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

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 $τ$, the residual energy under quantum annealing decays as $τ^{-α}$ 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.

Figures (3)

• Figure 1: Average energy density $\epsilon(\tau)$ at the end of various protocols as a function of protocol runtime $\tau$, for the pure model $f[Q] = Q^3$. Each set of points is fit to a power-law decay (Eq. \ref{['eq:power_law_decay']}), with the fitted curve shown as the solid colored line. The large-$\tau$ limit of the fitted curve is indicated by the dashed colored line, and the solid black line indicates the threshold energy $\epsilon_{\textrm{th}}$ (Eq. \ref{['eq:mixed_threshold_energy']}). Calculations for all protocols use $\Delta t = 0.1$.
• Figure 2: Average energy density $\epsilon(\tau)$ at the end of various protocols as a function of protocol runtime $\tau$, for the mixed model $f[Q] = Q^3 + Q^{14}$. Each set of points is fit to a power-law decay (Eq. \ref{['eq:power_law_decay']}), with the fitted curve shown as the solid colored line. The large-$\tau$ limit of the fitted curve is indicated by the dashed colored line, and the solid black line indicates the threshold energy $\epsilon_{\textrm{th}}$ (Eq. \ref{['eq:mixed_threshold_energy']}). Calculations for the quench and two-stage quench use $\Delta t = 0.02$, while those for the anneals use $\Delta t = 0.04$.
• Figure 3: Fitted exponents $\alpha$ (Eq. \ref{['eq:power_law_decay']}) for various protocols in the mixed model $f[Q] = Q^3 + Q^p$, as a function of $p$. Calculations for the quench and two-stage quench use timestep $\Delta t = 0.02$, while those for the anneals use $\Delta t = 0.04$.