Tunneling in projective quantum Monte Carlo simulations with guiding wave functions

TL;DR

This paper investigates how guiding wave functions influence tunneling rates in projective quantum Monte Carlo simulations across various models, revealing consistent linear scaling and implications for quantum optimization benchmarking.

Contribution

It demonstrates that guiding wave functions do not alter the linear scaling of tunneling rates in PQMC, supported by theoretical analysis and multiple model tests.

Findings

Tunneling rate scales linearly with energy gap in PQMC with guiding functions.

Guiding wave functions do not change the asymptotic tunneling scaling.

Semiclassical theory explains the observed linear scaling.

Abstract

Quantum tunneling is a valuable resource exploited by quantum annealers to solve complex optimization problems. Tunneling events also occur during projective quantum Monte Carlo (PQMC) simulations, and in a class of problems characterized by a double-well energy landscape their rate was found to scale linearly with the first energy gap, i.e., even more favorably than in physical quantum annealers, where the rate scales with the gap squared. Here we investigate how a guiding wave function --- which is essential to make many-body PQMC simulations computationally feasible --- affects the tunneling rate. The chosen testbeds are a continuous-space double-well problem, the ferromagnetic quantum Ising chain, and the recently introduced shamrock model. As guiding wave function, we consider an approximate Boltzmann-type ansatz, the numerically-exact ground state of the double-well model, and a neural-network wave function based on a Boltzmann machine. Remarkably, for each ansatz we find the same asymptotic linear scaling of the tunneling rate that was previously found in the PQMC simulations performed without a guiding wave function. We also provide a semiclassical theory for the double-well with exact guiding wave function that explains the observed linear scaling. These findings suggest that PQMC simulations guided by an accurate ansatz represent a valuable benchmark for physical quantum annealers and a potentially competitive quantum-inspired optimization technique.