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

2026-05-22

AI Summary

This work addresses the challenging problem of symbolic integration of polynomial functions over compact groups with respect to the Haar measure—a central task in quantum information and random matrix theory—by introducing IntegrateUnitary.jl, a Julia package that implements Weingarten calculus and trace polynomial integration with symbolic dimension \(d\). The package uniquely combines Wick contractions, the Murnaghan–Nakayama rule, symplectic-orthogonal duality, and tensor network techniques, while incorporating automated asymptotic expansions and index-agnostic reconstruction of Weingarten graphs. Its novel interface enables efficient computation of high-order pure trace moments (e.g., \(|\mathrm{tr}(U)|^{2k}\) for \(k>1\)), Harish-Chandra–Itzykson–Zuber integrals, and complex quantum information quantities, demonstrating excellent scalability in high-moment and large-scale regimes.

Abstract

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 $|\mathrm{tr}(U)|^{2k}$ for $k > 1$ 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.

Problem

• Haar measure
• symbolic integration
• compact groups
• Weingarten calculus
• random matrix theory

Innovation

• Weingarten calculus
• symbolic integration
• Haar measure
• tensor networks
• compact groups