Energy gap of quantum spin glasses: a projection quantum Monte Carlo study

TL;DR

This study investigates the energy gap scaling in quantum spin glasses using a novel quantum Monte Carlo method, revealing universal and model-specific behaviors that impact quantum annealing efficiency.

Contribution

It introduces an unbiased energy-gap estimator for continuous-time projection quantum Monte Carlo and applies it to analyze gap distributions in 2D-EA and SK models.

Findings

In 2D-EA, the inverse-gap distribution develops a fat tail with infinite variance as system size increases.

The SK model retains a finite-variance gap distribution with a slow power-law decay, approximately   N^{-1/3}.

Results suggest potential efficiency of quantum annealers for dense connectivity optimization problems.

Abstract

The performance of quantum annealing for combinatorial optimization is fundamentally limited by the minimum energy gap $\Delta$ encountered at quantum phase transitions. We investigate the scaling of $\Delta$ with system size $N$ for two paradigmatic quantum spin-glass models: the two-dimensional Edwards-Anderson (2D-EA) and the all-to-all Sherrington-Kirkpatrick (SK) models. Utilizing a newly proposed unbiased energy-gap estimator for continuous-time projection quantum Monte Carlo simulations, complemented by high-performance sparse eigenvalue solvers, we characterize the gap distributions across disorder realizations. It is found that, in the 2D-EA case, the inverse-gap distribution develops a fat tail with infinite variance as $N$ increases. This indicates that the unfavorable super-algebraic scaling of $\Delta$, recently reported for binary couplings [Nature 631, 749 (2024)], persists for the Gaussian disorder considered here, pointing to a universal feature of 2D spin glasses. Conversely, the SK model retains a finite-variance distribution, with the disorder-averaged gap following a rather slow power law, close to $\Delta \propto N^{-1/3}$. This finding provides a promising outlook for the potential efficiency of quantum annealers for optimization problems with dense connectivity.