Multilevel Monte Carlo for quantum mechanics on a lattice

TL;DR

This paper introduces multilevel Monte Carlo methods combined with hierarchical sampling to significantly reduce computational costs in lattice quantum field theory simulations, especially near the continuum limit.

Contribution

It develops a multilevel variance reduction technique tailored for quantum lattice simulations, addressing critical slowdown and autocorrelation issues.

Findings

Reduces simulation cost from polynomial to nearly logarithmic in inverse lattice spacing.

Demonstrates orders of magnitude speedup over standard methods on model systems.

Shows effectiveness in accelerating cluster algorithms for topological oscillators.

Abstract

Monte Carlo simulations of quantum field theories on a lattice become increasingly expensive as the continuum limit is approached since the cost per independent sample grows with a high power of the inverse lattice spacing. Simulations on fine lattices suffer from critical slowdown, the rapid growth of autocorrelations in the Markov chain. This causes a strong increase in the number of lattice configurations that have to be generated to obtain statistically significant results. This paper discusses hierarchical sampling methods to tame the growth in autocorrelations. Combined with multilevel variance reduction, this significantly reduces the computational cost of simulations for given tolerances $\epsilon_{\text{disc}}$ on the discretisation error and $\epsilon_{\text{stat}}$ on the statistical error. For observables with lattice errors of order $\alpha$ and integrated autocorrelation times that grow like $\tau_{\mathrm{int}}\propto a^{-z}$, multilevel Monte Carlo (MLMC) reduces the cost from $\mathcal{O}(\epsilon_{\text{stat}}^{-2}\epsilon_{\text{disc}}^{-(1+z)/\alpha})$ to $\mathcal{O}(\epsilon_{\text{stat}}^{-2}\vert\log \epsilon_{\text{disc}} \vert^2+\epsilon_{\text{disc}}^{-1/\alpha})$ or $\mathcal{O}(\epsilon_{\text{stat}}^{-2}+\epsilon_{\text{disc}}^{-1/\alpha})$. Higher gains are expected for simulations of quantum field theories in $D$ dimensions. The efficiency of the approach is demonstrated on two model systems, including a topological oscillator that is badly affected by critical slowdown from topological charge freezing. On fine lattices, the new methods are orders of magnitude faster than standard Hybrid Monte Carlo sampling. For high resolutions, MLMC can be used to accelerate even the cluster algorithm for the topological oscillator. Performance is further improved through perturbative matching which guarantees efficient coupling of theories on the multilevel hierarchy.