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  • Open Access

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

L. Brodoloni1,2, G. E. Astrakharchik3, S. Giorgini4, and S. Pilati1,2

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Phys. Rev. Lett. 137, 060403 – Published 3 August, 2026

DOI: https://doi.org/10.1103/fm8m-kz13

Abstract

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)], 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.

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Phys. Rev. Lett. 137, 060403– Published 3 August, 2026
  • Received 27 February 2026
  • Revised 18 June 2026
  • Accepted 13 July 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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