IntegrateUnitary.jl：用于Haar测度符号积分的Julia包
IntegrateUnitary.jl: A Julia package for symbolic integration over Haar measures
AI总结：
本文介绍IntegrateUnitary.jl，一个用于紧致群Haar测度上多项式函数精确期望的Julia包，基于Weingarten微积分和Wick收缩实现符号积分，支持多种群和随机系综。
摘要：
紧致群Haar测度上的符号积分是量子信息科学和随机矩阵理论中的计算基石。我们提出 exttt{IntegrateUnitary.jl}，一个全面的Julia包，用于计算多种紧致群（$U(d)$、$O(d)$、$Sp(d)$和$SU(d)$，针对平衡多项式）、圆系综和高斯系综、Ginibre系综、置换群、随机纯态以及酉$t$-设计的多项式函数的精确期望。该包提供了Weingarten微积分和Wick收缩的完全开源实现，对逐项和迹多项式积分具有广泛的符号-$d$支持，而选定的工作流目前需要具体的整数维度（包括$k>1$的更高纯迹矩$|\mathrm{tr}(U)|^{2k}$和带有 exttt{SymbolicMatrix}输入的HCIZ，以及 exttt{SymbolicMatrix}/ exttt{SymbolicMatrixProduct}表达式的直接矩阵值积分）、自动渐近展开、从无索引表达式重构Weingarten图的高级符号迹接口，以及用于张量网络平均的 exttt{ITensors.jl}桥接。我们讨论了底层算法，包括Murnaghan-Nakayama规则和辛-正交对偶性，并展示了该包能高效处理高阶矩和量子信息度量。
英文摘要：
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.
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