On the dynamics of Simulated Quantum Annealing in random Ising chains

TL;DR

This paper investigates the dynamics of Simulated Quantum Annealing in random Ising chains, highlighting how disorder causes sampling issues that affect the accuracy of SQA compared to true quantum annealing.

Contribution

It analyzes the effects of Trotter discretization and disorder on SQA's ability to replicate quantum annealing dynamics in one-dimensional Ising chains.

Findings

Proper continuum limit recovers correct Kibble-Zurek scaling in ordered chains

Disorder induces sampling crises in low-temperature phases

SQA results diverge from Schrödinger QA due to sampling issues

Abstract

Simulated Quantum Annealing (SQA), that is emulating a Quantum Annealing (QA) dynamics on a classical computer by a Quantum Monte Carlo whose parameters are changed during the simulation, is a well established computational strategy to cope with the exponentially large Hilbert space. It has enjoyed some early successes but has also raised more recent criticisms. Here we investigate, on the paradigmatic case of a one-dimensional transverse field Ising chain, two issues related to SQA in its Path-Integral implementation: the question of Monte Carlo vs physical (Schr\"odinger) dynamics and the issue of the imaginary-time continuum limit to eliminate the Trotter error. We show that, while a proper time-continuum limit is able to restitute the correct Kibble-Zurek scaling of the residual energy $\varepsilon_\mathrm{res}(\tau)\sim \tau^{-1/2}$ for the ordered case --- $\tau$ being the total annealing time ---, the presence of disorder leads to a characteristic sampling crisis for a large number of Trotter time-slices, in the low-temperature ordered phase. Such sampling problem, in turn, leads to SQA results which are apparently unrelated to the coherent Schr\"odinger QA even at intermediate $\tau$.