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arXiv:0810.0557 · Journal of Physics Conference Series

Re'class'ification of 'quant'ified classical simulated annealing

Toshiyuki Tanaka

Published Jan 7, 2009• 0 citations
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Abstract

We discuss a classical reinterpretation of quantum-mechanics-based analysis of classical Markov chains with detailed balance, that is based on the quantum-classical correspondence. The classical reinterpretation is then used to demonstrate that it successfully reproduces a sufficient condition for cooling schedule in classical simulated annealing, which has the inverse-logarithmic scaling.

DOI
10.1088/1742-6596/143/1/012020

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Quantum computers use quantum resources to carry out computational tasks and may outperform classical computers in solving certain computational problems. Special-purpose quantum computers such as quantum annealers employ quantum adiabatic theorem to solve combinatorial optimization problems. In this paper, we compare classical annealings such as simulated annealing and quantum annealings that are done by the D-Wave machines both theoretically and numerically. We show that if the classical and quantum annealing are characterized by equivalent Ising models, then solving an optimization problem, i.e., finding the minimal energy of each Ising model, by the two annealing procedures, are mathematically identical. For quantum annealing, we also derive the probability lower-bound on successfully solving an optimization problem by measuring the system at the end of the annealing procedure. Moreover, we present the Markov chain Monte Carlo (MCMC) method to realize quantum annealing by classical computers and investigate its statistical properties. In the numerical section, we discuss the discrepancies between the MCMC based annealing approaches and the quantum annealing approach in solving optimization problems.
Annealing schedule control provides opportunities to better understand the manner and mechanisms by which putative quantum annealers operate. By appropriately modifying the annealing schedule to include a pause (keeping the Hamiltonian fixed) for a period of time, we show that it is possible to more directly probe the dissipative dynamics of the system at intermediate points along the anneal and examine thermal relaxation rates, for example, by observing the repopulation of the ground state after the minimum spectral gap. We provide a detailed comparison of experiments from a D-Wave device, simulations of the quantum adiabatic master equation, and a classical analogue of quantum annealing, spin-vector Monte Carlo, and we observe qualitative agreement, showing that the characteristic increase in success probability when pausing is not a uniquely quantum phenomena. We find that the relaxation in our system is dominated by a single timescale, which allows us to give a simple condition for when we can expect pausing to improve the time to solution, the relevant metric for classical optimization. Finally, we also explore in simulation the role of temperature whilst pausing as a means to better distinguish quantum and classical models of quantum annealers.
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.
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