Direct comparison of quantum and simulated annealing on a fully-connected Ising ferromagnet

TL;DR

This paper compares quantum annealing and simulated annealing on a fully-connected Ising ferromagnet, revealing that SA outperforms QA at second-order transitions, while QA shows a residual energy decrease at first-order transitions, indicating limited quantum speedup.

Contribution

It provides the first direct comparison of QA and SA on the same model across different phase transition orders, demonstrating conditions where QA offers a quantum speedup.

Findings

SA outperforms QA at second-order transitions.

QA shows residual energy decrease at first-order transitions.

Limited quantum speedup observed in specific regimes.

Abstract

We compare the performance of quantum annealing (QA, through Schr\"odinger dynamics) and simulated annealing (SA, through a classical master equation) on the $p$-spin infinite range ferromagnetic Ising model, by slowly driving the system across its equilibrium, quantum or classical, phase transition. When the phase transition is second-order ($p=2$, the familiar two-spin Ising interaction) SA shows a remarkable exponential speed-up over QA. For a first-order phase transition ($p \ge 3$, i.e., with multi-spin Ising interactions), in contrast, the classical annealing dynamics appears to remain stuck in the disordered phase, while we have clear evidence that QA shows a residual energy which decreases towards $0$ when the total annealing time $\tau$ increases, albeit in a rather slow (logarithmic) fashion. This is one of the rare examples where a limited quantum speedup, a speedup by QA over SA, has been shown to exist by direct solutions of the Schr\"odinger and master equations in combination with a non-equilibrium Landau-Zener analysis. We also analyse the imaginary-time QA dynamics of the model, finding a $1/\tau^2$ behaviour for all finite values of $p$, as predicted by the adiabatic theorem of quantum mechanics. The Grover-search limit $p({\rm odd})=\infty$ is also discussed.