@article{bibcite_16181,
  keywords = {Symbolic integration; Haar measure; Weingarten calculus; Julia programming language; quantum information},
  author = {{\L}ukasz Pawela and Zbigniew Pucha{\l}a},
  title = {IntegrateUnitary.jl: A Julia package for symbolic integration over Haar measures},
  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.},
  year = {2026},
  journal = {arXiv:2605.23830},
  month = {06/2026},
  url = {https://arxiv.org/abs/2605.23830},
  doi = {10.48550/arXiv.2605.23830},
}