## Publication: Practical engineering of hard spin-glass instances

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2016-07-14

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American Physical Society

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Recent technological developments in the field of experimental quantum annealing have made prototypical annealing optimizers with hundreds of qubits commercially available. The experimental demonstration of a quantum speedup for optimization problems has since then become a coveted, albeit elusive goal. Recent studies have shown that the so far inconclusive results, regarding a quantum enhancement, may have been partly due to the benchmark problems used being unsuitable. In particular, these problems had inherently too simple a structure, allowing for both traditional resources and quantum annealers to solve them with no special efforts. The need therefore has arisen for the generation of harder benchmarks which would hopefully possess the discriminative power to separate classical scaling of performance with size from quantum. We introduce here a practical technique for the engineering of extremely hard spin-glass Ising-type problem instances that does not require "cherry picking" from large ensembles of randomly generated instances. We accomplish this by treating the generation of hard optimization problems itself as an optimization problem, for which we offer a heuristic algorithm that solves it. We demonstrate the genuine thermal hardness of our generated instances by examining them thermodynamically and analyzing their energy landscapes, as well as by testing the performance of various state-of-the-art algorithms on them. We argue that a proper characterization of the generated instances offers a practical, efficient way to properly benchmark experimental quantum annealers, as well as any other optimization algorithm.

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©2016 American Physical Society.
We thank Ehsan Khatami for useful comments and suggestions. This work was partially supported by MINECO (Spain) through Grants No. FIS2012-35719-C02, No. FIS2015-65078-C2-1-P. Computation for the work described in this paper was partially supported by the University of Southern California's Center for High-Performance Computing (http://hpcc.usc.edu).

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For the HFS algorithm, the most natural choice of TTS, so as to quantify the computational difficulty, is simply the physical, average solve time of an instance; due to the nature of the algorithm, it does not easily lend itself to a “universal,” system independent measure of TTS (e.g., number of algorithmic steps). The reader is referred to Refs. [16] and [51] for more details.
If one were to use a solver which suffers from intrinsic control errors (e.g., an analog device, such as the DW annealers), i.e., encoding errors in the Jij, one may have to perform some kind of averaging procedure to try to estimate the TTS more accurately (e.g., in the DW case, running over several different gauges [15]). The performance of our algorithm will of course be adversely affected in such a case.
To generate the hardest instances, k=4, we ran our algorithm 780 times with up to 1000 steps (that is, at least 100 of the 780 instances fell within the time interval [0.8,1.2] s). The total CPU time this took was approximately 400 h (though note that our code was not optimized). Also note that of these 780, about 120 have tHFS>1.2 s. The other k groups took significantly less time to generate.
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