Applicability of Quasi-Monte Carlo for lattice systems

Abstract

This project investigates the applicability of quasi-Monte Carlo methods to Euclidean lattice systems in order to improve the asymptotic error scaling of observables for such theories. The error of an observable calculated by averaging over random observations generated from ordinary Monte Carlo simulations scales like N −1/2 , where N is the number of observations. By means of quasi-Monte Carlo methods it is possible to improve this scaling for certain problems to N −1 , or even further if the problems are regular enough. We adapted and applied this approach to simple systems like the quantum harmonic and anharmonic oscillator and verified an improved error scaling of all investigated observables in both cases.

On the efficient numerical solution of lattice systems with low-order couplings

“…The harmonic and anharmonic quantum mechanical oscillators have been studied with QMC methods in [6,7,8].…”

“…Applications of QMC methods to the anharmonic oscillator model and comparison with MCMC techniques have been studied in [6,7,8]. In particular, the superiority of randomized QMC techniques applied to the anharmonic oscillator model has been observed for small time periods T ≤ 1.5 independently of the size of lattice spacing a.…”

“…In [6,7,8] we initiated a test of QMC methods for the harmonic and anharmonic quantum mechanical oscillator discretized on a Euclidean time lattice; see the next section for an introduction to these systems. The results of these investigations have been very promising.…”

“…As demonstrated in refs. [6,7,8] with QMC using the harmonic approximation this problem is completely overcome. This is not surprising since QMC avoids long autocorrelation times essentially by construction.…”

“…In this paper, we want to extend the work of refs. [6,7,8] in two directions. One direction is to apply recursive numerical integration methods for the anharmonic oscillator.…”