Improving Monte Carlo integration by symmetrization

The error scaling for Markov chain Monte Carlo (MCMC) techniques with N samples behaves like 1/√N. This scaling makes it often very time intensive to reduce the error of calculated observables, in particular for applications in 4-dimensional lattice quantum chromodynamics as our theory of the interaction between quarks and gluons. Even more, for certain cases, where the infamous sign problem appears, MCMC methods fail to provide results with a reliable error estimate. It is therefore highly desirable to have alternative methods at hand which show an improved error scaling and have the potential to overcome the sign problem. One candidate for such an alternative integration technique we used is based on a new class of polynomially exact integration rules on U(N) and SU(N) which are derived from polynomially exact rules on spheres. We applied these rules successfully to a non-trivial, zero-dimensional model with a sign problem and obtained arbitrary precision results. In this article we test a possible way to apply the integration rules for spheres to the case of a one-dimensional U(1) model, the topological rotor, which already leads to a problem of very high dimensionality.