  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.
