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

TL;DR

The paper investigates how the minimum energy gap $Δ$ scales with system size $N$ at spin-glass quantum phase transitions, a key factor in quantum annealing performance. It introduces an unbiased, parity-resolved PQMC gap estimator based on imaginary-time correlations, robust to the choice of guiding function, and validates it against sparse-eigenvalue solvers for both 2D-EA with Gaussian couplings and the all-to-all SK model. The results reveal a fat-tailed distribution of $η=1/Δ$ with infinite variance for the 2D-EA Gaussian case, indicating super-algebraic gap scaling, while the SK model shows a finite-variance $η$ and a power-law gap scaling with $[Δ] ∝ N^{-0.32}$. These findings suggest a qualitative advantage for quantum annealing in densely connected problems and establish the PQMC gap estimator as a robust tool for spectral properties in quantum many-body systems.

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 super-algebraic scaling of $Δ$, 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 $Δ\propto N^{-1/3}$. This finding provides a promising outlook for the potential efficiency of quantum annealers for optimization problems with dense connectivity.

Figures (12)

• Figure 1: Complementary empirical distribution function $1-F(\eta)$ of the inverse odd gap $\eta=1/\Delta$ of the Gaussian 2D-EA model at the spin-glass quantum phase transition. The various symbols correspond to the lattice sizes $L=\sqrt{N}$ reported in the legend. The three vertical lines indicate the smallest $\eta$ accounted for in the Hill estimator for $L=11$, for three sample fractions: $\kappa=0.8$, $\kappa=0.3$, and $\kappa=0.15$ (from left to right).
• Figure 2: Hill estimator for the tail index $\alpha$ (see Eq. \ref{['hill']}) for the Gaussian 2D-EA Hamiltonian as a function of the lattice size $L$. The three symbols correspond to the sampled fractions $\kappa$ reported in the legend. The continuous curves represent power-law fitting functions. The error bars are determined via a bootstrap method with $N_{\mathrm{boot}}=2000$ resamples. The arrows on the right point to the extrapolation $L\rightarrow \infty$. The two horizontal straight lines indicate the thresholds for finite mean and variance.
• Figure 3: Complementary empirical distribution function $1-F(\eta)$ for the SK model. Symbols and lines are defined as in Fig. \ref{['fig1']}.
• Figure 4: Hill estimator $\alpha$ as a function of the number of spins $N$ for the SK Hamiltonian. Symbols and lines are defined as in Fig. \ref{['fig3']}.
• Figure 5: Main panel: Odd energy gap $\Delta_o$, averaged over disorder realizations, as a function of the number of spins $N$, for the SK model. The black line represents a power-law fit $\Delta = C / N^\theta$, providing $C=3.06(3)$ and $\theta=0.32(1)$. Inset: Scaling exponent $\theta$ as a function of the transverse field $\Gamma$ for the odd gap $\Delta_o$ and the even gap $\Delta_e$. The dashed vertical line shows the position of the critical point.