Heavy Tails in the Distribution of Time to Solution for Classical and Quantum Annealing

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Heavy Tails in the Distribution of Time to Solution for Classical and Quantum Annealing^*

Damian S. Steiger, Troels F. Rønnow, and Matthias Troyer

Phys. Rev. Lett. 115, 230501 – Published 4 December 2015

Abstract

For many optimization algorithms the time to solution depends not only on the problem size but also on the specific problem instance and may vary by many orders of magnitude. It is then necessary to investigate the full distribution and especially its tail. Here, we analyze the distributions of annealing times for simulated annealing and simulated quantum annealing (by path integral quantum Monte Carlo simulation) for random Ising spin glass instances. We find power-law distributions with very heavy tails, corresponding to extremely hard instances, but far broader distributions—and thus worse performance for hard instances—for simulated quantum annealing than for simulated annealing. Fast, nonadiabatic, annealing schedules can improve the performance of simulated quantum annealing for very hard instances by many orders of magnitude.

Received 21 May 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.230501

^*The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright annotation thereon.

Authors & Affiliations

Damian S. Steiger¹, Troels F. Rønnow^1,2, and Matthias Troyer¹

¹Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland
²Nokia Technologies, Broers Building, 21 JJ Thomson Avenue, CB3 0FA Cambridge, United Kingdom

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Vol. 115, Iss. 23 — 4 December 2015

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Images

Figure 1
Scaling of the annealing time to find the ground state with 99% probability for a bimodal Ising spin glass with couplings $\pm 1$ on chimera graphs with $N$ spins using (a) simulated annealing (SA), (b) simulated quantum annealing (SQA), and (c) a D-Wave Two device (DW2). Results (with standard errors) are taken from Ref. [19]. We observe an enormous spread in the scaling of different percentiles, ranging from easy (1%) to hard (99%) problem instances.
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Figure 2
Distribution of mean number of repetitions $τ$ required to find the ground state for 20 000 problems with 200 spins on chimera topology for SA and SQA (with $β = 10$ ). The tails of these distribution functions are decaying polynomially slowly and there is a qualitative difference between SA and SQA: for SA the tail is decaying faster and the overall distribution function is more compact.
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Figure 3
Generalized Pareto probability density function $w_{ξ, 0, σ} (x) = (d / d x) W_{ξ, 0, σ} (x)$ plotted on a linear-log scale. The qualitative tail behavior depends on the shape parameter $ξ$ . For $ξ < 0$ the probability density is bounded, for $ξ = 0$ it decays exponentially, and for $ξ > 0$ the tail is heavy and decays as a power law.
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Figure 4
Distribution of mean number of repetitions $τ$ required to find the ground state for 20 000 problem instances with $N = 200$ spins for (a) SQA with $β = 10$ and (b) SA with either the optimal number or $10^{4}$ sweeps. While the distribution changes only slightly for SA, surprisingly the tails in SQA decrease much more rapidly when annealing faster.
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Figure 5
Correlation plot of (a) SQA with $β = 10$ and (b) SA for all 20 000 instances with $N = 200$ spins using $t_{a}^{opt}$ and $t_{a} = 10^{4}$ sweeps, respectively. Points below the white line ( $τ$ , $τ$ ) are those where the success probability $s$ increases upon annealing faster. This is the case for 2312 instances for SQA but only 11 for SA. This increase in $s$ for the hard instances explains the more rapid decrease of the tail in SQA. The gray line indicates points with equal total effort $t_{a} τ$ . Instances below the gray line (all but one for SQA and all but 51 for SA) are those that get solved more efficiently with a faster annealing time.
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Figure 6
Shown are the fitted shape parameters $ξ$ with standard errors for SA, MFA, and SQA. The different versions of SQA have different inverse temperature $β$ and $t_{a}$ , with $t_{a}^{opt} (N)$ varying from 20 to 400 sweeps depending on $N$ and $β$ . For $ξ > 0$ (indicated by a dashed line) the tail of the distribution function is heavy and for $ξ \geq 1$ (indicated by a dotted line) the mean diverges. The $k$ th moment diverges for $k \geq 1 / ξ$ .
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