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arXiv:2102.02570 · Physical Review Letters

Classical Simulation and Theory of Quantum Annealing in a Thermal Environment

Hiroki Oshiyama, Sei Suzuki, Naokazu Shibata

Published Apr 27, 2022• 11 citations
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Abstract

We study quantum annealing in the quantum Ising model coupled to a thermal environment. When the speed of quantum annealing is sufficiently slow, the system evolves following the instantaneous thermal equilibrium. This quasistatic and isothermal evolution, however, fails near the end of annealing because the relaxation time grows infinitely, therefore yielding excess energy from the thermal equilibrium. We develop a phenomenological theory based on this picture and derive a scaling relation of the excess energy after annealing. The theoretical results are numerically confirmed using a novel non-Markovian method that we recently proposed based on a path-integral representation of the reduced density matrix and the infinite time evolving block decimation. In addition, we discuss crossovers from weak to strong coupling as well as from the adiabatic to quasistatic regime, and propose experiments on the D-Wave quantum annealer.

DOI
10.1103/physrevlett.128.170502

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We study quantum annealing in the quantum Ising model coupled to a thermal environment. When the speed of quantum annealing is sufficiently slow, the system evolves following the instantaneous thermal equilibrium. This quasistatic and isothermal evolution, however, fails near the end of annealing because the relaxation time grows infinitely, therefore yielding excess energy from the thermal equilibrium. We develop a phenomenological theory based on this picture and derive a scaling relation of the excess energy after annealing. The theoretical results are numerically confirmed using a novel non-Markovian method that we recently proposed based on a path-integral representation of the reduced density matrix and the infinite time evolving block decimation. In addition, we discuss crossovers from weak to strong coupling as well as from the adiabatic to quasistatic regime, and propose experiments on the D-Wave quantum annealer.
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.
We study the statistics of the kink number generated by quantum annealing in a one-dimensional transverse Ising model coupled to a bosonic thermal bath. Using the freezing ansatz for quantum annealing in the thermal environment, we show the relation between the ratio of the second to the first cumulant of the kink number distribution and the average kink density. The theoretical result is confirmed thoroughly by numerical simulation using the non-Markovian infinite time-evolving block decimation which we proposed recently. The simulation using D-Wave's quantum annealer is also discussed. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
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.
Convergence conditions for quantum annealing are derived for optimization problems represented by the Ising model of a general form. Quantum fluctuations are introduced as a transverse field and/or transverse ferromagnetic interactions, and the time evolution follows the real-time Schrodinger equation. It is shown that the system stays arbitrarily close to the instantaneous ground state, finally reaching the target optimal state, if the strength of quantum fluctuations decreases sufficiently slowly, in particular inversely proportionally to the power of time in the asymptotic region. This is the same condition as the other implementations of quantum annealing, quantum Monte Carlo and Green's function Monte Carlo simulations, in spite of the essential difference in the type of dynamics. The method of analysis is an application of the adiabatic theorem in conjunction with an estimate of a lower bound of the energy gap based on the recently proposed idea of Somma et. al. for the analysis of classical simulated annealing using a classical-quantum correspondence.
When a nonintegrable system evolves out of equilibrium for a long time, local observables are in general expected to attain stationary expectation values, independent of the details of the initial state. But the thermalization of a closed quantum system is not yet well understood. Here we show that it presents indeed a much richer phenomenology than its classical counterpart. Using a new numerical technique, we identify two distinct regimes, strong and weak, occurring for different initial states. Strong thermalization, intrinsically quantum, happens when instantaneous local expectation values converge to the thermal ones. Weak thermalization, well known in classical systems, shows convergence to thermal values only after time averaging. Remarkably, we find a third group of states showing no thermalization, neither strong nor weak, to the time scales one can reliably simulate.
We compare the performance of quantum annealing (QA, through Schr\"odinger dynamics) and simulated annealing (SA, through a classical master equation) on the ppp-spin infinite range ferromagnetic Ising model, by slowly driving the system across its equilibrium, quantum or classical, phase transition. When the phase transition is second order (p=2p=2p=2, the familiar two-spin Ising interaction) SA shows a remarkable exponential speed-up over QA. For a first-order phase transition ($p\ensuremath{\ge}3$, i.e., with multispin Ising interactions), in contrast, the classical annealing dynamics appears to remain stuck in the disordered phase, while we have clear evidence that QA shows a residual energy which decreases towards zero when the total annealing time $\ensuremath{\tau}$ increases, albeit in a rather slow (logarithmic) fashion. This is one of the rare examples where a limited quantum speedup, a speedup by QA over SA, has been shown to exist by direct solutions of the Schr\"odinger and master equations in combination with a nonequilibrium Landau-Zener analysis. We also analyze the imaginary-time QA dynamics of the model, finding a $1/{\ensuremath{\tau}}^{2}$ behavior for all finite values of ppp, as predicted by the adiabatic theorem of quantum mechanics. The Grover-search limit $p(\mathrm{odd})=\ensuremath{\infty}$ is also discussed.