schedule adaptively so that Thus: it slows down as the gap decreases is an approach that is essential for obtaining a quadratic speedup using the d t S(t) 2E 1 x U † (t , t) g adiabatic Grover algorithm [recall Sec. III.A]. Here we dt ≤ − t h | R f | xi f x discuss how such ideas, including the condition for the X t locally optimized schedule [Eq. (24)] can be understood 2E 1 √Nk , (180a) ≤ − t as arising from a variational time-optimal strategy for de- f termining the interpolating Hamiltonian between H0 and where in Eq. (180a) we used the same trick as in H1 (Rezakhani et al., 2009). By time-optimal, we mean tf d √ Eq. (177). Therefore, 0 dt S(t) dt Etf Nk. Putting a strategy that gives rise to the shortest total evolution the upper and lower bound| for S|(0)≤ together, we have: time t while guaranteeing that the final evolved state R f ψ(t ) is close to the desired final ground state ε (t ) . | f i | 0 f i Etf √Nk 2N 1 1 p 2 Nkp , (181) The success of the strategy is judged by the trade-off be- ≥ − − − tween t and the fidelity F (t ) = ψ(t ) ε (t ) 2. We p p f f |h f | 0 f i| which yields Eq. (171). first discuss this method generally and then show how it As an example of the relevance of this result, consider applies to the adiabatic Grover case. the trivial case of n decoupled spins in a global mag- The interpolating Hamiltonian’s time-dependence netic field. For an initial Hamiltonian that reflects the comes from a set of control parameters ~x(t) = 1 M bit-structure of the problem, e.g., the standard H0 = x (t), . . . , x (t) , i.e., H(t) = H[~x(t)]. We can param- σx, the run time of the adiabatic algorithm scales eterize ~x(t) in terms of a dimensionless time parameter − i i P 49

ds 1 1 2 2 p 2 s(t) with s(0) = 0 and s(tf ) = 1, where v = dt character- Γ g− ∂g ∆− ∂∆, and R ∂ g + gΓ ∆− − . izes the speed with of motion along the control trajectory Thus,∼ the smaller∼ the gap, the higher∼ the curvature.∼ ~x[s(t)]. The total evolution time is then given by: Let us illustrate with a simple example. Consider the following Hamiltonian with a single control parameter 1 ds x1(s): tf = . (184) 0 v(s) Z 1 1 H(s) = 1 x (s) Pa⊥ + x (s)Pb⊥ , (189) Motivated by the form of the adiabatic condition, let us − define the following Lagrangian where we have defined the projector Pa⊥ = 11 a a and similarly for P . This includes the Grover−| problemih | as ∂ H(s) 2 b⊥ ˙ s HS the special case where a is the uniform superposition [~x(s), ~x(s)] k p k (185) L ≡ ∆ (s) and b is the marked state.| i We can always find a state Tr (∂ i H(s)∂ j H(s)) | i = x x ∂ xi(s)∂ xj(s) a⊥ such that b = α0 a + α1 a⊥ , where α0 = a b . ∆p(s) s s |Therefore,i the evolution| i | accordingi | i to H(s) occursh in| i a i,j X two dimensional subspace spanned by a and a⊥ , and: (p > 0), and adiabatic-time functional | i | i

1 ˙ ∂x1 H(s) = P ⊥ + P ⊥ (190a) [~x(s)] = ds [~x(s), ~x(s)] , (186) − a b T 0 L 2 Z Tr (∂x1 H(s)∂x1 H(s)) = 2 1 α0 (190b) − | | where A Tr (A A) is the Hilbert-Schmidt norm, 2 1 1 HS † ∆(s) = 1 4 (1 α0 ) x (s)(1 x (s)) (190c) chosenk tok ensure≡ analyticity (this choice is not unique, − − | | − p 2(1 α 2) but other choices may not induce a corresponding Rie- p g = − | 0| . (190d) 11 ∆(s)3 mannian geometry). The time-optimal curve ~xQAB(s) is the quantum adiabatic brachistochrone (QAB), and is The geodesic equation is then given by: the solution of the variational equation δ [~x(s)]/δ~x(s) = 0. T d2 x1(s) (191) Alternatively, the problem can be thought of in ds2 geometrical terms. The integral in Eq. (186) is 1 2 2 p 1 2x (s) 1 α0 d 1 i j − − | | of the form ds g (~x)∂ x ∂ x , which defines a + 1 1 2 x (s) = 0 . i,j ij s s 1 4x (s) (1 x (s)) (1 α0 ) ds reparametrization-invariant object. Therefore, using − − − | | results from differentialR P geometry, the Euler-Lagrange In the case of p = 4, we can solve this equation ana- equations derived from extremizing Eq. (186) are sim- lytically, and the solution with the boundary conditions x1(0) = 1 x1(1) = 0 is given by: ply the geodesic equations associated with the metric gij − ˙ i j appearing in [~x(s), ~x(s)] = gij(~x)x ˙ x˙ (Einstein sum- L 1 1 α0 1 mation convention): x (s) = + | | tan cos− ( α0 ) (2s 1) 2 2 2 1 α0 | | − 2 k k i j − | | (192) ∂s x + Γij∂sx ∂sx = 0 , (187) p 2 1 1 √1 α0 (note that cos− ( α0 ) = tan− −| | ). Remark- where Γk = 1 gkl (∂ g + ∂ g ∂ g ) are the Christof- | | α0 ij 2 j li i lj l ij | | fel symbols (connection coefficients)− and ably, this is equivalent to the expression we found for the Grover problem [Eq. (29)] if we take α0 = 1/√N, despite Tr[∂iH(~x)∂jH(~x)] the different choice of norm and value of p. This shows gij(~x) = . (188) ∆p(~x) that the optimal schedule for the Grover problem has a deep differential geometric origin. To find the variational time-optimal strategy associated We can extend the analysis to two control parameters with minimizing Eq. (186), the procedure is thus as such that the time-dependent Hamiltonian is given by: follows: (a) solve Eq. (187) to find the optimal path ~xQAB(s); (b) compute the adiabatic error using the 1 2 H(s) = x (s)Pa⊥ + x (s)Pb⊥ . (193) Schrödingerequation along this optimal path (or multi- parameter schedule). Note that to compute the metric The associated QAB (or geodesic) path can be found requires knowledge of the gap, or at least an estimate numerically, and it turns out that it is not of the form thereof. x2(s) = 1 x1(s), i.e., it is different from the (Roland The optimal path is a geodesic in the parameter mani- and Cerf, 2002− ) path given by Eq. (192). The optimal fold endowed with the Riemannian metric g. This metric two-parameter path reduces the adiabatic error relative gives rise to a curvature tensor R, which can be com- to the latter [see Fig. 9(a)], but can of course not reduce puted from the metric tensor and the connection us- the (already optimal) √N scaling. The two-parameter ing standard methods (Nakahara, M., 1990). Namely, QAB also has lower curvature than the Roland-Cerf path 50

[see Fig. 9(b)], which implies that it follows a path with It was then shown that in this case, by picking a ran- a larger gap and less entanglement than the latter (Reza- dom initial Hamiltonian of the form et al. khani , 2009), as mentioned in Sec. VI.G.2. n The diﬀerential geometric approach to AQC was fur- 1 x H0 = ci (11 σ ) , (194) ther explored in (Rezakhani et al., 2010a), where its con- 2 − i i=1 nections to quantum phase transitions were elucidated, X within a unifying information-geometric framework. See where ci is a random variable taking value 1/2 or 3/2 with also (Zulkowski and DeWeese, 2015). equal probability, it is possible to remove the small gap encountered by the standard adiabatic algorithm with !!!!!!! !!!!! !!! !!!! !! !!! !!! !!!!! !!!! !!! !!!! !!!! !! !4 !! !! !! ! !! ! ! ! ! ! !!!! !!!!! !!!! !! ! ! !! !""! !"! ! ! ! ! !! !!!! !!!!!! !!!! !!! ! ! " ! !! !! ! ! """" ! !"! !""! !""! !"! ! ! ! ! !! !! !! !!!!!! !!!!!! !!!!!!! !!!!!! !!!!! ! !""""! !"! !""! !""! !""! !"! ! ! ! ! ! !! ! !! ! !! ! !!!! !!!! high probability. Since this strategy does not rely on in- ! !" "! !""! !""! !""! !""! !""! !"! !! !! !! ! ! ! ! ! ! ! ! ! !" "! !""! !""! !""! !""! !""! !""! !! !! !! !! !! !! ! ! ! !" "! "!" !"!" !"!" !"!" !"!" !"!" !"! !! !! !"! !"! !"! !! !6 ! !" !" !!" !"!" !"!" !"!" !"!" !"!" !"! !! !"! !"! !"! !"! !"! ! !" !" !"!" !" !"!" !"!" !"!" !"!" !"!" !"! !"! !""! !""! !""! !""! ! !" !" !" !"!" !"!" !"!" !"!" !"!" !"!" !"! !"! !"!" !""! !""! !""! ! !" !" !" " !" !"" !"!" !"!" !"!" !"! !"!" !"!" !"!" !"!" !"!" formation about the speciﬁc instance, it appears to be ! !" !" "!" !"!" !" !! "" !" !"!" !"!" !"!" !"!" !"!" !"!" !"!" ! " " !" "" !"! !"!" !"! !!" "!" !"!" !"!" !"!" !"!" ! !" !" ! ! ! " !" !"!" !" "! !"" !"" !"!" !8 ! !" " ! !" ! !"!" "! " !"!" !" "! ! "" geodesic2 ! ! ! "" "! ! !" "! !" !" !! ! " " " ! ! " ! "! " ! ! " " " ! " quite general. Therefore, if the algorithm is to be run on ! " " ! " !" ! "" " !! ! ! ! ! !" !∆ ! !! ! " ! ! !10 ! ! " ! " ! " " ! ! " RC ! " a single instance of some optimization problem, the adi- ∆

! !12 ! ! ! abatic algorithm should be run repeatedly with different log !14 ! initial Hamiltonians (Farhi et al., 2011). 0 100 200 300 400 An alternative approach based on modifying the ini- T tial Hamiltonian, with a different goal, was proposed in (a) (Perdomo-Ortiz et al., 2011), whereby an initial guess for the solution (a computational basis state) is used as the initial state of the adiabatic algorithm. An initial Hamil- tonian is used with this state as its ground state. Evolu- tion to the final Hamiltonian then proceeds according to a standard schedule. If the final state that is measured is not the ground state of the final Hamiltonian (due to diabatic transitions), the algorithm can be repeated with the measured state as the new initial state. Such “warm start” repetitions of the algorithm exhibited improved performance compared to the standard approach (b) for 3-SAT problems, although the results were limited to p relatively small system sizes of 6 and 7 qubits. FIG. 9 (a) Final-time error δ(T ) = 1 − F (T )(T = Tf in our notation) for the single-control (denoted RC for Roland- Cerf) and two-parameter control (denoted geodesic2) geodesic paths for the Grover problem with n = 6. Squares (cyan) in- D. Modifying the final Hamiltonian dicate where the two-parameter geodesic path outperforms (i.e. has a lower error than) the single-parameter path; cir- The same problem can be specified by two or more dif- cles (red) correspond to the opposite case. (b) The curva- ferent final Hamiltonians, as we saw, e.g., in the case of ture tensor component R1212 for n = 3. The curves on the Exact Cover 3 (EC3), in terms of Eqs. (154) and (155). curvature surface show the case of the standard linear inter- It was claimed in (Altshuler et al., 2010) that adiabatic polation x = 1 − x (denoted Crit.), the path followed by 2 1 quantum optimization fails for random instances of EC3 the one-parameter geodesic (denoted RC), and the path followed by the two-parameter geodesic (denoted QAB). From because of Anderson localization. The claim, which we (Rezakhani et al., 2009). discuss in more detail in Sec. VII.G, was based on the form given in Eq. (155). However, as argued in (Choi, 2011), it is possible to reformulate the final Hamiltonian for EC3 such that the argument in (Altshuler et al., 2010) C. Modifying the initial Hamiltonian may not apply. Namely, for any pair of binary variables

xCi , xCj in the same clause C, add a term DijxCi xCj with Rather than modifying the adiabatic interpolation, one Dij > 0; this is permissible since in order for a clause may modify the initial Hamiltonian. Such a strategy was to be satisfied, exactly one variable must take value 1, pursued in (Farhi et al., 2011) and tried on a particu- whereas the other two are 0. Numerical evidence for lar set of 3-SAT instances, where the clauses are picked up to 15 bits suggests that the addition of the new set randomly subject to satisfying two disparate planted so- of arbitrary parameters Dij may avoid the Anderson lo- lutions and then penalizing one of them with a single calization issue (Choi, 2011). This example illustrates additional clause. This was done in order to generate in- a general principle, that it can be incorrect to conclude stances with an avoided crossing at the final time s = 1, from the failure of one specific choice of the final Hamil- reproducing the type of obstacle to AQC envisioned in tonian that all quantum adiabatic algorithms fail for the (Altshuler et al., 2010). same problem. 51

E. Adding a catalyst Hamiltonian to study the semi-classical potential associated with the Hamiltonian: We define a “catalyst” as a term that (1) vanishes at the initial and final times, but is present at intermediate V (s, θ, ϕ) = θ, ϕ H(s) θ, ϕ , (201) times, (2) is a sum of local terms with the same qubit- h | | i interaction graph as the final Hamiltonian H1, (3) does not use any other information specific to the particular where θ, ϕ is the spin-coherent state defined in | i instance. Eq. (162). In the large n limit we have (Farhi et al., Consider, e.g.: 2002b):

H(s) = (1 s)H + s(1 s)H + sH . (195) − 0 − C 1 lim V/(2/n)3 = 2(1 s)(1 sin θ cos ϕ) n →∞ − − The specific form of HC is of course important, but even 1 + s 13 + 3 cos θ 9 cos2 θ 7 cos3 θ a randomly chosen catalyst can help (Farhi et al., 2011, 6 − − 2002b; Zeng et al., 2016). We illustrate how HC can 8s(1 s) cos θ sin θ cos ϕ , (202) turn a slowdown (exponential run time) into a success (at − − worst polynomial run time) for a specific problem with a specific HC that is analytically tractable. Consider a where the three terms arise from the initial, final and final Hamiltonian of the form catalyst Hamiltonians, respectively. We display the behavior of this potential in Fig. 10. In the absence of HC, H = h(z) z z , (196) there is a value of s where the potential has degenerate 1 | ih | z minima, and the system must tunnel from the right well X to the left well in order to follow the global minimum. where z denotes an n-bit string, and h(z) = This point is associated with an exponentially small gap iz3 = 0 presence of HC the potential never exhibits such an ob- 3, z1 + z2 + z3 = 1 h3(z1, z2, z3) =  . (197) stacle; there is always a single global minimum that the  1, z1 + z2 + z3 = 2 system can follow from s = 0 to s = 1 with polynomial  1, z1 + z2 + z3 = 3 run time.  The all-zero bit string minimizes the final Hamiltonian Using this method of introducing a catalyst Hamilto- with energy 0. nian, improvements were generally observed on a large The cost function h(z) is bit-permutation symmetric number of MAX 2-SAT instances of size n = 20 (by di- and only depends on the Hamming weight z , which fa- rectly solving the Schrödingerequation) (Crosson et al., | | cilitates the analysis. Specifically (Farhi et al., 2002a): 2014). Both stoquastic and non-stoquastic HC were tried and improved the success rate, but the difference between 3 1 h(z) = z (n z )(n z 1) + z ( z 1) (n z ) stoquastic and non-stoquastic was not decisive. 2| | − | | − | | − 2| | | | − − | | A similar study was performed in (Hormozi et al., 1 + z ( z 1) ( z 2) . (198) 2017) on fully-connected Ising instances, H1 = 6| | | | − | | − n h σz + n J σzσz, of size n 17, where the i=1 i i i

2.5 2.5 Seoane and Nishimori, 2012) to analyze inﬁnite-range

2 2 Ising models with ferromagnetic as well as random interactions. These studies concluded that non-stoquastic 1.5 1.5 terms can sometimes modify ﬁrst-order quantum phase 1 1 transitions (with an exponentially small gap) in the sto- 0.5 0.5 quastic Hamiltonian to second-order transitions (with a 0 0 polynomially small gap) in the modiﬁed, non-stoquastic -0.5 -0.5 0 /4 /2 3 /4 0 /4 /2 3 /4 Hamiltonian. (a) (b) The Hamiltonian is of the form

(p) (k) 2.5 2.5 H(s, λ) = (1 s)H s λH + (1 λ)H (203) − 0 − 1,z − 1,x 2 2 n 1.5 1.5 where H = σx is a standard initial Hamiltonian, 0 − i=1 i 1 1 and 0.5 0.5 P n q 0 0 (q) 1 α H1,α = n σi , α x, z , q p, k , -0.5 -0.5 n ∈ { } ∈ { } 0 /4 /2 3 /4 0 /4 /2 3 /4 i=1 ! X (c) (d) (204) where λ [0, 1] controls the strength of the non- ∈ 2.5 2.5 (k) stoquastic term H1,x , and both p and k are integers 2 2 2 that determine the locality of the model. The pa- ≥ 1.5 1.5 rameter λ is increased to 1 along with s, so that the 1 1 final Hamiltonian is the infinite-range p-body ferromag- (p) 0.5 0.5 netic Ising model H(1, 1) = H . Also the r-pattern − 1,z 0 0 (p) Hopfield model was studied, where λH1,z is replaced by -0.5 -0.5 z z 0 /4 /2 3 /4 0 /4 /2 3 /4 1 i <

0 0 situation in the Hopﬁeld model case is identical to the

-0.5 -0.5 ferromagnetic case, for an extensive number of patterns 0 /4 /2 3 /4 /4 /2 3 /4 0 r n. For a fixed number of patterns p 5 is sufficient (g) (h) and∝ p > 3 is necessary in order to avoid first≥ order phase transitions (Seki and Nishimori, 2015). FIG. 10 The diamonds represent the minima followed by a polynomial run time. In the case with HC, the potential can follow the global minimum polynomial time. In the case with- G. Avoiding perturbative crossings out HC, there is an s value where the potential has a degenerate minimum, and the algorithm cannot tunnel to the new global minimum in polynomial time. An important slowdown mechanism we already al- luded to in Sec. VII.C is due to anti-crossings very close to the end of the evolution, that can result in an ex- F. Addition of non-stoquastic terms tremely small minimum gap. These crossings are often referred to as perturbative, because a perturbative ex- The addition of non-stoquastic terms was already con- pansion back in time from the final Hamiltonian [e.g., sidered numerically in the previous subsection; here we perturbation theory in Γ for Eq. (143)] yields perturbed focus on analytical results obtained for certain mean field models.

Quantum statistical-mechanical techniques (Trotter- 39 This addition does not result in a truly non-stoquastic Hamilto- Suzuki decomposition, replica method under the replica- nian; there exists a local unitary transformation that makes the symmetric ansatz, and the static approximation) were Hamiltonian stoquastic. Specifically, rotate σz to σx. In this used in (Nishimori, 2016; Seki and Nishimori, 2012, 2015; new basis, the Hamiltonian is stoquastic. 53 states that cross in energy very close to where the exact ysis, assume that there are no single bit-flip degenera- eigenstates anti-cross, with a gap that is exponentially cies, meaning that there are no degenerate states that are small in the Hamming weight of the unperturbed crossing Hamming distance 1 from each other. For each non-zero z states (Amin and Choi, 2009) [shown there in the context hi term that the ground state satisfies, i.e., hiσi = 1, of the weighted maximum independent set problem; see add a 1 ancilla qubits with an interaction of the form:− ≥ also (Farhi et al., 2011; Foini et al., 2010)]. This prob- a lem of perturbative crossings was demonstrated for the z z Hh = b (hiσi + 1) σik + 1 /2 , (206) NP-complete Exact Cover problem [recall Sec. VI.F.2] in kX=1 (Altshuler et al., 2010), who related the mechanism of ex- where i1, . . . , ia Mh. This term vanishes when the ponentially small spectral gaps to Anderson localization { } ∈ term hi is satisfied, regardless of the orientation of the a of the eigenfunctions of H(s) in the space of the solu- ancillas, whereas otherwise it gives an energy bn1, where tions. They showed that the Hamming weight between n1 is the number of ancillas pointing up. Note that when such states can be Θ(n), which is clearly problematic for the term is unsatisfied, when all ancilla spins point down the adiabatic algorithm. It was also claimed in (Altshuler the energy cost is zero. This is important because we et al., 2010) that these anti-crossings appear with high do not want to change the energy of the ground state probability as the transverse field goes to zero; however configuration. the latter claim did not survive a more accurate anal- Similarly, for each (non-zero) Jij, also add a ancillas ysis that took into account the extreme value statistics with the following interaction term: of the energy levels: the exponential degeneracy of the a ground state, which is a distinguishing feature of random z z z HJ = Jijσ σ + 1 σ + 1 /2 , (207) NP-complete problems with a discrete spectrum (such as i j (ij)k k=1 Exact Cover), dooms the proposed mechanism (Knysh, X where ij , . . . , ij M . This introduces 3-local 2016; Knysh and Smelyanskiy, 2010). { 1 a} ∈ J Nevertheless, this does not rule out the occurrence of terms; it is possible to use 2-local terms to achieve the exponentially small gaps close to the end of the evolu- same result at the expense of introducing an additional tion. Furthermore, it is plausible that the mechanism ancilla for each term [see the Appendix of (Dickson, 2011) for avoided level crossings presented in (Altshuler et al., for details]. 2010) may not necessarily be restricted to the end of The spectrum of the new Hamiltonian (with ma ad- the evolution, but may occur throughout a many-body- ditional ancilla qubits) is the original spectrum when all localized phase (Laumann et al., 2015). In light of this we ancilla qubits point down, and all the new energy states now discuss a rather general way to circumvent such per- correspond to flips of the ancilla qubits, with increased turbative crossings, that differs from the random initial energy. Note that this means that no new local minima Hamiltonian approach presented in Sec. VII.C. were introduced. Now consider the following adiabatic Using the NP-hard maximum independent set prob- algorithm Hamiltonian: lem, it was shown that this problem occurs only for one H = λH0 + H10 ,H10 = H1 + Hh + HJ , (208) particular implementation of the adiabatic algorithm, with λ decreased from (proportional to abm suffices) and different choices can avoid the problem (Choi, 2010). ∞ In fact, (Dickson and Amin, 2011) showed that there is to 0, and where the initial Hamiltonian includes trans- always some choice of the initial and final Hamiltonians verse fields on the ancilla qubits: that avoids such perturbative crossings (note that this x H0 = σi . (209) does not include non-perturbative crossings). Further- − i M Mh MJ more, this choice can be made efficiently, i.e., in polyno- ∈ ∪X ∪ Consider a non-degenerate classical state α with energy mial time, space and energy (Dickson, 2011), as we now E under the action of H . It becomes degenerate under summarize. α 1 the action of H . Let α denote the uniform superpo- The idea of (Dickson, 2011) is to cause the ground 10 sition over all these degenerate| i states with energy E . state to diverge from all other states by changing the α Introducing λ > 0 breaks the degeneracy, and from first degeneracy of the spectrum of the final Hamiltonian, such order degenerate perturbation theory (see Appendix E.1) that the ground state is the most degenerate, the first the state α is the new lowest energy eigenstate within excited state less degenerate, up to the highest excited the subspace| i spanned by the unperturbed degenerate state, which will be the least degenerate. Consider an states with energy Eα. The correction to its energy is n-qubit Ising Hamiltonian of the form (1) E α = Eα + λE α + ... , where z z z | i | i H1 = hiσi + Jijσi σj , (205) i M i,j M Eα = (# of terms in H10 satisfied by α) X∈ { X}∈ − + (# of terms in H10 unsatisfied by α) (210) where hi,Jij 0, 1 and M specifies the non-zero ∈ { ± } = 2(# of terms in H0 satisfied by α) + m terms, of which there are m. In order to simplify the anal- − 1 54

(recall that hi,Jij 0, 1 ), and avoided-level crossing. An instance of this behavior is ∈ { ± } shown in Fig. 11. (1) E α = α H0 α (211) | i h | | i = a (# of terms in H10 satisfied by α) 1.0 −a = (Eα m) . 2 − 0.8 2 ÈXΨHtLÈΨ0HtL\ Note that in Eα the contribution is entirely due to H1, (1) 2 0.6 ÈXΨHtLÈΨ1HtL\ while in E α the contribution is entirely due to Hh +HJ . | i Taking a = b = n2, it can be shown that higher order corrections do not depend on a, and hence the first or- 0.4 der correction dominates the behavior. Therefore, it is clear that a state α with a lower (final) energy than a 0.2 state β has a larger| i negative slope (first-order pertur- | i bation energy correction). Therefore, the states α and 0.0 β grow farther apart for λ > 0 according to first| i order 0.0 0.2 0.4 0.6 0.8 1.0 perturbation| i theory. This means that the perturbative t T crossing is avoided. FIG. 11 Overlap squared of the evolved wavefunction |ψ(t)i This method works in general for the problem of find- and the instantaneous ground state |ψ0(t)i and first excited ing the ground state of an arbitrarily-connected Ising state |ψ1(t)i for an instance of MAX-2-SAT with n = 20 and model with local fields, and is fully stoquastic. Thus, all with a total time T = 10. Because of the rapid evolution, NP-complete problems can be attacked using StoqAQC population leaks out of the ground state and hence the de- without encountering perturbative crossings. Of course, crease in the ground state population. There is an avoided this does not prove that StoqAQC can solve NP-complete level crossing at approximately t/T = 0.65, where the pop- problems in polynomial time. However, it does mean that ulation between the ground state and first excited state are effectively swapped. Therefore, if more substantial leaking proving otherwise requires identifying some effect other into the first excited state occurs, this will lead to an increase than perturbative crossings that unavoidably results in in probability of finding the ground state at the end of the exponentially long adiabatic run times. evolution. From (Crosson et al., 2014).

H. Evolving non-adiabatically A similar result was observed in (Muthukrishnan et al., 2016) for a large class of Perturbed Hamming Weight Our discussion so far has been restricted to adiabatic problems (recall Sec. VI.E.1), but with the difference evolutions, where the minimum gap controls the effi- that the rapid evolution diabatically pushes population ciency of the quantum algorithm. However, as we have to higher excited states and then returns to the ground seen with the glued-trees problem in Sec. III.D, the quan- state through a series of avoided-level crossings, a phe- tum evolution can take advantage of the presence of nomenon called “diabatic cascade”. two avoided-level crossings (and their associated exponentially small gaps) to leave and return to the ground These results raise the question of whether adiabatic state with high probability in polynomial time, whereas evolution is in fact the most efficient choice for running an adiabatic evolution would have required exponential a quantum adiabatic algorithm. After all, the goal is to time. Setting aside the fascinating and intricate field of find the ground state once, with the highest probability open-system AQC where relaxation can play a beneficial and in the shortest amount of time. Therefore, rather role in returning the computation to the ground state [the than maximizing the probability by increasing the evolu- subject of a separate review (Albash and Lidar, 2017)], tion time tf , we can instead use many rapid repetitions of this is one among several cases where non-adiabatic, i.e., the algorithm to simultaneously shorten tf and increase diabatic evolution enhances the performance of a quan- the success probability. Let pS(tf ) denote the single-run tum algorithm based on Hamiltonian computation. An- success probability of the algorithm with evolution time other example is (Crosson et al., 2014) [see also (Hor- tf . The probability of failing to find the ground state R mozi et al., 2017)] where it was observed that evolving after R independent repetitions is (1 pS) , so the prob- rapidly (as well as starting from excited states) increased ability of succeeding at least once is 1− (1 p )R, which − − S the success probability on the hardest instances of ran- we set equal to the desired probability pd. The trade- domly generated n = 20 MAX-2-SAT instances with a off between success probability and run time is there- unique ground state. When evolving rapidly, population fore well captured by the time-to-solution (TTS) metric, leaks into the first excited state before the avoided-level which measures the time required to find the ground state crossing and then returns to the ground state after the at least once with probability pd (typically taken to be 55

99%): ground state of the mapped Hamiltonian can then be prepared using adiabatic evolution followed by appropriate ln(1 p ) TTS(t ) = t − d . (212) measurements to determine the energy spectrum. How- f f ln[1 p (t )] − S f ever, the scaling of the minimum gap for such a preparation procedure is not known, and hence this is an exam- Other metrics exist that quantify this tradeoff, e.g., with- ple of AQC with a non-stoquastic Hamiltonian for which out insisting on finding the ground state (King et al., it is unknown whether a quantum speedup is possible. 2015), or that make use of optimal stopping theory and A variety of other interesting AQC results with an un- assign a cost to each run (Vinci and Lidar, 2016). known speedup, and which we did not have the space For p 1 (close to the adiabatic limit), only a single S . to review here in detail, can be found in (Behrman and (or few) repetitions of the algorithm are necessary and Steck, 2016; Cao et al., 2016; Chancellor, 2016; Dulny the TTS scales linearly with t . As t is lowered, the f f and Kim, 2016; Durkin, 2016; Goto, 2016; Hashizume success probability typically decreases and more repeti- et al., 2015; Inack and Pilati, 2015; Karimi and Rosen- tions are necessary, but the TTS may in fact be lower berg, 2016; Karimi and Ronagh, 2016; Kurihara et al., because of the smaller t value. The optimal t for the f f 2014; Miyahara and Tsumura, 2016; O’Gorman et al., algorithm minimizes the TTS, and is defined as: 2015; Rajak and Chakrabarti, 2014; Raymond et al., 2016; Rosenberg et al., 2016; Santra et al., 2016; Sato TTSopt = min TTS(tf ) . (213) tf >0 et al., 2014). Moreover, to make the review comprehensive and de- Benchmarking of algorithms then proceeds as follows. tailed enough to be self-contained, we focused only on For a specific class of problem instances of varying sizes the closed-system setting, thus completely ignoring the n, TTS is calculated for each size n. The scaling of opt important problem of AQC in open systems, with the the algorithm with n is then determined from the scaling associated questions of error correction and fault toler- with n of TTS , as, e.g., in (Boixo et al., 2014). opt ance. We also left out the experimental work on AQC and One benefit of this approach is in obtaining a quantum quantum annealing. These important topics will be the scaling advantage over specific classical algorithms.. For subject of a separate review (Albash and Lidar, 2017). example, the constant gap perturbed Hamming weight oracular problems (Reichardt, 2004) [Sec. VI.D.1] and Due to the prominence of stoquastic Hamiltonians the “spike” problem of (Farhi et al., 2002a) [Sec. VI.E.1] in the body of work on AQC, we coined a new term, with a polynomially closing quantum gap, can be solved StoqAQC, which is roughly what was meant when the in O(1) time using a classical algorithm. However, QA term “quantum adiabatic algorithm” was first intro- exhibits a scaling advantage over SA for these problems in duced. Correspondingly, we devoted a substantial part the sense that QA offers a TTS that scales better than SA of this review to StoqAQC, despite the fact that there with single-spin updates (Muthukrishnan et al., 2016). are indications that this model of computation may not be more powerful than classical computing. Its prominence is explained by the fact that it is easier to ana- VIII. OUTLOOK AND CHALLENGES lyze than universal AQC, which requires non-stoquastic terms, and by the fact that it is easier to implement ex- Adiabatic quantum computing has blossomed from a perimentally [see, e.g., (Bunyk et al., 2014; Weber et al., speculative alternative approach for solving optimization 2017)]. The relatively short history of AQC has wit- problems, to a formidable alternative to other universal nessed a fascinating battle of sorts between attempts to models of quantum computing, with deep connections show that StoqAQC fails to deliver quantum speedups, to both classical and quantum complexity theory, and and corresponding refutations by clever tweaks. To put condensed matter physics. this and other results we have discussed in the proper In this review we have given an account of most of the perspective, we conclude with a list of 10 key theoretical major theoretical developments in the field. Of course, challenges for the field of AQC: some omissions were inevitable. For example, a poten- tially promising application of AQC is in quantum chem- 1. Prove or disprove that StoqAQC is classically effi- istry, where the calculation of molecular energies can ciently simulatable. be formulated in terms of a second-quantized fermionic 2. Find an NP-hard optimization for which AQC gives Hamiltonian that is mapped, via a generalized Jordan- a quantum speedup in the worst case. Wigner transformation (Bravyi and Kitaev, 2002; P. Jor- dan and E. Wigner, 1928), to a non-stoquastic qubit 3. Find a class of non-oracular, physically realizable Hamiltonian (Aspuru-Guzik et al., 2005; Seeley et al., optimization problems for which AQC gives a quan- 2012). This mapping generates k-local interactions, but tum speedup. perturbative gadgets can be used to reduce the problem to only 2-local interactions (Babbush et al., 2014b). The 4. Identify a subset of non-stoquastic Hamiltonians 56

for which ground state preparation can be done ef- Appendix A: Technical calculations ficiently using adiabatic evolution. 1. Upper bound on the adiabatic path length L 5. Formulate every quantum algorithm that gives a speedup in the circuit model natively as an AQC Right below Eq. (14) we claimed that an upper bound algorithm (i.e., directly, without using perturbative on L is max H˙ (s) /∆. To see this differentiate the s k k gadgets). eigenstate equation H εa = εa εa for the normalized instantaneous eigenstate| iε and| inner-multiplyi by ε , 6. Find a problem that can be solved with a quantum a b with b = a, to get (ε | εi ) ε ε˙ = ε H˙ ε .h Let| speedup using AQC, that was not previously known a b b a b a ∆ = ε6 ε and ∆ =− minh min| i ∆ h(s).| Using| i our from other models of quantum computing. ba b a a b s ba phase choice:− 7. Give a way to decide whether adiabatic evolution ε˙ = ε ε ε˙ = ε ε ε˙ gives rise to a stronger or weaker speedup than non- | ai | bih b| ai | bih b| ai b b=a adiabatic (diabatic) evolution for a given problem. X X6 = ε ε H˙ ε /∆ . (A1) 8. Predict the optimal adiabatic schedule for a given − | bih b| | ai ba b=a problem without a priori knowledge of the size X6 and/or position of its spectral gap. Thus 9. Prove or disprove that tunneling can generate a 1 ε˙a εb εb H˙ εa (scaling, not prefactor) quantum speedup in AQC. k| ik ≤ ∆a k | ih | | ik b=a X6 10. Establish the relation between entanglement and 1 1 εb εb H˙ εa H˙ , (A2) quantum speedups using AQC. ≤ ∆a k | ih |kk | ik ≤ ∆a k |k b=a X6 Solving these problems will likely keep researchers busy where in the last equality we used the definition of the for years to come, require interdisciplinary collabora- operator norm and the fact that b=a εb εb is a pro- tions, and will significantly advance our understanding jector. Integration just multiplies by6 1. | ih | of AQC. We hope that this review will catalyze new and P productive approaches, enhancing our repertoire of algorithms that give rise to quantum speedups from the 2. Proof of the inequality given in Eq. (25) unique perspective of AQC. Note that

ACKNOWLEDGMENTS ∂ H[A(s)] = ∂ A H H 2 ∂ A (A3a) k s k | s | k 1 − 0k ≤ | s | ∂2H[A(s)] 2 ∂2A (A3b) We are grateful to Vicky Choi, Elizabeth Crosson, s ≤ | s | Itay Hen, Milad Marvian, Siddharth Muthukrishnan, for the interpolating Hamiltonian (16). Also note that Hidetoshi Nishimori, and Rolando Somma for useful discussions. This work was supported under ARO 2 p 1 d∆ ∂ A(s) = cp∆ − [A(s)] ∂ A(s) (A4a) Grant No. W911NF-12-1-0523, ARO MURI Grants No. s dA s W911NF-11-1-0268 and No. W911NF-15-1-0582, and 2 d∆ 2p 1 = c p ∆ − [A(s)] . (A4b) NSF Grant No. INSPIRE-1551064. The research is dA based upon work (partially) supported by the Office of the Director of National Intelligence (ODNI), Intelligence Thus: Advanced Research Projects Activity (IARPA), via the 1 ∂2H[A(s)] ∂ H[A(s)] 2 U.S. Army Research Office contract W911NF-17-C-0050. s s 2 + k 3 k ds The views and conclusions contained herein are those of 0 ∆ [A(s)] ∆ [A(s)] ! Z the authors and should not be interpreted as necessar- 1 ∂2A 2 ∂ A 2 2 | s | + | s | ds (A5a) ily representing the official policies or endorsements, ei- ≤ ∆2[A(s)] ∆3[A(s)] Z0 ther expressed or implied, of the ODNI, IARPA, or the 1 2 d∆ 2p 3 U.S. Government. The U.S. Government is authorized to = 2c p + 2 ∆ − [A(s)]ds (A5b) dA reproduce and distribute reprints for Governmental pur- Z0 1 poses notwithstanding any copyright annotation thereon. d∆ p 3 = 2c p + 2 ∆ − (u)du (A5c) du Z0 1 p 3 = 4c ∆ − (u)du , (A5d) Z0 57

where in line (A5c) we used the change of variables Let us now sum over all m and m0: p u = A(s), so that ds = du/∂sA = du/[c∆ (u)], and d 1 p 3 D (t) 4 ψ H ψ (B5) in line (A5d) we used B(p) 2c ∆ − d∆ = 0, mm0 m 1m m0 ≡ 0 dt ≤ |h | | i| since B(2) = 2c ln[∆(1)/∆(0)] = 0 and B(p = 2) = m,mX0 m,mX0 2c p 2 p 2 R 6 p 2 ∆ − (1) ∆ − (0) = 0, due to the boundary con- 4 H1m ψm ψm = 4 H1m ψm , − − ≤ k | 0 ikk| ik k | 0 ik ditions ∆(0) = ∆(1) = 1 [Eq. (18a)]. m,m m,m X0 X0 where we first used the fact that under the sum the Appendix B: A lower bound for the adiabatic Grover search two terms in the last line of Eq. (B4) are identical, and problem then we used the Cauchy-Schwartz inequality ( x y x y ). Now we note that: |h | i| ≤ k kk k Here we show that there is no schedule that gives a H ψ 2 = ψ H H ψ (B6) better scaling for the adiabatic Grover problem than the k 1m| m0 ik h m0 | 1m 1m| m0 i m m one discussed in Sec. III.A.2, resulting in a quadratic X X = A2(t) ψ m m ψ = A2(t) , quantum speedup. The argument is due to (Roland and h m0 | ih | m0 i m Cerf, 2002), which in turn is based on the general Hamil- X tonian quantum computation argument by (Farhi and so that Gutmann, 1998). 2 To show this, consider two different searches, one for H ψ = (~x ~y)2 (~x ~x)(~y ~y) = NA2(t) , k 1m| m0 ik · ≤ · · m and another for m0. We do not allow the schedule to m ! depend on m, i.e., the same schedule must apply to all X (B7) marked states. Let us denote the states for each at the where ~x = ( H ψ , H ψ ,..., H ψ ) k 11| m0 ik k 12| m0 ik k 1N | m0 ik end of the algorithm by ψm(tf ) and ψm (tf ) . In order and ~y = (1, 1,..., 1). Therefore, we have: | i | 0 i to be able to distinguish if the search gave m or m0, we d must require that ψm(tf ) and ψm (tf ) are sufficiently 0 Dmm0 (t) 4 H1m ψm0 different. Let us define| thei distance| (or infidelity)i dt ≤ k | ik m,mX0 m,mX0 D (t) 1 ψ (t) ψ (t) 2 , (B1) 4 √NA(t) = 4N√NA(t) . (B8) mm0 ≡ − |h m | m0 i| ≤ Xm0 [note that D (0) = D (t) = 0] and demand that: mm0 mm If we integrate both sides we have:

t D (t ) , m = m0 . (B2) f mm0 f √ √ ≥ 6 Dmm0 (tf ) 4N N A(t)dt 4N Ntf . ≤ 0 ≤ First, we have a lower bound on the sum: m,mX0 Z (B9) D (t ) = D (t ) mm0 f mm0 f Combining this with Eq. (B3), we have N(N 1) m,m0 m=m0 − ≤ X X6 4N√Ntf , and hence: = N(N 1) . (B3) ≥ − N 1 m=m0 tf − , (B10) X6 ≥ 4 √N Next, let us ﬁnd an upper bound on the sum. We so that the computation must last a minimum time of write the Hamiltonian (16) explicitly as H(t) = 11 [1 O(√N) if the schedule is to be agnostic about the identity A(t)] φ φ + H (t) where H (t) = A(t) m −m .40− 1m 1m of the marked state. Therefore, the solution using the Then:| ih | − | ih | locally optimized schedule is asymptotically optimal. d D (t) = 2 [ ψ (H H ) ψ ψ ψ ] dt mm0 = h m| 1m − 1m0 | m0 ih m0 | mi Appendix C: Technical details for the proof of the 2 ψ (H H ) ψ ψ ψ ≤ |h m| 1m − 1m0 | m0 ih m0 | mi| universality of AQC using the History State construction 2 ψ H ψ + 2 ψ H ψ . ≤ |h m| 1m| m0 i| |h m| 1m0 | m0 i| (B4) We add details to the proof sketch given in Sec. IV.B.

1. |γ(0)i is the ground state of Hinit 40 Optimality applies to arbitrary driving Hamiltonians. Hence the Let us first check that γ(0) is the ground state of lower bound holds more generally, and does in fact not require | i the initial Hamiltonian to be a projector onto the uniform super- Hinit with eigenvalue 0. Note that Hinit is a sum of pro- position as we have done here for simplicity. jectors, so it is positive semi-definite. Therefore if we find 58 a state with energy 0, then it is definitely a ground state. using Eq. (C3), is: The all-zero clock state is annihilated by Hc, Hinput, and Hc init, so we have Hinit γ(0) = 0, i.e., γ(0) is a ground − | i | i 0 0 0 ... state of Hinput. We shall show later that it is a unique 0 1 0 ground state.  1  H 0 (s) = (1 s)  .  (C4) S −  ..    2. |ηi is a ground state of Hfinal  1 0     1    Next we check that η is a ground state of Hfinal with 1 1  | i 2 2 0 0 ... 0 eigenvalue 0. First, we wish to show that Hfinal is positive 1 − 1 2 1 2 semi-definite. We already know that Hinput and Hc are  − 1 − 1  0 2 1 2 positive semi-definite, so we only need to show this to +s − − .  ......  be the case for the H`’s. This follows since it is easily  . . . .  1 2  1 1  † checked that H` = H` = 2 H` , so that X H` X =  2 1 2  1 1 2 h | | i  − 1 − 1  X H†H X = H X 0. Thus H(s) is positive  0  2 ` ` 2 `  − 2 2  semi-definite,h | | i sincek it is| aik sum≥ of positive semi-definite   terms. Let us now check that Hfinal annihilates η . First, because η only involves legal clock states, it| isi annihilated a. Bound for s < 1/3 | i by Hc. Next, Let us first bound the gap for s < 1/3. The Ger- 1 H η = H α(0) 0L = 0 . (C1) schgorin circle theorem states (Gershgorin, 1931): input input √ c | i L + 1| i ⊗ | i Let A be any matrix with entries aij. Consider the disk D (for 1 i n) in the complex plane defined as: L i ≤ ≤ Finally, note that the only non-zero terms in `=0 H` η | i Di = z z aii j=i aij . Then the eigenvalues are of the form: | − | ≤ 6 | | P of A aren contained in iDi and anyo connected component ` 1 L `+1 ∪P H α(` 1) 1 − 0 − = (C2a) of iDi contains as many eigenvalues of A as the number `| − i ⊗ | ic ∪ ` 1 L `+1 ` L ` of disks that form the component. α(` 1) 1 − 0 − α(`) 1 0 − | − i ⊗ | ic − | i ⊗ | ic ` L ` Consider the cases i = 1, i = L + 1, and i = 1,L + 1 H α(`) 1 0 − = (C2b) 6 `| i ⊗ | ic separately. Note that when s < 1/3: ` 1 L `+1 ` L ` α(` 1) 1 − 0 − + α(`) 1 0 − , − | − i ⊗ | ic | i ⊗ | ic 1 1 1 1 which cancel. Therefore, η has eigenvalue 0 and is a [H 0 (s)]11 = s < , a1j = s < , S 2 6 | | 2 6 | i j=1 ground state of Hfinal. X6 (C5a) 1 L 3. Gap bound in the space spanned by {|γ(`)i} [H 0 (s)]L+1,L+1 = 1 s > 5/6 `=0 S − 2 1 L aL+1,j = s < , (C5b) Let 0 be the space spanned by γ(`) `=0. Let us | | 3 S {| i} j=L+1 show that H(s) acting on any state in 0 keeps it in this 6 X S subspace: [H 0 (s)] = 1 , i = 1,L + 1 S ii 6 1 1 a = s < . (C5c) Hc γ(`) = 0 , (C3a) | 1j| 2 3 | i j=i H γ(`) = 0 , (C3b) X6 input| i 0 , ` = 0 1 Hc init γ(`) = (C3c) Therefore, we can identify a disk D1 centered at z 6 on − | i γ(`) , ` = 0 1 ≤ | i 6 the real line with radius . The closest possible disk ≤ 6 H` γ(`0) = δ` ,` ( γ(` 1) γ(`) ) to it which does not overlap it, is the disk D centered | i 0 | − i − | i L+1 + δ ( γ(`) γ(` 1) ) . (C3d) at z 5/6 with a radius 1/3. Therefore, since no disks `0,` | i − | − i ≥ ≤ intersect D1, it covers the smallest values on the real line, Since the initial state γ(0) , the dynamics gener- and it follows that the ground state for s < 1/3 is unique. | i ∈ S0 ated by H(s) keep the state in 0. Thus, it is sufficient Furthermore, we have learned that the minimum gap is a to bound the gap in this subspace.S In the basis given by constant of at least 1/6, since that is the closest distance L γ(`) `=0, we can write an (L+1) (L+1) matrix rep- between D1 and DL+1. (This also proves that γ(0) is resentation{| i} of the Hamiltonian in the× subspace, which, the unique ground state at s = 0.) | i S0 59 b. Bound for s ≥ 1/3 P to the energy gap between the ground state energy of H (given by 1 µ = λ) and the first excited state (given Now let s 1/3 and consider the matrix representa- by 1 δ) − ≥ tion of G(s) 11 H (s) in the same basis: − ≡ − S0 δ µ δ ∆(H 0 ) 1 1 ∆largest(P ) = 1 = − = S . (C10) 1 2 s 2 s − µ µ 1 λ − 1 1 − 2 s 0 2 s  . . .  where “largest” denotes the gap from the largest eigen- G(s) = ...... (C6)   value of P to the next largest eigenvalue. We wish to  0 1 s   2  bound the gap of P and hence of H(s). Let us de-  1 s 1 s   2 2  fine a non-empty set 1, 2,...,L + 1 satisfying   1 B ⊆ { ~} This matrix is Hermitian and has all non-negative real i Πi 2 , where Πi are the entries of Π. Then the conductance∈B ≤ of P , ϕ(P ) is defined as: entries for 0 < s 1. Note that increasing powers of P G(s) fill more of the≤ matrix, and G(s)L+1 has all posi- F ( ) tive entries for 0 < s 1. We can thus invoke Perron’s ϕ(P ) = min B , (C11) ≤ Π( ) theorem: B B Let G be a Hermitian matrix with real non-negative en- where tries. If there exists a finite k such that all entries of Gk are positive, then G’s largest eigenvalue is positive and all F ( ) = Π P , (C12a) B i ij other eigenvalues are strictly smaller in absolute value. i j / The eigenvector corresponding to the largest eigenvalue X∈B X∈B Π( ) = Π . (C12b) is unique, and all its entries are positive. B i i Therefore, by Perron’s theorem, G(s)’s largest eigen- X∈B value µ must be positive, and the associated unique eigen- The Conductance bound (Sinclair and Jerrum, 1989) vector ~α = (α1, . . . , αL+1) has αi > 0. Let us use this to then states that αj define a matrix P with entries Pij = Gij 0, such µαi ≥ 1 that ∆ (P ) ϕ(P )2 . (C13) largest ≥ 2 1 P = G α = 1 , (C7) ij µα ij j To use the result of the Conductance bound, we would j i j X X like to show that the ground state of H(s) [and hence the where we used that ~α is an eigenvector of G with eigen- eigenstate associated with the largest eigenvalue of G(s)] is monotone, i.e., that α α α 0. The value µ. Thus P is a stochastic matrix (it has only non- 1 ≥ 2 ≥ · · · ≥ L+1 ≥ negative entries and its rows sum to 1). Now note that if case s = 0 is obvious, so consider s > 0. First note that (α1v1, . . . αL+1vL+1) is a left eigenvector of P with eigen- G(s) applied to a monotone vector generates a monotone value ν/µ, then (v1, . . . vL+1) is an eigenvector of G with vector, i.e., G(s) preserves monotonicity. To see, this eigenvalue ν: consider G(s)~v = ~w with ~v monotone. The components of ~w are given by: ν α α α v = α v P = v j G = j G v µ j j i i ij i µ ij µ ji i i i i 1 1 X X X w1 = 1 s v1 + sv2 (C14a) − 2 2 = νvj = Gjivi . (C8) ⇒ i 1 1 X wk = svk 1 + svk+1 , 2 k L (C14b) 2 − 2 ≤ ≤ It is straightforward to check that the reverse also holds: 1 1 if (v , . . . v ) is an eigenvector of G with eigenvalue ν, wL+1 = svL + svL+1 (C14c) 1 L+1 2 2 then (α1v1, . . . αL+1vL+1) is a left eigenvector of P with eigenvalue ν/µ. By taking ~v = ~α, which corresponds to Therefore we have: the largest eigenvalue (ν = µ) of G, it then follows that 1 2 2 α~2 = α , . . . α is a left eigenvector of P with the w1 w2 = (1 s)v1 + s (v2 v3) (C15a) 1 L+1 − − 2 − ~2 maximal eigenvalue 1. If we normalize α , i.e., define 1 wk wk+1 = s (v1 v2 + v3 v4) , 2 k L 1 1 − 2 − − ≤ ≤ − Π~ = α2, . . . α2 ,Z = α2 , (C9) (C15b) Z 1 L+1 i i 1 X wL wL+1 = s (vL 1 vL) , (C15c) − 2 − − then Π~ is the limiting distribution of P , i.e., P Π~ = Π.~ We can then relate the energy gap between the highest which clearly are all 0 by the monotonicity of ~v and and second highest eigenvalue (let us denote it by δ/µ) of s 1. Therefore ~w is≥ also monotone. ≤ 60

Recall that G(s) is Hermitian, so it has an orthonormal size of ¯ is L (recall that 1 ), it follows that Π( ¯) L+1 B ∈ B B ≤ set of eigenvectors v with eigenvalues µ , where LΠk+1, so that Πk+1 1/(2L), and hence: i i=1 i ≥ v = ~α and µ = µ{|.41i}Because these eigenvectors form | 1i 1 F ( = 1, others, k ) 1 a basis we can always find a set of coefficients c L+1 B { } . (C21) { i}i=1 Π( ) ≥ 6L such that: B Next consider the case where 1 / . Now let k be the c v = (1,..., 1) = ~1 . (C16) ∈ B i| ii smallest index such that k / but k + 1 . Then: i ∈ B ∈ B X F ( ) = Π P + Π P (C22a) Then: B i ij k+1 k+1,k i ,i=k+1 j / k k ∈BX6 X∈B 1 µi Πk+1 G(s) ci vi = ci vi (C17a) Π P . (C22b) µ | i µ | i k+1 k+1,k 1 i i 1 ≥ ≥ 6 X X 1 k µ k In this case, since the maximum size of is L but it = G(s) ~1T = i ~1T . (C17b) B excludes the index 1, we have Π( ) LΠk+1, so that ⇒ µ1 µ1 i F ( ) Π( )/(6L). Therefore, weB again≤ find the condi- X B ≥ B Using µi < µ1 by Perron’s theorem, we have from tion (C21). Thus, by the conductance bound [Eq. (C13)]: Eq. (C17a| |) that: ∆(H ) 1 1 2 ∆(P ) = S0 . (C23) 1 k 1 λ ≥ 2 6L lim G(s) ci vi = c1 v1 , (C18) − k µ1 | i | i →∞ i Now since λ is the ground state of H 0 , for any state in X S v 0, we must have v H v λ. In particular: k | i ∈ S h | S0 | i ≥ Since the quantities 1 G(s) (for k L + 1), µ1 ≥ 1 γ(0) H 0 γ(0) = s λ , (C24) ci vi = ~1, and v1 = ~α are all positive, it follows h | S | i 2 ≥ i | i | i that also c > 0. Since ~1 is monotone and G(s) preserves P 1 i.e., λ 1/2. This finally yields Eq. (79). monotonicity, we have finally established that v = ~α is ≤ | 1i monotone. This then implies that Π~ [Eq. (C9)] is monotone. 4. Gap bound in the entire Hilbert space We are ready to calculate the conductance ϕ(P ). First, consider the case where the first index (of Π)~ is in the set Let us now go a step further and show how the global , i.e., 1 . Let k be the smallest index such that gap (i.e., not restricted to the 0 subspace) scales with S Bk but∈k B+ 1 / . (Note that from the form of P , L. Let denote the subspace spanned by all legal clock S only∈ BP , P , P∈ B are nonzero.) Then we have states. The dimensions of this subspace will be dim( ) = 11 j,j+1 L+1,L+1 S for F ( ): (L + 1)2n, since we have L + 1 legal clock states and 2n B computational states. H(s) acting on any state in does S F ( ) = ΠiPij + ΠkPk,k+1 ΠkPk,k+1 not generate any illegal clock states, so for any v B ≥ | i ∈ S i ,i=k j / we have H(s) v . Similarly, for any state in the ∈BX6 X∈B | i ∈ S Π Π Π orthogonal subspace ⊥, i.e., the subspace of illegal clock k+1 k k+1 S = Πk [G(s)]k,k+1 = [G(s)]k,k+1 states, for any state v⊥ ⊥, we have H(s) v⊥ ⊥. µ√Πk 1 λ p p − Therefore, the eigenstates| i ∈ of S H(s) below either| toi ∈ S or Πk+1 S [G(s)] , (C19) to ⊥, and H(s) is block diagonal with blocks H (s) and ≥ 1 λ k,k+1 S S − H ⊥ (s) that can be diagonalized independently. where the last inequality follows from the monotonicity S Let us first restrict to H ⊥ (s). Hc penalizes all illegal of Π.~ Because 0 < 1 λ 1, and [G(s)] = 1 s 1/6 clock states by at least oneS unit of energy and acts as the − ≤ k,k+1 2 ≥ for s 1/3, it follows that: identity on the computational qubits. Therefore, it shifts ≥ the entire spectrum of by at least one unit of energy. Π ⊥ F ( = 1, others, k ) k+1 . (C20) Since the remaining termsS are positive semi-definite, they 6 B { } ≥ cannot lower the energy. Therefore, regardless of the Since by definition Π( ) 1/2, then Π( ¯) 1/2 where form of the ground state in the subspace, it has an energy ¯ is the complement ofB ≤, but since theB largest≥ possible of at least one unit. B B Let us now restrict to H (s) and define: S ` L ` γj(`) = αj(`) 1 0 − c , (C25) 41 Note that we abuse notation and mix kets with standard vec- | i | i ⊗ | i tors here, and also do not use transpose notation to distinguish where αj(`) is the state of the circuit at time ` had the column from row vectors. input state| beeni given by the binary representation of j 61

L (e.g., if j = 4, the input configuration of the circuit would H2 can be written as a linear combination of γj(`) `=1, n 3 {| i} have been 0 − 130201 ). Note that γ0(`) = γ(`) . whereas the (unique) ground state of H1 can be written | i | i L | i L Let j denote the space spanned by γj(`) `=0. Since as a monotone vector in γj(`) `=0. Therefore: H (sS) cannot mix states with different{| j subindicesi} (it {| i} canS only propagate forward or backward in `), H (s) L L S cos θ = max α` γj(`) c` γj(`0) is block diagonal in the subspaces j. Therefore, we c 0 `0 h | | i S { } `=0 `0=1 only need to find the minimum ground state energy of X X L H j>0 (s) to determine the minimum gap from H 0 (s). S S = max α`c` γj(`) γj(`) (C29) In order to determine the ground state energy of H , c j `0 h | i S { } `=1 we notice that we can write: X L L 1 max α` c` 1 , H j (s) = H0(s) + H j ,input , j > 0 (C26) ≤ c | | ≤ L + 1 ≤ − 2L S S { `0 } `=1 r X where H0(s) has exactly the same spectral properties as where we have used that ~α is monotone so that c` = H except in the subspace. The reason for this de- 0 j α L+1 maximizes the sum. Therefore, the global compositionS is becauseS H is zero in and hence is ` L input 0 3 absent from H (s). Note that: S gapq can be bounded from below by Ω(1/L ), which is S0 Eq. (80).

k γj(0) , ` = 0 H ,input γj(`) = | i (C27) Sj | i 0 , ` > 0 Appendix D: Proof of the Ampliﬁcation Lemma (Claim1)

(recall that Hinput projects onto the 0 clock state, which is why for ` > 0 we have zero.) Therefore, in the basis To prove Claim1, define a new verifier V ∗(η, X) γ (`) L , we can write the matrix representation of which amounts to repeating V (η, X) K times, where {| j i}`=0 K = poly( η ) to keep the verifier efficient, and taking H j ,input as: | | S a majority vote on the output, i.e., Pr(V ∗(η, X) = 1) = K k 0 Pr Vi > K/2 , where Vi 0, 1 is the random i=1 ∈ { } 0 number associated with the i-th run of V (η, X).   P H j ,input = . , (C28) S .. Now recall the multiplicative Chernoff bound:    0 0  K   β2Kp/2   Pr Y (1 β) Kp e− , 0 < β < 1 i ≤ − ≤ where k 1 denotes the number of 1’s in the binary i=1 ! ≥ X representation of j > 0. In particular, note that it is K β2Kp/(2+β) diagonal in this basis. We now use the Geometrical Pr Y (1 + β) Kp e− , 0 < β , i ≥ ≤ Lemma (Aharonov and Naveh, 2002; Kitaev et al., 2000): i=1 ! X (D1)

Lemma 4 (Geometrical Lemma). Let H1 and H2 be two for p = E(Y ) where Y 0, 1 is a random variable. ∈ { } Hamiltonians with ground state energies g1 and g2 respec- Consider first take the case where Q(η) = 1. In that tively. Both Hamiltonians have a ground state energy gap case, p 2/3. If we now pick β = 1 1/(2p) (i.e., to the first excited state that is larger than Λ. Let the an- 1/4 β ≥ 1/2) in the Chernoff bound, then:− gle between the two ground subspaces be θ.42. Then the ≤ ≤ K K ground state energy (g0) of H0 = H1 + H2 is at least K K Pr Vi > = 1 Pr Vi (D2) g1 + g2 + Λ(1 cos θ). 2 − ≤ 2 − i=1 ! i=1 ! X X (p 1/2)2K (2/3 1/2)2K Let H = H and H = H . We saw that the − − 1 0 2 1,input 2p − 4/3 S 2 1 e− 1 e . ground state gap of H1 is Ω(1/L ) and that of H2 is ≥ − ≥ − 1, so we can take Λ = Ω(1/L2). The ground state en- For the case where Q(η) = 0, p 1/3, take β = 1/(2p) ≤ − ergy of H2 is g2 = 0. Therefore, using the Geometrical 1 > 0, so that: Lemma, we have g0 g1 Λ(1 cos θ). It remains to K K bound the angle between− ≥ the two− ground spaces. From K K Pr V > Pr V (D3) Eq. (C28), it is clear that the (degenerate) ground state of i 2 ≤ i ≥ 2 i=1 ! i=1 ! X X (p 1/2)2K (1/3 1/2)2K − − e− p(p+1/2) e− (1/3+1/2)/3 . ≤ ≤ g g 42 η η The angle θ is defined via cos θ = maxv1,v2 |hv1|v2i|, where |vii This shows that MA(2/3, 1/3) = MA(1 e−| | , e−| | ), belongs to space i. since K = poly( η ). − | | 62

g To show that MA(c, c 1/ η ) MA(2/3,1/3), it is where γ is the gap to the first excited state of H0. We sufficient to show that when− Q|(η| ) =⊆ 0 it is exponentially first show how to construct this effective Hamiltonian in unlikely that Merlin is able to fool Arthur that Q(η) = terms of other, more convenient operators. 1. Therefore consider the probability of fooling Arthur, Let P be the projection onto , and define: 0 E0 i.e., Pr (V ∗(η, X) = 1) > c when Q(η) = 0. Take p = g Pr (V (η, X) = 1) = c 1/ η . Then: αj = P0 ψj , j = 1, . . . , d . (E3) − | | | i | i K For λ sufficiently small [this will amount to satisfying Pr (Arthur fooled) = Pr Vi Kc (D4) d ≥ Eq. (E2)], the states αj are linearly indepen- i=1 ! {| i}j=1 X d K dent since the states ψ are only slightly per- 1 {| ji}j=1 = Pr Vi p + , turbed from the eigenstates of H0. Note that the states K ≥ d i=1 ! α are not necessarily orthogonal or normalized. X {| ji}j=1 g There exists an operator such that: where we take = 1/ η . Recall the additive Chernoff U bound: | | αj = ψj , j = 1, . . . , d (E4a) K U| i | i 1 KD(p+ p) φ = 0 , φ ⊥ . (E4b) Pr Y p + e− k , (D5) 0 K i ≥ ≤ U| i ∀| i ∈ E i=1 ! X This means that: where P α = P ψ = α x 1 x 0U| ji 0| ji | ji D(x y) = x ln + (1 x) ln − (D6) = P 2 α = P α = P α k y − 1 y ⇒ 0 U| ji 0U| ji 0| ji − = P = P . (E5) is the Kullback-Leibler divergence. Expanding D(p + ⇒ 0U 0 2 3 2 ε p) = + O( ), we see that if K = − − , where Let ˜ be the operator satisfying: k 2p(1 p) U 0 < ε −1, then we can exponentially suppress the ˜ ψ = α , j = 1, . . . , d (E6a) probability that Arthur is fooled by Merlin while keeping U| ji | ji K = poly( η ). ˜ φ = 0 , φ ⊥ . (E6b) | | U| i ∀| i ∈ E Note that ˜ is not the inverse of because is not Appendix E: Perturbative Gadgets U U U invertible on the entire Hilbert space. Also, ˜ is not P0 because of Eq. (E6b) (it annihilates all statesU outside of In this section we review the subject of perturbative ). Note that: gadgets, which have played an important role in the re- E duction of the locality of interactions in the proofs of P0 ˜ ψj = ψj . (E7) QMA completeness and the universality of AQC. These U U| i | i tools are generally useful. Our discussion is based pri- Now define: marily on (Jordan and Farhi, 2008) [see also (Bravyi et al., 2008a)]. To set up the appropriate tools we first = λP0V . (E8) briefly review degenerate perturbation theory. A U Note that the states α d are right eigenvectors of {| ii}i=1 A with eigenvalues E1,...,Ed respectively: 1. Degenerate Perturbation Theory àla (Bloch, 1958)

αj = λP0V ψj = P0 (H0 + λV ) ψj = P0Ej ψj Consider H = H0 + λV where H0 has a d-dimensional A| i | i | i | i = Ej αj , (E9) degenerate ground subspace 0 with energy 0. Let | i ψ ,... ψ be the lowest d energyE eigenstates of H with 1 d where we used that P0H0 = 0 because the eigenvalue of |energiesi |E i,...,E , and let their span define the sub- 1 d the ground subspace of H0 is zero. The effective Hamil- space . The goal is to define a perturbative expansion E tonian associated with H can now be constructed using (in λ) for the effective Hamiltonian Heff of H, defined as: , ˜, : U U A d ˜ H (H, d) = E ψ ψ . (E1) Heff (H, d) = . (E10) eff j| jih j| UAU j=1 X To see this note that: We will show that this expansion converges provided λ satisfies ˜ φ = 0 , φ ⊥ (E11a) UAU| i ∀| i ∈ E ˜ λV < γ/4 , (E2) ψj = αj = Ej αj = Ej ψj , (E11b) k k UAU| i UA| i U| i | i 63

2m 1 2m 1 which is identical to the action of Heff on a complete set clear that m− 2 − . Therefore, we can upper of vectors. The strategy is now to find a perturbative bound the sum with≤ this value: expansion for (we will not need the explicit expansion U m of ˜, so we do not provide it here), construct using ∞ 2m 1 λV 1 + 2 − k k . (E17) U d d A kUk ≤ γm Eq. (E8), find αj j=1 and Ej j=1 as, respectively, m=1 the right eigenvectors{| i} and eigenvalues{ } of , and apply X to α to get a perturbative expansion forA ψ . U This series converges if the condition for λ in Eq. (E2) is | ji | ji It can be shown that the desired perturbative expan- satisfied. sion of and is given by: U A

∞ 2. Perturbative Gadgets = P + (E12a) U 0 Um m=1 X For a k-local target Hamiltonian HT, the goal is to ∞ ∞ construct a 2-local “gadget” Hamiltonian HG, whose low = P V = , (E12b) A 0 Um Am energy spectrum (captured by an effective Hamiltonian m=1 m=1 T X X Heff ) approximates the spectrum of H . In order to do where so, we shall use the expression in Eq. (E10) for the effective Hamiltonian in terms of the operators and and = (S λV )(S λV ) ... (S λV )P , use their perturbative expansion from the previousU A sub- Um `1 `2 `m 0 `1 1,`2 0,...,`m 0 section. We shall show that for our gadget Hamiltonian, ≥ ≥ ≥ `1+ X+`m=m ` + +` ··· p, 1 p m 1 the perturbative expansion of the effective Hamiltonian 1 ··· p≥ ≤ ≤ − (E13a) matches that of the target Hamiltonian. 1 The perturbative gadget we review here uses a strongly ` (11 P0) , ` > 0 S = ( H0) − . (E13b) bound set of ancillas, coupled to the target qubits via ` − P , ` = 0 − 0 weaker interactions, where the latter are treated as a perturbation. HT is then generated in low order pertur- The series in Eq. (E12) converges for λV < γ/4. To k k bation theory of the combined system consisting of both see this note that: ancilla and target qubits. Such gadgets first appeared in

∞ ∞ the proof of QMA-completeness of the 2-local Hamilto- = + + nian problem via a reduction from 3-local Hamiltonian, kUk kU0 Umk ≤ kU0k kUmk m=1 m=1 where they were used to construct effective 3-body inter- X X ∞ actions from 2-body ones (Kempe et al., 2006). m 0 1 + λ S` VS` ...S` VP0 (E14) ≤ k 1 2 m k Let Hs denote a k-local term. For the i-th qubit in m=1 3 X X Hs, we associate an arbitrary direction in R denoted by ∞ nˆ . A general k-local target Hamiltonian acting on n 1 + λm 0 S ... S V , s,i ≤ k `1 k k `m kk k qubits can then be expressed as: m=1 X X r where the sum 0 involves summing all the different ways T H = csHs , (E18) to add up to m while satisfying convexity, i.e., `1 + `2 + s=1 . . . ` p. BecauseP of the form of S [Eq. (E13b)], we X p ≥ ` have: with Hs = σs,1σs,2 . . . σs,k where σs,j =n ˆs,j ~σs,j. The goal is to simulate HT using only 2-local interactions.· ` 1 1 Toward this end, introduce k ancilla qubits for each Hs, S = = , (E15) k `k (0) γ` for a total of rk ancilla qubits. Define E1 ! r r (0) G A A where E1 is the energy of the first excited state of H0 H = H + λV = Hs + λ Vs (E19a) s=1 s=1 (corresponds to the state that minimizes H0Q0 to calcu- X X late the operator norm). Therefore, we have: k 1 HA = (11 Z Z ) (E19b) s 2 − s,i s,j ∞ V m i 2. From so the ancilla qubits must be initialized to be in the state the perturbationA expansion [Eq. (E12b)] we have: s r=1 + s. 2 | i G 2 = λP0VP0 + λ P0VS1VP0 , (E24) We wish to show that the low energy spectrum of H+ A≤ N T approximates the spectrum of H . Our task is to cal- but P0VP0 = 0 since V + is orthogonal to + . On the G n | i | i culate Heff (H+ , 2 ) [in the notation of Eq. (E1)] pertur- other hand, VP0 takes the system to a state with energy A batively to k-th order in λ. λV will perturb the ground k 1 for H , so S1VP0 = VP0/(k 1). Therefore: subspace of HA in two ways: − − − 2 λ 2 2 = P0V P0 . (E25) 1. It shifts the energy of the entire subspace; A≤ −k 1 − 2. It splits the degeneracy of the ground subspace be- Now note that: ginning at k-th order in perturbation theory. It is V 2 = (σ X )2 + (σ X )(σ X ) . (E26) this splitting that will allow us to mimic the spec- i ⊗ i i ⊗ i j ⊗ j i i=j trum of HT. X X6 The cross-terms are annihilated by P P since they take We analyze the shift and splitting separately. To do this, 0 0 the state out of . The diagonal term· is proportional to define: 0 the identity on theE ancilla qubits, so we have: ˜ Heff (H, d, ∆) Heff (H, d) ∆Π , (E21) λ2 ≡ − 2 = ΩP0 , (E27) A≤ −k 1 where Π is the projection onto the space spanned by − E d . Note that the eigenstates of H˜ and H where Ω is an operator that depends on the particular {| ji}j=1 eff eff are identical, and the energy gaps between energy levels orientation of the σj’s. This argument extends to order are identical too. k 1 so that: T − Let us start with the case where r = 1, i.e., H = m A k k 1 k 1 = λ ΩmP0 . (E28) σ1σ2 . . . σk, so that H = (1 ZiZj), A≤ − i=1 j=i+1 2 m even k − X and V = j=1 σj Xj. WeP firstP wish to construct G ⊗ A A At order k, something new happens. There are now cross- [Eq. (E8)] for H+ . Note that H has a ground state P term that involve all Xi’s once, i.e., of zero energy (corresponds to all qubits with Zi = 1 or all qubits with Z = 1), and the first excited state has λkP (σ X ) S (σ X ) S ...S (σ X ) P . i − 0 1 ⊗ 1 1 2 ⊗ 2 1 1 k ⊗ k 0 energy γ = k 1 (let Z1 = 1, all the rest are +1). Furthermore: − − The S1 operator measures the successive change in energy of the system to give an overall constant of: k k 1 1 1 V = σj Xj σj Xj = k . (E22) ... ( ) k k k ⊗ k ≤ k ⊗ k j=1 j=1 −k 1 −2(k 2) −(k 1)1 X X − − − k 1 k 1 − 1 ( 1) − Therefore, by Eq. (E2), the perturbative expansion will = = − . (E29) k 1 −j(k j) (k 1)!2 converge if λ < − . Because of the form of j=1 4k A Y − − 65

Therefore, the cross-terms (of which there are k!, since previous result is recovered, namely, Eq. (E33) continues the operators can be multiplied in any order) then take to hold with HT replaced by the sum over r terms as in the form: Eq. (E18), where again f(λ) is some polynomial in λ of order k with coefficients that depend on csHs, and where ( λ)kk! − P (σ . . . σ X) P . (E30) P+ is the projector onto s + s. − (k 1)!2 0 1 k ⊗ 0 ⊗ | i − Note that the convergence condition (E34) requires the Thus: interaction term V to be stronger than the effective interaction it generates, which scales as λk [as can be seen k k( λ) T from Eq. (E33)]. This may pose implementation diffi- k = f(λ)P0 − P0 H X P0 , (E31) A≤ − (k 1)! ⊗ culties, since a practical device is likely to have only a − limited range of interaction strengths. Weaker gadgets where f(λ) is some k-th order polynomial in λ, with co- can be implemented that circumvent this problem, al- T efficients that depend on H . Using the form of Heff in beit at the cost of a larger overhead of ancillary qubits Eq. (E10), we have: (Cao and Nagaj, 2015). The idea is to replace strong G n interactions by repetition of interactions with “classical” Heff (H , 2 ) = f(λ) P0 ˜ (E32) − U U ancillas. Additional gadgets simplifications and resource k k( λ) T k+1 reductions were proposed in (Cao et al., 2015). − P H X P + O(λ ) ˜ . −U (k 1)! 0 ⊗ 0 U − Recall that P ˜ ψ = ψ [Eq. E7] so P ˜ acts as U 0U| ji | ji U 0U the identity in so that the first term in Eq. (E32) can REFERENCES be dropped. Furthermore,E we can replace and ˜ in the second term by their λ0 counterpart sinceU we areU only Aaronson, S. (2010), arXiv:1009.5104. keeping terms to order k and the term in the parenthesis Aaronson, S. (2015), Nat Phys 11 (4), 291. is already of order k. Therefore: Adachi, S. H., and M. P. Henderson (2015), arXiv:1510.06356 . k( λ)k Aharonov, D., W. van Dam, J. Kempe, Z. Landau, S. Lloyd, H˜ (HG, 2n, f(λ)) = − P HT X P and O. Regev (2007), SIAM J. Comput. 37 (1), 166. eff + −(k 1)! 0 ⊗ 0 − Aharonov, D., D. Gottesman, S. Irani, and J. Kempe (2009), + O(λk+1) (E33) Communications in Mathematical Physics 287 (1), 41. Aharonov, D., and T. Naveh (2002), arXiv:quant-ph/0210077 k k( λ) T k+1 . = − H P+ + O(λ ) . −(k 1)! ⊗ Aharonov, D., and A. Ta-Shma (2003), in Proceedings of the − Thirty-fifth Annual ACM Symposium on Theory of Com- This shows that the target Hamiltonian HT appears as puting, STOC ’03 (ACM, New York, NY, USA) pp. 20–29. the leading order term in the effective Hamiltonian that Albash, T., and D. Lidar (2017), Rev. Mod. Phys. , to be describes the 2n-Hilbert space of the n target qubits, al- published. Altshuler, B., H. Krovi, and J. Roland (2010), Proceedings beit with a diminished magnitude of order λk/(k 1)!. − of the National Academy of Sciences 107 (28), 12446. Let us now consider the general r case, i.e., the Hamil- Amara, P., D. Hsu, and J. E. Straub (1993), The Journal tonian in Eq. (E18). We note that just as in the r = 1 of Physical Chemistry, The Journal of Physical Chemistry case, HA again has an energy gap of k 1. General- 97 (25), 6715. izing from Eq. (E22), the perturbative expansion− then Ambainis, A., and O. Regev (2004), arXiv:quant-ph/0411152 converges for: . Amico, L., R. Fazio, A. Osterloh, and V. Vedral (2008), Re- k 1 views of Modern Physics 80 (2), 517. λ < − . (E34) Amin, M. H., E. Andriyash, J. Rolfe, B. Kulchytskyy, and 4 V k k R. Melko (2016), arXiv:1601.02036. A r Amin, M. H. S. (2009), Phys. Rev. Lett. 102, 220401. In the sector where Xs = +1, H has the state s=1 + s Amin, M. H. S., and V. Choi (2009), Physical Review A A ⊗ | i as a ground state. Since H acts as the identity on the 80 (6), 062326. computational qubits, the ground state is 2n-fold degen- Andriyash, E., and M. H. Amin (2017), ArXiv e-prints erate. arXiv:1703.09277 [quant-ph]. In the perturbation expansion for , products of V Apolloni, B., C. Carvalho, and D. de Falco (1989), Stochastic A Processes and their Applications 33 (2), 233. again appear. Each Vs acts on a different ancilla register. Therefore, at order k, cross-terms of different V ’s cannot Apolloni, B., N. Cesa-Bianchi, and D. de Falco (1988), in s Proceedings of the Ascona/Locarno Conference, p. 97. flip all k ancilla qubits in a register, so they are annihi- Arecchi, F. T., E. Courtens, R. Gilmore, and H. Thomas lated by P P . The only cross-terms that contribute 0 · 0 (1972), Phys. Rev. A 6, 2211. are k products of a given s where each ancilla qubit ap- Aspuru-Guzik, A., A. D. Dutoi, P. J. Love, and M. Head- pears once. Therefore, the natural generalization of the Gordon (2005), Science 309 (5741), 1704. 66

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