Adiabatic Quantum Computing
Abstract
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 closedsystem 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. To be submitted to Reviews of Modern Physics.
Contents
 I Introduction
 II Adiabatic Theorem
 III Algorithms

IV Universality of AQC
 IV.1 The circuit model can efficiently simulate AQC
 IV.2 AQC can efficiently simulate the circuit model: history state proof
 IV.3 Fermionic ground state quantum computation
 IV.4 Spacetime CircuittoHamiltonian Construction
 IV.5 Universal AQC in 1D with state particles
 IV.6 Adiabatic gap amplification
 V Hamiltonian quantum complexity theory and universal AQC

VI Stoquastic Adiabatic Quantum Computation
 VI.1 Why it might be easy to simulate stoquastic Hamiltonians
 VI.2 Why it might be hard to simulate stoquastic Hamiltonians
 VI.3 QMAcomplete problems and universal AQC using stoquastic Hamiltonians

VI.4 Examples of slowdown by StoqAQC
 VI.4.1 Perturbed Hamming Weight Problems with Exponentially Small Overlaps
 VI.4.2 2SAT on a Ring
 VI.4.3 Weighted 2SAT on a chain with periodicity
 VI.4.4 Topological slowdown in a dimer model or local Ising ladder
 VI.4.5 Ferromagnetic Meanfield Models
 VI.4.6 3Regular 3XORSAT
 VI.4.7 SherringtonKirkpatrick and TwoPattern Gaussian Hopfield Models
 VI.5 StoqAQC algorithms with speedup over simulated annealing
 VI.6 StoqAQC algorithms with undetermined speedup
 VI.7 Speedup mechanisms?

VII Circumventing slowdown mechanisms for AQC
 VII.1 Avoiding poor choices for the initial and final Hamiltonians
 VII.2 Quantum Adiabatic Brachistochrone
 VII.3 Modifying the initial Hamiltonian
 VII.4 Modifying the final Hamiltonian
 VII.5 Adding a catalyst Hamiltonian
 VII.6 Addition of nonstoquastic terms
 VII.7 Avoiding perturbative crossings
 VII.8 Evolving nonadiabatically
 VIII Outlook and Challenges
 A Technical details for the proof of the universality of AQC using the History State construction
 B Proof of the Amplification Lemma (Claim 1)
 C Perturbative Gadgets
I Introduction
Quantum computation (QC) originated with Benioff’s proposals for quantum Turing machines Benioff (1980, 1982) and Feynman’s ideas for circumventing the difficulty of simulating quantum mechanics by classical computers Feynman (1982). This led to Deutsch’s proposal for universal QC in terms of what has become the “standard” model: the circuit, or gate model of QC Deutsch (1989). Adiabatic quantum computation (AQC) is based on an idea that is quite distinct from the circuit model. Whereas in the latter a computation evolves in the entire Hilbert space and is encoded into a series of unitary quantum logic gates, in AQC the computation proceeds from an initial Hamiltonian whose ground state is easy to prepare, to a final Hamiltonian whose ground state encodes the solution to the computational problem. The adiabatic theorem guarantees that the system will track the instantaneous ground state provided the Hamiltonian varies sufficiently slowly. It turns out that this approach to QC has deep connections to condensed matter physics, computational complexity theory, and heuristic algorithms.
In its first incarnation, the idea of encoding the solution to a computational problem in the ground state of a quantum Hamiltonian appeared as early as 1988, in the context of solving classical combinatorial optimization problems, where it was called quantum stochastic optimization Apolloni et al. (1989).^{1}^{1}1Even though Apolloni et al. (1989) was published in 1989, it was submitted in 1988, before Apolloni et al. (1988), which referenced it. It was renamed quantum annealing (QA) in Apolloni et al. (1988) and reinvented several times Somorjai (1991); Amara et al. (1993); Finnila et al. (1994); Kadowaki and Nishimori (1998).^{2}^{2}2It was called “quasiquantal method” in Somorjai (1991), “imaginarytime algorithm” in Amara et al. (1993), and quantum annealing in Finnila et al. (1994); Kadowaki and Nishimori (1998). The latter term has become widely accepted. These early papers emphasized that QA was to be understood as an algorithm that exploits simulated quantum (rather than thermal) fluctuations and tunneling, thus providing a quantuminspired version of simulated annealing (SA) Kirkpatrick et al. (1983).
A very different approach was taken via an experimental implementation of QA in a disordered quantum ferromagnet Brooke et al. (1999, 2001). This provided the impetus to reconsider QA from the perspective of quantum computing, i.e., to consider a dedicated device that solves optimization problems by exploiting quantum evolution. Thus was born the idea of the quantum adiabatic algorithm (QAA) Farhi et al. (2000, 2001) [also referred to as adiabatic quantum optimization (AQO) Smelyanskiy et al. (2001); Reichardt (2004)], wherein a physical quantum computer solves a combinatorial optimization problem by evolving adiabatically in its ground state. The term adiabatic quantum computation we shall use here was introduced in van Dam et al. (2001), though the context was still optimization.^{3}^{3}3The 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.
Adiabatic quantum algorithms for optimization problems typically use “stoquastic” Hamiltonians, characterized by having only nonpositive offdiagonal elements in the computational basis. Adiabatic quantum computation with nonstoquastic Hamiltonians is as powerful as the circuit model of quantum computation Aharonov et al. (2007). In other words, nonstoquastic AQC and all other models for universal quantum computation can simulate one another with at most polynomial resource overhead. For this reason the contemporary use of the term AQC typically refers to the general, nonstoquastic setting, thus extending beyond optimization to any problem in the complexity class BQP (bounded error quantum polynomial time). When discussing the case of stoquastic Hamiltonians we will use the term “stoquastic AQC” (StoqAQC).
For most of this review we essentially adopt the definition of AQC from Aharonov et al. (2007), as this definition allows for the proof of the equivalence with the circuit model, and is thus used to establish the universality of AQC. Interestingly, this proof builds on one of the first QC ideas due to Feynman, which was later shown to allow a general purpose quantum computation to be embedded in the ground state of a quantum system Feynman (1985); Kitaev et al. (2000). A related ground state embedding approach was independently pursued in Mizel et al. (2001, 2002), around the same time as the original development of the QAA. To define AQC, we first need the concept of a local Hamiltonian, which is a Hermitian matrix acting on the space of state particles that can be written as where each acts nontrivially on at most particles, i.e., where is a Hamiltonian on at most particles, and denotes the identity operator.
Definition 1 (Adiabatic Quantum Computation).
A local adiabatic quantum computation is specified by two local Hamiltonians, and , acting on state particles, . The ground state of is unique and is a product state. The output is a state that is close in norm to the ground state of . Let (the “schedule”) and let be the smallest time such that the final state of an adiabatic evolution generated by for time is close in norm to the ground state of .
Several comments are in order. (1) A uniqueness requirement was imposed on the ground state of in Aharonov et al. (2007), but this is not necessary. E.g., in the setting where represents a classical optimization problem, multiple final ground states do not pose a problem as any of the final states represents a solution to the optimization problem. (2) In Aharonov et al. (2007) the runtime of the adiabatic algorithm was defined to be ,^{4}^{4}4Unless otherwise stated, we shall always use to denote the operator norm for operators:
The runtime of an adiabatic algorithm scales at worst as , where is the minimum eigenvalue gap between the ground state and the first excited state of the Hamiltonian of the adiabatic algorithm Jansen et al. (2007). If the Hamiltonian is varied sufficiently smoothly, one can improve this to up to a polylogarithmic factor in Elgart and Hagedorn (2012). While these are useful sufficient conditions, they involve bounding the minimum eigenvalue gap of a complicated manybody Hamiltonian, a notoriously difficult problem. This is one reason that AQC has generated so much interest among physicists: it has a rich connection to well studied problems in condensed matter physics. For example, because of the dependence of the runtime on the gap, the performance of quantum adiabatic algorithms is strongly influenced by the type of quantum phase transition the same system would undergo in the thermodynamic limit Latorre and Orus (2004).
Nevertheless, a number of examples are known where the gap analysis can be carried out. For example, adiabatic quantum computers can perform a process analogous to Grover search Grover (1997), and thus provide a quadratic speedup over the best possible classical algorithm for the Grover search problem Roland and Cerf (2002). Other examples are known where the gap analysis can be used to demonstrate that AQC provides a speedup over classical computation, including adiabatic versions of some of the keystone algorithms of the circuit model. However, much more common is the scenario where either the gap analysis reveals no speedup over classical computation, or where a clear answer to the speedup question is unavailable. In fact, least is known about adiabatic quantum speedups in the original setting of solving classical combinatorial optimization problems. This remains an area of very active research, partly due to the original (still unmaterialized) hope that the QAA would deliver quantum speedups for NPcomplete problems Farhi et al. (2001), and partly due the availability of commercial quantum annealing devices manufactured by DWave Systems Inc. Johnson et al. (2011), designed to solve optimization problems using stoquastic Hamiltonians.
The goal of this article is to review the field of AQC from its inception, with a focus on the closed system case. That is, we omit the fascinating topic of AQC in open systems coupled to an environment. This includes all experimental work on AQC, and all work on quantum error correction and suppression methods for AQC, as these topics deserve a separate review and including them here would limit our ability to do justice to the many years of work on AQC in closed systems, an extremely rich topic with many elegant results. To achieve our goal we organized this review 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 begin by reviewing the adiabatic theorem in Sec. II. The adiabatic theorem forms the backbone of AQC: it provides a sufficient condition for the success of the computation, and in doing so provides the runtime of a computation in terms of the eigenvalue gap of the Hamiltonian and the Hamiltonian’s timederivative. In fact there is not one single adiabatic theorem, and we review a number of different variants that provide different runtime requirements, under different smoothness and differentiability assumptions about the Hamiltonian.
Next, we review in Sec. III the handful of explicit algorithms for which AQC is known to give a speedup over classical computation. The emphasis is on “explicit”, since Sec. IV provides several proofs for the universality of AQC in terms of its ability to efficiently simulate the circuit model, and vice versa. This means that every quantum algorithm that provides a speedup in the circuit model [many of which are known Jordan (2016)] can in principle be implemented with up to polynomial overhead in AQC. That the number of explicit AQC algorithms is still small is therefore likely to be a reflection of the relatively modest amount of effort that has gone into establishing such results compared to the circuit model. However, there is also a real difficulty, in that performing the gap analysis in order to establish the actual scaling (beyond the polynomialtime equivalence) is, as already mentioned above, in many cases highly nontrivial. A second nontrivial aspect of establishing a speedup by AQC is that when such a speedup is polynomial, relying on universality is insufficient, since the polynomial overhead involved in implementing the transformation from the circuit model to AQC can then swamp the speedup. A good example is the case of Grover’s algorithm, where a direct use of the equivalence to the circuit model would not suffice; instead, what is required is a careful analysis and choice of the adiabatic schedule in order to realize the quantum speedup.
In Sec. V we go beyond universality into Hamiltonian quantum complexity theory. This is an active contemporary research area, that started with the introduction of the complexity class QMA (“quantum MerlinArthur”) as the natural quantum generalization of the classical complexity classes NP (nondeterministic polynomial time) and MA Kitaev et al. (2000). The theory of QMAcompleteness deals with decision problems that are efficiently checkable using quantum computers. It turns out that these decision problems can be formulated naturally in terms of local Hamiltonians, of the same type that appear in the proofs of the universality of AQC. Thus universality and Hamiltonian quantum complexity studies are often pursued handinhand, and a reduction of as well as the dimensionality of the particles appearing in these constructions is one of the main goals. For example, we will see that already and leads to both universal AQC and QMAcomplete Hamiltonians in two spatial dimensions, while in one spatial dimension is needed for both.
We turn our attention to StoqAQC in Sec. VI. This is the setting of the vast majority of AQC work to date. The final Hamiltonian is assumed to be a classical Ising model Hamiltonian, typically (but not always) representing a hard optimization problem such as a spin glass. The initial Hamiltonian is typically assumed to be proportional to a transverse field, i.e., , whose ground state is the uniform superposition state in the computational basis. AQC with stoquastic Hamiltonians is probably less powerful than universal quantum computation, but examples can be constructed which show that it may nevertheless be more powerful than classical computation. Moreover, if we relax the definition of AQC to allow for computation using excited states, it turns out that stoquastic Hamiltonians can even be QMAcomplete and support universal AQC. To do justice to this mixed and complicated picture, we first review examples where it is known that StoqAQC does not outperform classical computation (essentially because the eigenvalue gap decreases rapidly with problem size but classical algorithms do not suffer a slowdown), then discuss examples where StoqAQC offers a “limited” quantum speedup in the sense that it outperforms classical simulated annealing but not necessarily other classical algorithms, and finally point out examples where it is currently not known whether StoqAQC offers a quantum speedup, but one might hope that it does. We also discuss the role of potential quantum speedup mechanisms, in particular tunneling and entanglement.
The somewhat bleak picture regarding StoqAQC should not necessarily be a cause for pessimism. Some of the obstacles in the way of a quantum speedup can be overcome or circumvented, as we discuss in Sec. VII. In all cases this involves modifying some aspect of the Hamiltonian, either by optimizing the schedule , or by adding certain terms to the Hamiltonian such that small gaps are avoided. This can result in a nonstoquastic Hamiltonian whose final ground state is the same as that of the original Hamiltonian, with an exponentially small gap (corresponding to a first order quantum phase transition) changing into a polynomially small gap (corresponding to a second order phase transition). Another type of modification is to give up adiabatic evolution itself, and allow for diabatic transitions. While this results in giving up the guarantee of convergence to the ground state provided by the adiabatic theorem, it can be a strategy that results in better runtime scaling for the same Hamiltonian than an adiabatic one.
We conclude with an outlook and discussion of future directions in Sec. VIII. Various technical details are provided in the Appendix.
Ii Adiabatic Theorem
The origins of the celebrated quantum adiabatic approximation date back to Einstein’s “Adiabatenhypothese”: “If a system be affected in a reversible adiabatic way, allowed motions are transformed into allowed motions” Einstein (1914). Ehrenfest was the first to appreciate the importance of adiabatic invariance, guessing—before the advent of a complete quantum theory— that quantum laws would only allow motions which are invariant under adiabatic perturbations Ehrenfest (1916). The more familiar, modern version of the adiabatic approximation was put forth by Born and Fock already in 1928 for the case of discrete spectra Born and Fock (1928), after the development of the BornOppenheimer approximation for the separation of electronic and nuclear degrees of freedom a year earlier Born and Oppenheimer (1927). Kato put the approximation on a firm mathematical foundation in 1950 Kato (1950) and arguably proved the first quantum adiabatic theorem.
The adiabatic approximation states, roughly, that for a system initially prepared in an eigenstate (e.g., the ground state) of a timedependent Hamiltonian , the time evolution governed by the Schrödinger equation
(1) 
(we set from now on) will approximately keep the actual state of the system in the corresponding instantaneous ground state of , provided that varies “sufficiently slowly”. Quantifying the exact nature of this slow variation is the subject of the Adiabatic Theorem (AT), which exists in many variants. In this section we provide an overview of these variants of the AT, emphasizing aspects that are pertinent to AQC. We discuss the “folklore” adiabatic condition, that the total evolution time should be large on the timescale set by the square of the inverse gap, and the question of how to ensure a high fidelity between the actual state and the ground state. We then discuss a variety of rigorous versions of the AT, emphasizing different assumptions and consequently different performance guarantees.
ii.1 Approximate versions
Let () denote the instantaneous eigenstate of with energy such that , i.e., and denotes the (possibly degenerate) ground state. Assume that the initial state is .
The simplest as well as one of the oldest traditional versions of the adiabatic approximation states that a system initialized in an eigenstate will remain in the same instantaneous eigenstate for all , where denotes the final time, provided Messiah, A. (1962):
(2) 
This version has been critiqued Marzlin and Sanders (2004); Tong et al. (2005); Du et al. (2008) on the basis of arguments and examples involving a separate, independent timescale. Indeed, if the Hamiltonian includes an oscillatory driving term then the eigenstate population will oscillate with a timescale determined by this term, that is independent of , even if the adiabatic criterion (2) is satisfied.^{5}^{5}5For example, it is easily checked that when , the adiabatic condition (2) reduces to . However, even if this condition is satisfied the population can oscillate between the two eigenstates: at resonance (when ) the system undergoes Rabi oscillations with period , a timescale that is independent of .
A more careful statement of the adiabatic condition that excludes such additional timescales is thus required. The first step is to assume that the Hamiltonian in the Schrödinger equation can be written as , where is the dimensionless time, and is independent. This includes the “interpolating” Hamiltonians of the type often considered in AQC, i.e., [where and are monotonically decreasing and increasing, respectively] and excludes cases with multiple timescales.^{6}^{6}6For example, a case such as is now excluded since after a change of variables we have and evidently still depends on . The Schrödinger equation then becomes
(3) 
which is the starting point for all rigorous adiabatic theorems.
A more careful adiabatic condition subject to this formulation is given by Amin (2009):
(4) 
The conditions (2) and (4) give rise to the widely used criterion that the total adiabatic evolution time should be large on the timescale set by the minimum of the square of the inverse spectral gap . In most cases one is interested in the ground state, so that is replaced by
(5) 
However, arguments such as those leading to Eqs. (2) and (4) are approximate, in the sense that they do not result in strict inequalities and do not result in bounds on the closeness between the actual timeevolved state and the desired eigenstate. We discuss this next.
ii.2 Rigorous versions
The first rigorous adiabatic condition is due to Kato Kato (1950), and was followed by numerous alternative derivations and improvements giving tighter bounds under various assumptions, e.g., Teufel (2003); Nenciu (1993); Avron and Elgart (1999); Hagedorn and Joye (2002); Reichardt (2004); Ambainis and Regev (2004); Jansen et al. (2007); Lidar et al. (2009); Elgart and Hagedorn (2012); Ge et al. (2015). All these rigorous results are more severe in the gap condition than the traditional criterion, and they involve a power of the norm of time derivatives of the Hamiltonian, rather than a transition matrix element.
We summarize a few of these results here, and refer the reader to the original literature for their proofs. For simplicity we always assume that the system is initialized in its ground state and that the gap is the ground state gap (5). We also assume that for all the Hamiltonian has an eigenprojector with eigenenergy , and that the gap never vanishes, i.e., .^{7}^{7}7There is a weaker form of the AT, where one does not require a nonvanishing gap Avron and Elgart (1999). In this case, as in Theorem 2, the estimate on the error term is as . The ground state, and hence the projector , is allowed to be (even infinitely) degenerate. represents the “ideal” adiabatic evolution.
Let . This is the projector onto the timeevolved solution of the Schrödinger equation, i.e., the “actual” state. Adiabatic theorems are usually statements about the “instantaneous adiabatic distance” between the projectors associated with the actual and ideal evolutions, or the “finaltime adiabatic distance” . Typically, adiabatic theorems give a bound of the form for the instantaneous case, and a bound of the form for any for the finaltime case. After squaring, these projectordistance bounds immediately become bounds on the transition probability, defined as , where , with .
ii.2.1 Inverse cubic gap dependence with generic
Kato’s work on the perturbation theory of linear operators Kato (1950) introduced techniques based on resolvents and complex analysis that have been widely used in subsequent work. Jansen, Ruskai, and Seiler (JRS) proved several versions of the AT that build upon these techniques Jansen et al. (2007), and that rigorously establish the gap dependence of , without any strong assumptions on the smoothness of . Their essential assumption is that the spectrum of has a band associated with the spectral projection which is separated by a nonvanishing gap from the rest. Here we present one their theorems:
Theorem 1.
Suppose that the spectrum of restricted to consists of eigenvalues (each possibly degenerate, crossing permitted) separated by a gap from the rest of the spectrum of , and that is twice continuously differentiable. Assume that , , and are bounded operators, an assumption that is always fulfilled in finitedimensional spaces.^{8}^{8}8We use the notation throughout. Then for any ,
(6) 
The numerator depends on the norm of the first or second time derivative of , rather than the matrix element that appears in the traditional versions of the adiabatic condition.
Ignoring the dependence for simplicity, this result shows that the adiabatic limit can be approached arbitrarily closely if (but not only if)
(7) 
Similar techniques based on Kato’s approach can be used to prove a rigorous adiabatic theorem for open quantum systems, where the evolution is generated by a nonHermitian Liouvillian instead of a Hamiltonian Venuti et al. (2016).
ii.2.2 Rigorous inverse gap squared
A version of the AT that yields a scaling of with the inverse of the gap squared (up to a logarithmic correction) was given in Elgart and Hagedorn (2012). All other rigorous AT versions to date have a worse gap dependence (cubic or higher). The proof introduces assumptions on that go beyond those of Theorem 1. Namely, it is assumed that is bounded and infinitely differentiable, and the higher derivatives cannot have a magnitude that is too large, or more specifically, that belongs to the Gevrey class :
Definition 2 (Gevrey class).
if and there exist constants , such that for all ,
(8) 
An example is , where if , and if . The constant is chosen so that . For this family , so that .
The AT due to Elgart and Hagedorn (2012) can now be stated as follows:
Theorem 2.
Assume that is bounded and belongs to the Gevrey class with , and that , where . If
(9) 
for some independent constant (with units of energy), then the distance is .
This result is remarkable in that it rigorously gives an inverse gap squared dependence, which is essentially tight due to existence of a lower bound of the form for Hamiltonians satisfying Cao and Elgart (2012). However, the error bound is not tight, and we address this next.
ii.2.3 Arbitrarily small error
Building on work originating with Nenciu (1993) [see also Hagedorn and Joye (2002)], Ge et al. (2015) proved a version of the AT that results in an exponentially small error bound in . The inverse gap dependence is cubic.
Assume for simplicity that and choose the phase of so that .
Theorem 3.
Assume that all derivatives of the Hamiltonian vanish at , and moreover that it satisfies the following Gevrey condition: there exist constants such that for all ,
(10) 
Then the adiabatic error is bounded as
(11) 
where and .
Thus, as long as , the adiabatic error is exponentially small in .
The idea of using vanishing boundary derivatives dates back at least to Garrido, L. M. and Sancho, F. J. (1962). It was also used in Lidar et al. (2009) for a different class of functions than the Gevrey class: functions that are analytic in a strip of width in the complex time plane and have a finite number of vanishing boundary derivatives, i.e., . The adiabatic error is then upperbounded by as along as , where is a parameter that can be optimized given knowledge of . Thus, the adiabatic error can be made arbitrarily small in the number of vanishing derivatives, while the scaling of with is encoded into .^{9}^{9}9This corrects an omission in Lidar et al. (2009), where the dependence of on was ignored since the supremum of was taken over instead of over the region of analyticity of , as noted in Ge et al. (2015). An example of a function whose first derivatives vanish at the boundaries is the regularized function Rezakhani et al. (2010b). It is possible to further reduce the error quadratically in using an interference effect that arises from imposing an additional boundary symmetry condition Wiebe and Babcock (2012).
Note that an important difference between Theorems 2 and 3 is that the former applies for all times (“instantaneous AT”), while the latter applies only at the final time (“finaltime AT”), which typically gives rise to tighter error bounds.
Also note that Landau and Zener already showed that the transition probability out of the ground state is Landau, L. D. (1932); C. Zener (1932) [see Joye (1994) for a rigorous proof for analytic Hamiltonians], thus combining an inverse gap square dependence with an exponentially small error bound. However, this result only holds for twolevel systems.
ii.2.4 Lower bound
Let , with , be a given continuous Hamiltonian path and the corresponding nondegenerate eigenstate path (eigenpath). In the socalled blackbox model the only assumption is to be able to evolve with for some schedule (here is allowed to be a general function of ), without exploiting the unknown structure of . Define the path length as:
(12) 
where dot denotes . Assuming, without loss of generality, that the phase of is chosen so that , is the only natural length in projective Hilbert space (up to irrelevant normalization factors).
It was shown in Boixo and Somma (2010) that there is a lower bound on the time required to prepare from with bounded precision:
(13) 
Since an upper bound on is ,^{10}^{10}10To see this differentiate the eigenstate equation for the normalized instantaneous eigenstate and innermultiply by , with , to get . Let and . Using our phase choice: . Thus one obtains the estimate , reminiscent of the approximate versions of the adiabatic condition [e.g., Eq. (4)]. The proof of the lower bound is essentially based on the optimality of the Grover search algorithm.
The lower bound is nearly achievable using a “digital”, nonadiabatic method proposed in Boixo et al. (2010), that does not require path continuity or differentiability. The time required scales as , where is a specified bound on the error of the output state . is the angular length of the path and is suitably defined to generalize Eq. (12) to the nondifferentiable case.
Armed with an arsenal of adiabatic theorems we are now well equipped to start surveying AQC algorithms.
Iii Algorithms
In this section we we review the algorithms which are known to provide quantum speedups over classical algorithms. However, to make the idea of a quantum 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 classification of quantum speedup types proposed in Rønnow et al. (2014). The classification is the following, in decreasing order of strength.

A “provable” quantum speedup is the case where there exists a proof that no classical algorithm can outperform a given quantum algorithm. The best known example is Grover s search algorithm Grover (1997), which, in the query complexity setting, exhibits a provable quadratic speedup over the best possible classical algorithm Bennett et al. (1997).

A “strong” quantum speedup was originally defined in Papageorgiou and Traub (2013) by comparing a quantum algorithm against the performance of the best classical algorithm, whether such a classical algorithm is explicitly known or not. This aims to capture computational complexity considerations allowing for the existence of yettobe discovered classical algorithms. Unfortunately, the performance of the best possible classical algorithm is unknown for many interesting problems (e.g., for factoring).

A “quantum speedup” (unqualified, without adjectives) is a speedup against the best available classical algorithm [for example Shor’s polynomial time factoring algorithm Shor (1994)]. Such a speedup may be tentative, in the sense that a better classical algorithm may eventually be found.

Finally, a “limited quantum speedup” is a speedup obtained when compared specifically with classical algorithms that ‘correspond” to the quantum algorithm in the sense that they implement the same algorithmic approach, but on classical hardware. This definition allows for the existence of other classical algorithms that are already better than the quantum algorithm. The notion of a limited quantum speedup will turn out to be particularly useful in the context of StoqAQC.
A refinement of this classification geared at experimental quantum annealing was given in Mandrà et al. (2016).
Using this classification, this section collects most of the adiabatic quantum algorithms known to give a provable quantum speedup (Grover, DeutschJozsa, BernsteinVazirani, and glued trees^{11}^{11}11The glued trees case is, strictly, not an adiabatic quantum algorithm, since it explicitly makes use of excited states.), or just a quantum speedup (PageRank).
Many other adiabatic algorithms have been proposed, and we review a large subset of these in Sec. VI. In a few of these cases there is a limited quantum speedup against classical simulated annealing, while in some cases there are definitely faster classical algorithms.
iii.1 Adiabatic Grover
The adiabatic Grover algorithm Roland and Cerf (2002) is perhaps the hallmark example of a provable quantum speedup using AQC, so we review it in detail. As in the circuit model Grover algorithm Grover (1997), informally the objective is to find the marked item (or possibly multiple marked items) in an unsorted database of items by accessing the database as few times as possible. More formally, one is allowed to call a function (where is the number of bit strings) with the promise that and , and the goal is to find the unknown index in the smallest number of calls. This is an oracular problem Nielsen and Chuang (2000), in that the algorithm can make queries to an oracle that recognizes the marked items. The oracle remains a black box, i.e., the details of its implementation and its complexity are ignored. This allows for an uncontroversial determination of the complexity of the algorithm in terms of the number of queries to the oracle.
For a classical algorithm, the only strategy is to query the oracle until the marked item is found. Whether the classical algorithm uses no memory, i.e., the algorithm does not keep track of items that have already been checked, or uses an exponential amount of memory to store all the items that have been checked, the classical algorithm will have an average number of queries that scales linearly in .
In the AQC algorithm we denote the marked item by the binary representation of . The oracle is defined in terms of the final Hamiltonian , where is the marked state associated with the marked item. In this representation, the binary representations give the eigenvalues under , i.e., and . The marked state is the ground state of this Hamiltonian with energy 0, and all other computational basis states have energy 1.
iii.1.1 Setup for the adiabatic quantum Grover algorithm
We use the initial Hamiltonian , where is the uniform superposition state,
(14) 
where . We take the timedependent Hamiltonian to be an interpolation:
(15)  
where is the dimensionless time, is the total computation time, and is a “schedule” that can be optimized. For simplicity, we first consider a linear annealing schedule: .
If the initial state is initialized in the ground state of , i.e., , then the evolution of the system is restricted to a twodimensional subspace, defined by the span of and . In this twodimensional subspace can be written as:
(16) 
where:
(17a)  
(17b)  
(17c) 
The eigenvalues and eigenvectors in this subspace are then given by:
(18a)  
(18b)  
(18c) 
The remaining eigenstates of have eigenvalue throughout the evolution. The minimum gap occurs at and scales exponentially with :
(19) 
(This can be viewed as a special case of Lemma 1 below.)
In our discussion of the adiabatic theorem we saw that without special assumptions on except that it is twice differentiable, the adiabatic condition is inferred from Eq. (1), which requires setting , where we have accounted for the boundary conditions and used the positivity of the integrand to extend the upper limit to .^{12}^{12}12Whenever we use the symbol we mean that the larger quantity should be larger by some large multiplicative constant, such as . Differentiating Eq. (16) yields
(20) 
which has eigenvalues , so that .^{13}^{13}13The other integrand in Eq. (1), involving , vanishes after differentiating Eq. (20). The ground state degeneracy throughout. Since that approaches as , the adiabatic condition becomes , which is a monotonically increasing function of
(21) 
This suggests the disappointing conclusion that the quantum adiabatic algorithm scales in the same way as the classical algorithm.
However, by imposing the adiabatic condition globally, i.e., to the entire time interval , the evolution rate is constrained throughout the whole computation, while the gap only becomes small around . Thus, it makes sense to use a schedule that adapts and slows down near the minimum gap, but speeds up away from it van Dam et al. (2001); Roland and Cerf (2002) [this is related to the idea of rapid adiabatic passage, which has a long history in nuclear magnetic resonance Powles (1958)]. By doing so the quadratic quantum speedup can be recovered, as we address next.
iii.1.2 Quadratic quantum speedup
Consider again the adiabatic condition (1), which we can rewrite as:
(22) 
where now and depend on a schedule . Note that and for the interpolating Hamiltonian (15). Let us now use the ansatz Jansen et al. (2007); Roland and Cerf (2002)
(23) 
This schedule slows down as the gap becomes smaller, as desired. The normalization constant [using ] is chosen to ensure that . Also note that . We thus have
(24a)  
(24b)  
(24c) 
where in the last equality we again used the change of variables , so that . Finally, the boundary term in Eq. (22) yields .
The case serves to illustrate the main point. In this case the boundary term is and evaluating the integrals yields
where the asymptotic expressions are for . Substituting this into Eq. (24c) yields the adiabatic condition
(25) 
which is a sufficient condition for the smallness of the adiabatic error, and nearly recovers the quadratic speedup expected from Grover’s algorithm.
The appearance of the logarithmic factor latter is actually an artifact of using bounds that are not tight.^{14}^{14}14A detailed analysis of the adiabatic Grover algorithm along with tighter error bounds than we have given here was presented in Rezakhani et al. (2010b). The quadratic speedup, i.e., the scaling of with , can be fully recovered by solving for the schedule from Eq. (23) in the case Roland and Cerf (2002). We first rewrite Eq. (23) in dimensional time units as , with the boundary conditions and ;. To solve this differential equation we rewrite it as . After integration we obtain
One may be tempted to conclude that can be made arbitrarily small since so far is arbitrary and can be chosen to be large. However, the adiabatic error bound (25) shows that this is not the case: while it is not tight, it suggests that if scales as then must scale as in order to keep the adiabatic error small. Thus, the general conclusion is that increasing results in a larger adiabatic error.^{15}^{15}15Note that the scaling conclusion reported in Roland and Cerf (2002) is based on the interpretation of Eq. (23) as a heuristic “local” adiabatic condition and does not constitute a proof that the adiabatic error is small. The evidence that the rigorous bound (25) is not tight and that suffices to achieve a small adiabatic error for the schedule (28) is numerical.
iii.1.3 Lower bound
Since the choice in Eq. (23) is not unique, we may wonder if there exists a schedule that gives an even better scaling. Given that Grover’s algorithm is known to be optimal in the circuit model setting Bennett et al. (1997); Zalka (1999), this is, unsurprisingly, not the case Farhi and Gutmann (1998); Roland and Cerf (2002).
To show this, consider two different searches, one for and another for . We do not allow the schedule to depend on , i.e., the same schedule must apply to all marked states. Let us denote the states for each at the end of the algorithm by and . In order to be able to distinguish if the search gave or , we must require that and are sufficiently different. Let us define the distance (or infidelity)
(29) 
[note that ] and demand that:
(30) 
First, we have a lower bound on the sum:
(31) 
Next, let us find an upper bound on the sum. We write the Hamiltonian (15) explicitly as where . Then:
(32) 
Let us now sum over all and :
(33)  
where we first used the fact that under the sum the two terms in the last line of Eq. (III.1.3) are identical, and then we used the CauchySchwartz inequality (). Now we note that:
so that
(35) 
where