Energy Gap of Quantum Spin Glasses: A Projection Quantum Monte Carlo Study

The performance of quantum annealing for combinatorial optimization is fundamentally limited by the minimum energy gap Δ encountered at quantum phase transitions. We investigate the scaling of Δ 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 superalgebraic scaling of Δ, recently reported for binary couplings [M. Bernaschi et al., The quantum transition of the two-dimensional Ising spin glass, Nature (London) 631, 749 (2024)NATUAS0028-083610.1038/s41586-024-07647-y], 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 Δ∝N^{-1/3}. This finding provides a promising outlook for the potential efficiency of quantum annealers for optimization problems with dense connectivity.