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arXiv:cond-mat/0608420 · Physical Review E

Quantum annealing of an Ising spin-glass by Green’s function Monte Carlo

Lorenzo Stella, Giuseppe E. Santoro

Published Mar 13, 2007• 19 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.75.036703

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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.
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.
The path integral Monte Carlo simulated quantum annealing algorithm is applied to the optimization of a large hard instance of the random satisfiability problem (N = 10,000). The dynamical behavior of the quantum and the classical annealing are compared, showing important qualitative differences in the way of exploring the complex energy landscape of the combinatorial optimization problem. At variance with the results obtained for the Ising spin glass and for the traveling salesman problem, in the present case the linear-schedule quantum annealing performance is definitely worse than classical annealing. Nevertheless, a quantum cooling protocol based on field-cycling and able to outperform standard classical simulated annealing over short time scales is introduced.
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.
Quantum annealing was recently found experimentally in a disordered spin-12\frac{1}{2}21​ magnet to be more effective than its classical, thermal counterpart. We use the random two-dimensional Ising model as a test example and perform on it both classical and quantum (path-integral) Monte Carlo annealing. A systematic study of the dependence of the final residual energy on the annealing Monte Carlo time quantitatively demonstrates the superiority of quantum relative to classical annealing in this system. In order to determine the parameter regime for optimal efficiency of the quantum annealing procedure we explore a range of values of Trotter slice number P and temperature T. This identifies two different regimes of freezing with respect to efficiency of the algorithm, and leads to useful guidelines for the optimal choice of quantum annealing parameters.
Programmable quantum annealers can realize real-time dynamics of frustrated transverse-field Ising models on large, nontrivial graphs. Recent work by King et al. argued that the classical simulation of such experiments would require exponential computational resources for classical methods such as tensor networks and neural quantum states. Here, we numerically simulate the D-Wave spin-glass annealing protocol with time-dependent variational Monte Carlo (t-VMC) using a correlator state tailored to spin-glass dynamics. For the two-dimensional cylinder, three-dimensional dimer, diamond, and biclique instances considered, at annealing times of 7 and 20 ns, we show that systematically increasing the variational ansatz size enables t-VMC to approximate the final two-spin correlation errors of the quantum processing unit (QPU). We also perform an accurate large-scale simulation of a challenging biclique instance for which no other variational method is known to work, and we find close agreement with the quantum annealer. Through ablation studies, we identify poor Markov-chain mixing, high-variance local-energy estimators, and stochastic Runge-Kutta error estimates as the principal numerical failure modes. We address these numerical issues by using parallel tempering, blurred sampling and an importance-weighted differential equation solver, thereby clarifying the numerical requirements for stable, large-scale t-VMC simulations. Our results extend the frontier of classical simulation while providing a realistic assessment of the computational costs of simulating quantum dynamics at this scale.
Abstract : Many computationally challenging problems can be reduced to Quadratic Unconstrained Binary Optimization (QUBO), which can be solved by a quantum evolution from a strong transverse field to a spin glass Hamiltonian (also known as quantum annealing or QA). We have examined open system QA, using theoretical and experimental techniques to deepen our understanding of open system QA by situating it in the context of related, well studied classical and quantum problems. We have explored and elucidated the significance of tunneling in QA with Path Integral Monte Carlo. We have also provided signatures of quantum behavior of QA against closely related simulated annealing implementations, analytically, numerically, and experimentally. We have carried out a comprehensive comparison of geometrically-local open-system QA and classical solvers for computationally hard problems such as MAX-2-SAT. We have exploited and further developed a graph-theoretical mapping between the Ising spin glass partition function and circuit model decision problems, discovered in a previous ARO Quantum Algorithms funded project. We have made significant progress on error correction techniques for QA, both theoretically and experimentally, and in demonstration of quantum speedup.