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arXiv:2212.08634 · Physical Review A

Diabatic quantum and classical annealing of the Sherrington-Kirkpatrick model

Artem Rakcheev, Andreas Martin Läuchli

Published Jun 1, 2023• 3 citations
arXivHTMLPDF

Abstract

Quantum annealing is a contender to solve combinatorial optimization problems based on quantum dynamics. While significant efforts have been undertaken to investigate the quality of the solutions and the required run times, much less attention has been paid to understanding the dynamics of quantum annealing and the process leading to the solution during the sweep itself. In this comprehensive study, we investigate various aspects of the quantum annealing dynamics using different approaches. We perform quantum annealing, simulated quantum annealing, and classical annealing on several hundred instances of the Sherrington-Kirkpatrick model with intermediate system sizes up to 22 spins using numerical simulations. We observe qualitative differences between the quantum and classical methods, in particular at intermediate times, where a peak in the fidelity, also known as diabatic bump, appears for hard instances. Furthermore, we investigate the two-point correlation functions, which feature differences at intermediate times as well. At short times, however, the methods are similar again, which can be explained by relating the short-time expansion of quantum annealing to a high-temperature expansion, thus allowing one in principle to find the classical solution already at short times, albeit at prohibitive sampling cost.

DOI
10.1103/physreva.107.062602

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Quantum annealing is a contender to solve combinatorial optimization problems based on quantum dynamics. While significant efforts have been undertaken to investigate the quality of the solutions and the required run times, much less attention has been paid to understanding the dynamics of quantum annealing and the process leading to the solution during the sweep itself. In this comprehensive study, we investigate various aspects of the quantum annealing dynamics using different approaches. We perform quantum annealing, simulated quantum annealing, and classical annealing on several hundred instances of the Sherrington-Kirkpatrick model with intermediate system sizes up to 22 spins using numerical simulations. We observe qualitative differences between the quantum and classical methods, in particular at intermediate times, where a peak in the fidelity, also known as diabatic bump, appears for hard instances. Furthermore, we investigate the two-point correlation functions, which feature differences at intermediate times as well. At short times, however, the methods are similar again, which can be explained by relating the short-time expansion of quantum annealing to a high-temperature expansion, thus allowing one in principle to find the classical solution already at short times, albeit at prohibitive sampling cost.
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Boltzmann sampling is a central component of many computational frameworks, including numerous algorithms in machine learning. Although quantum annealers have been investigated as potential fast Boltzmann samplers, their dependence on environmental noise makes precise control of the effective temperature difficult, introducing uncertainty into the sampling process. As an alternative, we propose diabatic quantum annealing-a faster, purely unitary process-as a controllable Boltzmann sampler in which the effective temperature is determined by the annealing rate. Using the ferromagnetic Ising model and the Sherrington-Kirkpatrick model as test cases, we demonstrate that this method achieves rapid and accurate sampling in the high-temperature regime.
Simulated quantum annealing based on the path-integral Monte Carlo is one of the most common tools to simulate quantum annealing on classical hardware. Nevertheless, it is in principle highly nontrivial whether or not this classical algorithm can correctly reproduce the quantum dynamics of quantum annealing, particularly in the diabatic regime. We study this problem numerically through the generalized Kibble-Zurek mechanism of defect distribution in the simplest ferromagnetic one-dimensional transverse-field Ising model with and without coupling to the environment. We find that, in the absence of coupling to the environment, simulated quantum annealing correctly describes the annealing-time dependence of the average number of defects, but a detailed analysis of the defect distribution shows clear deviations from the theoretical prediction. When the system is open (coupled to the environment), the average number of defects does not follow the theoretical prediction but is qualitatively compatible with the numerical result by the infinite-time-evolving block decimation combined with the quasiadiabatic propagator path integral, which is valid in a very-short-time region. The distribution of defects in the open system turns out to be not far from the theoretical prediction. It is surprising that the classical stochastic dynamics of simulated quantum annealing ostensibly reproduces some aspects of the quantum dynamics. However, a serious problem is that it is hard to predict for which physical quantities in which system it is reliable. Those results suggest the necessity to exert a good amount of caution in using simulated quantum annealing to study the detailed quantitative aspects of the dynamics of quantum annealing.
The correspondence between long-range interacting quantum spin glasses and combinatorial optimization problems underpins the physical motivation for adiabatic quantum computing. On one hand, in disordered (quantum) spin systems, the focus is on exact methods such as the replica trick that allow the calculation of system quantities in the limit of infinite system and ensemble size. On the other hand, when solving a given instance of an optimization problem, disorder-averaged quantities are of no relevance, as one is solely interested in instance-specific, finite-size properties, in particular the optimal solution. Here, we apply the nonequilibrium Green's function formalism to the spin coherent-state path integral to obtain the statistical fluctuations and the collective-excitation spectrum along the annealing path. For the example of the quantum Sherrington-Kirkpatrick spin glass, by comparing to extensive numerically exact results, we show that this method provides access to the instance-specific bottlenecks of the annealing protocol.
Relations of simulated annealing and quantum annealing are studied by a mapping from the transition matrix of classical Markovian dynamics of the Ising model to a quantum Hamiltonian and vice versa. It is shown that these two operators, the transition matrix and the Hamiltonian, share the eigenvalue spectrum. Thus, if simulated annealing with slow temperature change does not encounter a difficulty caused by an exponentially long relaxation time at a first-order phase transition, the same is true for the corresponding process of quantum annealing in the adiabatic limit. One of the important differences between the classical-to-quantum mapping and the converse quantum-to-classical mapping is that the Markovian dynamics of a short-range Ising model is mapped to a short-range quantum system, but the converse mapping from a short-range quantum system to a classical one results in long-range interactions. This leads to a difference in efficiencies that simulated annealing can be efficiently simulated by quantum annealing but the converse is not necessarily true. We conclude that quantum annealing is easier to implement and is more flexible than simulated annealing. We also point out that the present mapping can be extended to accommodate explicit time dependence of temperature, which is used to justify the quantum-mechanical analysis of simulated annealing by Somma, Batista, and Ortiz. Additionally, an alternative method to solve the nonequilibrium dynamics of the one-dimensional Ising model is provided through the classical-to-quantum mapping.
A combinatorial optimization problem is the problem of minimizing the energy function among many combinations of variables, which is often difficult to solve with conventional classical computers. Recently, Ising machines, including quantum annealers, have gained attention as a promising architecture for efficiently solving combinatorial optimization problems. Among various methods for solving such problems using Ising machines, one prominent approach is the three-stage annealing method. The approach effectively solves a combinatorial optimization problem, utilizing an initial solution, but it performs the annealing process only once. Repeating the annealing process several times may enhance the solution more efficiently. In this paper, we propose a novel hybrid iterative annealing method that consists of an initial process using a classical computer, an annealing process using a quantum annealer, and a correction process/selection process using a classical computer. The proposed method repeats the annealing process and the correction process/selection process until the solution is sufficiently converged. In the experimental evaluations through the two types of typical combinatorial optimization problems, the proposed method shows improvements by up to 50.5% compared to the three-stage annealing method.