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arXiv:1706.06416

Proposal of a new quantum annealing schedule for studying transverse-field-based quantum versus classical annealing of the Ising model: a case study of the Ising spin glass model on the square lattice by the Monte Carlo simulation

Chiaki Yamaguchi

Published Jun 20, 2017• 0 citations
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Abstract

Recently, Heim, R\o nnow, Isakov and Troyer [{\it Science} \textbf{348} (2015) 215] have reported that Monte Carlo simulations for the Ising spin glass model on the square lattice in the physically relevant continuous-imaginary-time limit do not show superiority of quantum annealing (QA) using transverse field against classical annealing (CA). Although the QA schedule that they had used has been using conventionally, however the QA schedule mathematically has no guarantee that the used schedule is the best QA schedule for performance of optimization. We propose a new QA schedule for studying transverse-field-based quantum versus classical annealing of the Ising model. The present QA schedule is made based on the concept of perturbation theory. This QA schedule utilizes a smallest effective transverse field derived in this article. As a case study, we study QA of the Ising spin glass model on the square lattice at low but finite temperature. A new Monte Carlo algorithm using the physically relevant continuous-imaginary-time limit is performed. As the simulation results, we show superiority of QA against CA when the annealing time is spent enough.

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Recently, Heim, R\o nnow, Isakov and Troyer [{\it Science} \textbf{348}
(2015) 215] have reported that Monte Carlo simulations for the Ising spin glass
model on the square lattice in the physically relevant
continuous-imaginary-time limit do not show superiority of quantum annealing
(QA) using transverse field against classical annealing (CA). Although the QA
schedule that they had used has been using conventionally, however the QA
schedule mathematically has no guarantee that the used schedule is the best QA
schedule for performance of optimization. We propose a new QA schedule for
studying transverse-field-based quantum versus classical annealing of the Ising
model. The present QA schedule is made based on the concept of perturbation
theory. This QA schedule utilizes a smallest effective transverse field derived
in this article. As a case study, we study QA of the Ising spin glass model on
the square lattice at low but finite temperature. A new Monte Carlo algorithm
using the physically relevant continuous-imaginary-time limit is performed. As
the simulation results, we show superiority of QA against CA when the annealing
time is spent enough.
We present an implementation of quantum annealing (QA) via lattice Green's function Monte Carlo (GFMC), focusing on its application to the Ising spin glass in transverse field. In particular, we study whether or not such a method is more effective than the path-integral Monte Carlo- (PIMC) based QA, as well as classical simulated annealing (CA), previously tested on the same optimization problem. We identify the issue of importance sampling, i.e., the necessity of possessing reasonably good (variational) trial wave functions, as the key point of the algorithm. We performed GFMC-QA runs using such a Boltzmann-type trial wave function, finding results for the residual energies that are qualitatively similar to those of CA (but at a much larger computational cost), and definitely worse than PIMC-QA. We conclude that, at present, without a serious effort in constructing reliable importance sampling variational wave functions for a quantum glass, GFMC-QA is not a true competitor of PIMC-QA.
Quantum annealers use quantum fluctuations to escape local minima and find low-energy configurations of a physical system. Strong evidence for superiority of quantum annealing (QA) has come from comparing QA implemented through quantum Monte Carlo (QMC) simulations to classical annealing. Motivated by recent experiments, we revisit the question of when quantum speedup may be expected. Although a better scaling is seen for QA in two-dimensional Ising spin glasses, this advantage is due to time discretization artifacts and measurements that are not possible on a physical quantum annealer. Simulations in the physically relevant continuous time limit, on the other hand, do not show superiority. Our results imply that care must be taken when using QMC simulations to assess the potential for quantum speedup.
We discuss an Ising spin glass where each S=1/2 spin is coupled antiferromagnetically to three other spins (3-regular graphs). Inducing quantum fluctuations by a time-dependent transverse field, we use out-of-equilibrium quantum Monte Carlo simulations to study dynamic scaling at the quantum glass transition. Comparing the dynamic exponent and other critical exponents with those of the classical (temperature-driven) transition, we conclude that quantum annealing is less efficient than classical simulated annealing in bringing the system into the glass phase. Quantum computing based on the quantum annealing paradigm is therefore inferior to classical simulated annealing for this class of problems. We also comment on previous simulations where a parameter is changed with the simulation time, which is very different from the true Hamiltonian dynamics simulated here.
We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. The idea is tested by the two models, the transverse Ising model and the traveling salesman problem (TSP). Adding the transverse field to the Ising model is a simple way to introduce quantum fluctuations. The strength of the transverse field is controlled as a function of time similarly to the temperature in the conventional method. TSP can be described by the Ising spin, so that we also apply the same technique to TSP. We directory solve the time-dependent Schrödinger equation for small-size systems and perform the quantum Monte Carlo simulation for large-size systems. Comparison with the results of the corresponding classical (thermal) method reveals that the quantum method leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule of the control parameters. We also find that the relaxation time is quite short for quantum systems by numerical simulations. We consider this is one of the reasons why the annealing in quantum systems have a better performance of finding the optimal state in comparison with classical systems.
Probing the lowest energy configuration of a complex system by quantum annealing was recently found to be more effective than its classical, thermal counterpart. By comparing classical and quantum Monte Carlo annealing protocols on the two-dimensional random Ising model (a prototype spin glass), we confirm the superiority of quantum annealing relative to classical annealing. We also propose a theory of quantum annealing based on a cascade of Landau-Zener tunneling events. For both classical and quantum annealing, the residual energy after annealing is inversely proportional to a power of the logarithm of the annealing time, but the quantum case has a larger power that makes it faster.