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arXiv:1611.02585 · Physical Review A

Necessary adiabatic run times in quantum optimization

Lucas T. Brady, Wim van Dam

Published Mar 30, 2017• 18 citations
arXivHTMLPDF

Abstract

Quantum annealing is guaranteed to find the ground state of optimization problems provided it operates in the adiabatic limit. Recent work [S. Muthukrishnan et al., Phys. Rev. X 6, 031010 (2016)] has found that for some barrier tunneling problems, quantum annealing can be run much faster than is adiabatically required. Specifically, an $n$-qubit optimization problem was presented for which a nonadiabatic, or diabatic, annealing algorithm requires only a constant run time, while an adiabatic annealing algorithm requires a run-time polynomial in $n$. Here we show that this nonadiabatic speedup is the direct result of a specific symmetry in the studied problem. In the more general case, no such nonadiabatic speedup occurs and we show why the special case achieves this speedup compared to the general case. We also prove that the adiabatic annealing algorithm has a necessary and sufficient run time that is quadratically better than the standard quantum adiabatic condition suggests. We conclude with an observation about the required precision in timing for the diabatic algorithm.

DOI
10.1103/physreva.95.032335

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Quantum annealing is guaranteed to find the ground state of optimization problems provided it operates in the adiabatic limit. Recent work [S. Muthukrishnan et al., Phys. Rev. X 6, 031010 (2016)] has found that for some barrier tunneling problems, quantum annealing can be run much faster than is adiabatically required. Specifically, an nnn-qubit optimization problem was presented for which a nonadiabatic, or diabatic, annealing algorithm requires only a constant run time, while an adiabatic annealing algorithm requires a run-time polynomial in nnn. Here we show that this nonadiabatic speedup is the direct result of a specific symmetry in the studied problem. In the more general case, no such nonadiabatic speedup occurs and we show why the special case achieves this speedup compared to the general case. We also prove that the adiabatic annealing algorithm has a necessary and sufficient run time that is quadratically better than the standard quantum adiabatic condition suggests. We conclude with an observation about the required precision in timing for the diabatic algorithm.
Abstract Quantum annealing (QA) is a continuous-time heuristic quantum algorithm for solving or approximately solving classical optimization problems. The algorithm uses a schedule to interpolate between a driver Hamiltonian with an easy-to-prepare ground state and a problem Hamiltonian whose ground state encodes solutions to an optimization problem. The standard implementation relies on the evolution being adiabatic: keeping the system in the instantaneous ground state with high probability and requiring a time scale inversely related to the minimum energy gap between the instantaneous ground and excited states. However, adiabatic evolution can lead to evolution times that scale exponentially with the system size, even for computationally simple problems. Here, we study whether non-adiabatic evolutions with optimized annealing schedules can bypass this exponential slowdown for one such class of problems called the frustrated ring model. For sufficiently optimized annealing schedules and system sizes of up to 39 qubits, we provide numerical evidence that we can avoid the exponential slowdown. Our work highlights the potential of highly-controllable QA to circumvent bottlenecks associated with the standard implementation of QA.
We provide and analyze examples that counter the widely made claim that tunneling is needed for a quantum speedup in optimization problems. The examples belong to the class of perturbed Hamming-weight optimization problems. In one case, featuring a plateau in the cost function in Hamming weight space, we find that the adiabatic dynamics that make tunneling possible, while superior to simulated annealing, result in a slowdown compared to a diabatic cascade of avoided level-crossings. This, in turn, inspires a classical spin vector dynamics algorithm that is at least as efficient for the plateau problem as the diabatic quantum algorithm. In a second case whose cost function is convex in Hamming weight space, the diabatic cascade results in a speedup relative to both tunneling and classical spin vector dynamics.
Quantum annealing (QA) is a method for solving combinatorial optimization problems. We can estimate the computational time for QA using what is referred to as the adiabatic condition derived from the adiabatic theorem. The adiabatic condition consists of two parts: an energy gap and a transition matrix. Most past studies have focused on the relationship between the energy gap and computational time. The success probability of QA is considered to decrease exponentially owing to the exponentially decreasing energy gap at the first-order phase-transition point. In this study, through a detailed analysis of the relationship between the energy gap, transition matrix, and computational cost during QA, we propose a general method for constructing counterintuitive models wherein QA with a constant annealing time fails despite a constant energy gap, based on polynomial energy. We assume that the energy of the total Hamiltonian is at most $\mathrm{\ensuremath{\Theta}}(L)$, where LLL is the number of qubits. In our formalism, we choose a known model that exhibits an exponentially small energy gap during QA, and modify the model by adding a specific penalty term to the Hamiltonian. In the modified model, the transition matrix in the adiabatic condition becomes exponentially large as the number of qubits increases, while the energy gap remains constant. Moreover, we achieve a quadratic speedup, for which the upper bound for improvement in the adiabatic condition is determined by the polynomial energy. For concrete examples, we consider the adiabatic Grover search and the ferromagnetic ppp-spin model. In these cases, with the addition of the penalty term, although the success probability of QA on the modified models becomes exponentially small despite a constant energy gap, we are able to achieve a success probability considerably higher than that of conventional QA. Moreover, in concrete examples, we numerically show the scaling of the computational cost is quadratically improved compared to the conventional QA. Our findings pave the way for a better understanding of QA performance.
Quantum annealing is a general purpose optimization algorithm that is based on the quantum adiabatic theorem. Quantum annealing involves an evolving Hamiltonian that is local. Thus, we expect that short-time quantum annealing algorithms to be inherently local and limited as well. In this paper, we validate this intuition by proving some limitations of short-time quantum annealing algorithms. We show that the distribution of the measurement output of short-time (at most logarithmic) quantum annealing computations are \emph{concentrated} and satisfy an \emph{isoperimetric inequality}. To showcase explicit applications, we also study the \textsc{MaxCut} problem and conclude that quantum annealing needs at least a run-time that scales logarithmically in the problem size to beat classical algorithms. To establish our results, we also prove a Lieb-Robinson bound that works for time-dependent Hamiltonians which might be of independent interest.
Adiabatic quantum optimization is a procedure to solve a vast class of optimization problems by slowly changing the Hamiltonian of a quantum system. The evolution time necessary for the algorithm to be successful scales inversely with the minimum energy gap encountered during the dynamics. Unfortunately, the direct calculation of the gap is strongly limited by the exponential growth in the dimensionality of the Hilbert space associated to the quantum system. Although many special-purpose methods have been devised to reduce the effective dimensionality, they are strongly limited to particular classes of problems with evident symmetries. Moreover, little is known about the computational power of adiabatic quantum optimizers in real-world conditions. Here we propose and implement a general purposes reduction method that does not rely on any explicit symmetry and which requires, under certain general conditions, only a polynomial amount of classical resources. Thanks to this method, we are able to analyze the performance of ``nonideal'' quantum adiabatic optimizers to solve the well-known Grover problem, namely the search of target entries in an unsorted database, in the presence of discrete local defects. In this case, we show that adiabatic quantum optimization, even if affected by random noise, is still potentially faster than any classical algorithm.
Fixed-point quantum search algorithms succeed at finding one of MMM target items among NNN total items even when the run time of the algorithm is longer than necessary. While the famous Grover's algorithm can search quadratically faster than a classical computer, it lacks the fixed-point property---the fraction of target items must be known precisely to know when to terminate the algorithm. Recently, Yoder, Low, and Chuang [Phys. Rev. Lett. 113, 210501 (2014)] gave an optimal gate-model search algorithm with the fixed-point property. Previously, it had been discovered by Roland and Cerf [Phys. Rev. A 65, 042308 (2002)] that an adiabatic quantum algorithm, operating by continuously varying a Hamiltonian, can reproduce the quadratic speedup of gate-model Grover search. We ask, can an adiabatic algorithm also reproduce the fixed-point property? We show that the answer depends on what interpolation schedule is used, so as in the gate model, there are both fixed-point and non-fixed-point versions of adiabatic search, only some of which attain the quadratic quantum speedup. Guided by geometric intuition on the Bloch sphere, we rigorously justify our claims with an explicit upper bound on the error in the adiabatic approximation. We also show that the fixed-point adiabatic search algorithm can be simulated in the gate model with neither loss of the quadratic Grover speedup nor of the fixed-point property. Finally, we discuss natural uses of fixed-point algorithms such as preparation of a relatively prime state and oblivious amplitude amplification.