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.
Highlights
The Euclidean path integral formulation of quantum mechanics [1] allows the calculation of observable quantities as expectation values with respect to infinite-dimensional and highly peaked probability distributions
While this paper focuses on the application of these new methods in quantum mechanics, the ultimate goal is to apply them in D-dimensional quantum field theories, such as lattice quantum chromodynamics (QCD) with D 1⁄4 4 and α 1⁄4 2
We have described a hierarchical sampling algorithm and applied it for simulations in quantum mechanics
Summary
The Euclidean path integral formulation of quantum mechanics [1] allows the calculation of observable quantities as expectation values with respect to infinite-dimensional and highly peaked probability distributions. After discretizing the theory on a lattice with finite spacing a, expectation values are computed with Markov Chain Monte Carlo methods (see e.g., [2] for a highly accessible introduction). State-of-the-art techniques [5] are routinely used to accelerate the MetropolisHastings algorithm [6,7] and in particular the hybrid Monte Carlo (HMC) method [8] has proved to be highly successful in lattice QCD simulations. Lattice calculations with HMC methods still become prohibitively expensive as the continuum limit is approached. The reasons for this are twofold: 2470-0010=2020=102(11)=114512(23)
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
