We apply the Quasi Monte Carlo (QMC) and recursive numerical integration methods to evaluate the Euclidean, discretized time path-integral for the quantum mechanical anharmonic oscillator and a topological quantum mechanical rotor model. For the anharmonic oscillator both methods outperform standard Markov Chain Monte Carlo methods and show a significantly improved error scaling. For the quantum mechanical rotor we could, however, not find a successful way employing QMC. On the other hand, the recursive numerical integration method works extremely well for this model and shows an at least exponentially fast error scaling.

a b s t r a c tWe investigate the applicability of quasi-Monte Carlo methods to Euclidean lattice systems for quantum mechanics in order to improve the asymptotic error behavior of observables for such theories. In most cases the error of an observable calculated by averaging over random observations generated from an ordinary Markov chain Monte Carlo simulation behaves like N −1/2 , where N is the number of observations. By means of quasi-Monte Carlo methods it is possible to improve this behavior 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. Programming language: C and C++. Program summary Computer: PC.Operating system: Tested on GNU/Linux, should be portable to other operating systems with minimal efforts. Has the code been vectorized or parallelized?: No RAM:The memory usage directly scales with the number of samples and dimensions: Bytes used = ''number of samples'' × ''number of dimensions'' × 8 Bytes (double precision).Classification: 4.13, 11.5, 23.External routines: FFTW 3 library (http://www.fftw.org) Nature of problem:Certain physical models formulated as a quantum field theory through the Feynman path integral, such as quantum chromodynamics, require a non-perturbative treatment of the path integral. The only known approach that achieves this is the lattice regularization. In this formulation the path integral is discretized to a finite, but very high dimensional integral. So far only Monte Carlo, and especially Markov chain-Monte Carlo methods like the Metropolis or the hybrid Monte Carlo algorithm have been used to calculate approximate solutions of the path integral. These algorithms often lead to the undesired effect of autocorrelation in the samples of observables and suffer in any case from the slow asymptotic error behavior proportional to N −1/2 , if N is the number of samples. Solution method:This program applies the quasi-Monte Carlo approach and the reweighting technique (respectively the weighted uniform sampling method) to generate uncorrelated samples of observables of the anharmonic oscillator with an improved asymptotic error behavior. Unusual features:The application of the quasi-Monte Carlo approach is quite revolutionary in the field of lattice field theories. Running time:The running time depends directly on the number of samples N and dimensions d. On modern computers a run with up to N = 2 16 = 65536 (including 9 replica runs) and d = 100 should not take much longer than one minute.

In this paper we describe a new integration method for the groups UðNÞ and SUðNÞ, for which we verified numerically that it is polynomially exact for N ≤ 3. The method is applied to the example of onedimensional QCD with a chemical potential. We explore, in particular, regions of the parameter space in which the sign problem appears due the presence of the chemical potential. While Markov chain Monte Carlo fails in this region, our new integration method still provides results for the chiral condensate on arbitrary precision, demonstrating clearly that it overcomes the sign problem. Furthermore, we demonstrate that also in other regions of parameter space our new method leads to errors which are reduced by orders of magnitude.

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.

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