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New polynomially exact integration rules on U(N) and SU(N)

In lattice Quantum Field Theory, we are often presented with integrals over polynomials of coefficients of matrices in U(N) or SU(N) with respect to the Haar measure. In some physical situations, e.g., in presence of a chemical potential, these integrals are numerically very difficult since their integrands are highly oscillatory which manifests itself in form of the sign problem. In these cases, Monte Carlo methods often fail to be adequate, rendering such computations practically impossible. We propose a new class of integration rules on U(N) and SU(N) which are derived from polynomially exact rules on spheres. We will examine these quadrature rules and their efficiency at the example of a 0+1 dimensional QCD for a non-zero quark mass and chemical potential. In particular, we will demonstrate the failure of Monte Carlo methods in such applications and that we can obtain polynomially exact, arbitrary precision results using the new integration rules.