01790nas a2200133   4500000000100000008004100001260001200042653010900054100001900163700002200182245008500204856003700289520133000326       2026                            d  c06/202610aSymbolic integration; Haar measure; Weingarten calculus; Julia programming language; quantum information1 aŁukasz Pawela1 aZbigniew Puchała00aIntegrateUnitary.jl: A Julia package for symbolic integration over Haar measures  uhttps://arxiv.org/abs/2605.238303 aSymbolic integration over the Haar measure of compact groups is a computational cornerstone in quantum information science and random matrix theory. We present IntegrateUnitary.jl, a comprehensive Julia package for computing exact expectations of polynomial functions over a wide range of compact groups (U(d), O(d), Sp(d), and SU(d) for balanced polynomials), circular and Gaussian ensembles, Ginibre ensembles, permutation groups, random pure states, and unitary t-designs. The package provides a fully open-source implementation of the Weingarten calculus and Wick contractions with broad symbolic-d support for entry-wise and trace-polynomial integrals, while selected workflows currently require concrete integer dimensions (including higher pure trace moments |tr(U)|^2k for k > 1 and HCIZ with SymbolicMatrix inputs, and direct matrix-valued integration of SymbolicMatrix/SymbolicMatrixProduct expressions), automatic asymptotic expansions, a high-level symbolic trace interface that reconstructs Weingarten graphs from index-free expressions, and a bridge to ITensors.jl for tensor network averaging. We discuss the underlying algorithms, including the Murnaghan-Nakayama rule and symplectic-orthogonal duality, and demonstrate that the package efficiently handles high-degree moments and quantum information metrics.