IntegrateUnitary.jl: A Julia package for symbolic integration over Haar measures
| Author | Pawela Ł.; Puchała Z. |
|---|---|
| Title | IntegrateUnitary.jl: A Julia package for symbolic integration over Haar measures |
| Journal | arXiv:2605.23830 |
| Year | 2026 |
| Status | Submitted |
| DOI | 10.48550/arXiv.2605.23830 |
| URL | https://arxiv.org/abs/2605.23830 |
| Abstract | Symbolic 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. |
| Keywords | Symbolic integration; Haar measure; Weingarten calculus; Julia programming language; quantum information |
| 2605.23830v2.pdf |