IntegrateUnitary.jl: A Julia package for symbolic integration over Haar measures

Łukasz Pawela, Zbigniew Puchała (May 25 2026).

Abstract: Symbolic integration over the Haar measure of compact groups is a computational cornerstone in quantum information science and random matrix theory. We present \textttIntegrateUnitary.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 $|\mathrm{tr}(U)|^{2k}$ for $k > 1$ and HCIZ with \textttSymbolicMatrix inputs, and direct matrix-valued integration of \textttSymbolicMatrix/\textttSymbolicMatrixProduct expressions), automatic asymptotic expansions, a high-level symbolic trace interface that reconstructs Weingarten graphs from index-free expressions, and a bridge to \textttITensors.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.

Arxiv: https://arxiv.org/abs/2605.23830