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

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