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arXiv:2203.11417 · Physical Review A

Quantum and classical annealing in a continuous space with multiple local minima

Yang Wei Koh, Hidetoshi Nishimori

Published Jun 21, 2022• 13 citations
arXivHTMLPDF

Abstract

The protocol of quantum annealing is applied to an optimization problem with a one-dimensional continuous degree of freedom, a variant of the problem proposed by Shinomoto and Kabashima. The energy landscape has a number of local minima, and the classical approach of simulated annealing is predicted to have a logarithmically slow convergence to the global minimum. We show by extensive numerical analyses that quantum annealing yields a power-law convergence, thus an exponential improvement over simulated annealing. The power is larger, and thus the convergence is faster, than a prediction by an existing phenomenological theory for this problem. Performance of simulated annealing is shown to be enhanced by introducing quasiglobal searches across energy barriers, leading to a power-law convergence but with a smaller power than in the quantum case and thus a slower convergence classically even with quasiglobal search processes. We also reveal how diabatic quantum dynamics, quantum tunneling in particular, steers the systems toward the global minimum by a meticulous choice of annealing schedule. This latter result explicitly contrasts the role of tunneling in quantum annealing against the classical counterpart of stochastic optimization by simulated annealing.

DOI
10.1103/physreva.105.062435

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The protocol of quantum annealing is applied to an optimization problem with a one-dimensional continuous degree of freedom, a variant of the problem proposed by Shinomoto and Kabashima. The energy landscape has a number of local minima, and the classical approach of simulated annealing is predicted to have a logarithmically slow convergence to the global minimum. We show by extensive numerical analyses that quantum annealing yields a power-law convergence, thus an exponential improvement over simulated annealing. The power is larger, and thus the convergence is faster, than a prediction by an existing phenomenological theory for this problem. Performance of simulated annealing is shown to be enhanced by introducing quasiglobal searches across energy barriers, leading to a power-law convergence but with a smaller power than in the quantum case and thus a slower convergence classically even with quasiglobal search processes. We also reveal how diabatic quantum dynamics, quantum tunneling in particular, steers the systems toward the global minimum by a meticulous choice of annealing schedule. This latter result explicitly contrasts the role of tunneling in quantum annealing against the classical counterpart of stochastic optimization by simulated annealing.
We investigate the basic behavior and performance of simulated quantum annealing (QA) in comparison with classical annealing (CA). Three simple one-dimensional case study systems are considered: namely, a parabolic well, a double well, and a curved washboard. The time-dependent Schr\"odinger evolution in either real or imaginary time describing QA is contrasted with the Fokker-Planck evolution of CA. The asymptotic decrease of excess energy with annealing time is studied in each case, and the reasons for differences are examined and discussed. The Huse-Fisher classical power law of double-well CA is replaced with a different power law in QA. The multiwell washboard problem studied in CA by Shinomoto and Kabashima and leading classically to a logarithmic annealing even in the absence of disorder turns to a power-law behavior when annealed with QA. The crucial role of disorder and localization is briefly discussed.
Finding the global minimum in a rugged potential landscape is a computationally hard task, often equivalent to relevant optimization problems. Annealing strategies, either classical or quantum, explore the configuration space by evolving the system under the influence of thermal or quantum fluctuations. The thermal annealing dynamics can rapidly freeze the system into a low-energy configuration, and it can be simulated well on a classical computer, but it easily gets stuck in local minima. Quantum annealing, on the other hand, can be guaranteed to find the true ground state and can be implemented in modern quantum simulators; however, quantum adiabatic schemes become prohibitively slow in the presence of quasidegeneracies. Here, we propose a strategy which combines ideas from simulated annealing and quantum annealing. In such a hybrid algorithm, the outcome of a quantum simulator is processed on a classical device. While the quantum simulator explores the configuration space by repeatedly applying quantum fluctuations and performing projective measurements, the classical computer evaluates each configuration and enforces a lowering of the energy. We have simulated this algorithm for small instances of the random energy model, showing that it potentially outperforms both simulated thermal annealing and adiabatic quantum annealing. It becomes most efficient for problems involving many quasidegenerate ground states.
Quantum annealers aim at solving nonconvex optimization problems by exploiting cooperative tunneling effects to escape local minima. The underlying idea consists of designing a classical energy function whose ground states are the sought optimal solutions of the original optimization problem and add a controllable quantum transverse field to generate tunneling processes. A key challenge is to identify classes of nonconvex optimization problems for which quantum annealing remains efficient while thermal annealing fails. We show that this happens for a wide class of problems which are central to machine learning. Their energy landscapes are dominated by local minima that cause exponential slowdown of classical thermal annealers while simulated quantum annealing converges efficiently to rare dense regions of optimal solutions.
Can quantum computers solve optimization problems much more quickly than classical computers? One major piece of evidence for this proposition has been the fact that Quantum Annealing (QA) finds the minimum of some cost functions exponentially more quickly than classical Simulated Annealing (SA). One such cost function is the simple “Hamming weight with a spike” function in which the input is an n-bit string and the objective function is simply the Hamming weight, plus a tall thin barrier centered around Hamming weight n/4. While the global minimum of this cost function can be found by inspection, it is also a plausible toy model of the sort of local minima that arise in realworld optimization problems. It was shown by Farhi, Goldstone and Gutmann [1] that for this example SA takes exponential time and QA takes polynomial time, and the same result was generalized by Reichardt [2] to include barriers with width n^ζ and height n^α for ζ + α ≤ 1/2. This advantage could be explained in terms of quantummechanical “tunneling.” Our work considers a classical algorithm known as Simulated Quantum Annealing (SQA) which relates certain quantum systems to classical Markov chains. By proving that these chains mix rapidly, we show that SQA runs in polynomial time on the Hamming weight with spike problem in much of the parameter regime where QA achieves an exponential advantage over SA. While our analysis only covers this toy model, it can be seen as evidence against the prospect of exponential quantum speedup using tunneling. Our technical contributions include extending the canonical path method for analyzing Markov chains to cover the case when not all vertices can be connected by low-congestion paths. We also develop methods for taking advantage of warm starts and for relating the quantum state in QA to the probability distribution in SQA. These techniques may be of use in future studies of SQA or of rapidly mixing Markov chains in general.
Can quantum computers solve optimization problems much more quickly than classical computers? One major piece of evidence for this proposition has been the fact that Quantum Annealing (QA) finds the minimum of some cost functions exponentially more quickly than classical Simulated Annealing (SA). One such cost function is the simple “Hamming weight with a spike” function in which the input is an n-bit string and the objective function is simply the Hamming weight, plus a tall thin barrier centered around Hamming weight n/4. While the global minimum of this cost function can be found by inspection, it is also a plausible toy model of the sort of local minima that arise in realworld optimization problems. It was shown by Farhi, Goldstone and Gutmann [1] that for this example SA takes exponential time and QA takes polynomial time, and the same result was generalized by Reichardt [2] to include barriers with width nζ and height nαfor ζ + α ≤ 1/2. This advantage could be explained in terms of quantummechanical “tunneling.” Our work considers a classical algorithm known as Simulated Quantum Annealing (SQA) which relates certain quantum systems to classical Markov chains. By proving that these chains mix rapidly, we show that SQA runs in polynomial time on the Hamming weight with spike problem in much of the parameter regime where QA achieves an exponential advantage over SA. While our analysis only covers this toy model, it can be seen as evidence against the prospect of exponential quantum speedup using tunneling. Our technical contributions include extending the canonical path method for analyzing Markov chains to cover the case when not all vertices can be connected by low-congestion paths. We also develop methods for taking advantage of warm starts and for relating the quantum state in QA to the probability distribution in SQA. These techniques may be of use in future studies of SQA or of rapidly mixing Markov chains in general.