Multilevel Monte Carlo for quantum mechanics on a lattice

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 with decreasing lattice spacing. 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 this 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 an observable with lattice errors of order $\alpha$ and an integrated autocorrelation time that grows like $\tau_{\mathrm{int}}\propto a^{-z}$, multilevel Monte Carlo (MLMC) can reduce the cost from $\mathcal{O}(\epsilon_{\text{stat}}^{-2}\epsilon_{\text{disc}}^{-(1+z)/\alpha})$ to $\mathcal{O}(\epsilon_{\text{stat}}^{-2}\vert\log \epsilon_{\text{disc}} \vert^2+\epsilon_{\text{disc}}^{-1/\alpha})$. Even higher performance 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 due to freezing of the topological charge. On fine lattices, the new methods are orders of magnitude faster than standard sampling based on Hybrid Monte Carlo. 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.