Understanding Quantum Tunneling using Diffusion Monte Carlo Simulations

TL;DR

This paper demonstrates that diffusion Monte Carlo simulations can simulate quantum tunneling more efficiently than incoherent quantum tunneling in certain models, but still face exponential scaling issues with system size.

Contribution

It shows that DMC simulations scale as 1/Δ for tunneling time, outperforming PIMC, but still encounter exponential cost growth for larger systems.

Findings

DMC scales as 1/Δ in simple ferromagnetic models.

DMC scales as 1/Δ in shamrock models with topological obstructions.

Ground-state energy estimation with DMC exhibits exponential cost scaling.

Abstract

In simple ferromagnetic quantum Ising models characterized by an effective double-well energy landscape the characteristic tunneling time of path-integral Monte Carlo (PIMC) simulations has been shown to scale as the incoherent quantum-tunneling time, i.e., as $1/\Delta^2$, where $\Delta$ is the tunneling gap. Since incoherent quantum tunneling is employed by quantum annealers (QAs) to solve optimization problems, this result suggests there is no quantum advantage in using QAs w.r.t. quantum Monte Carlo (QMC) simulations. A counterexample is the recently introduced shamrock model, where topological obstructions cause an exponential slowdown of the PIMC tunneling dynamics with respect to incoherent quantum tunneling, leaving the door open for potential quantum speedup, even for stoquastic models. In this work, we investigate the tunneling time of projective QMC simulations based on the diffusion Monte Carlo (DMC) algorithm without guiding functions, showing that it scales as $1/\Delta$, i.e., even more favorably than the incoherent quantum-tunneling time, both in a simple ferromagnetic system and in the more challenging shamrock model. However a careful comparison between the DMC ground-state energies and the exact solution available for the transverse-field Ising chain points at an exponential scaling of the computational cost required to keep a fixed relative error as the system size increases.