Reaching states below the threshold energy in spin glasses via quantum annealing

TL;DR

This paper demonstrates that quantum annealing can efficiently find low-energy states below the threshold in spin glasses, outperforming classical methods in speed and energy minimization, with results derived from large-system limit equations.

Contribution

It shows quantum annealing's ability to reach sub-threshold states faster than classical algorithms in mean-field spin-glass models, using a novel analytical approach.

Findings

Quantum annealing finds states below the threshold energy in O(1) time.

Residual energy decays as a power law with a larger exponent than simulated annealing.

Results are derived from equations valid in the thermodynamic limit, free from finite-size effects.

Abstract

Although quantum annealing is usually considered as a method for locating the ground states of difficult spin-glass and optimization problems, its use in approximate optimization -- finding low- but not zero-energy states in a reasonably short amount of time -- is no less important. Here we investigate the behavior of quantum annealing at approximate optimization in the canonical mean-field spin-glass models, the spherical $p$-spin models, and find that it performs surprisingly well. Whereas it had long been assumed that infinite-range spin glasses have a unique ``threshold'' energy at which all quench and annealing dynamics become trapped until exponential timescales, recent work has shown that two-stage quenches can in fact reach states below the naive threshold in more generic situations. We demonstrate that quantum annealing is also capable of exploiting this effect to locate sub-threshold states in $O(1)$ time. Not only can it attain energies as far below the threshold as classical annealing algorithms, but it can do so significantly faster: for an annealing schedule taking time $\tau$, the residual energy under quantum annealing decays as $\tau^{-\alpha}$ with an exponent up to twice as large as that of simulated annealing in the cases considered. Importantly, by deriving and numerically solving closed integro-differential equations that hold in the thermodynamic limit, our results are free from finite-size effects and hold for annealing times that are unambiguously independent of system size.