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arXiv:2605.23830

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

Łukasz Pawela, Zbigniew Puchała

quant-phcs.MSSubmitted May 22, 2026• Announced May 25, 2026• 1 citation• Revised May 29, 2026 (v2)
v1 · May 22, 2026v2 · May 29, 2026
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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.

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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)U(d)U(d), O(d)O(d)O(d), Sp(d)Sp(d)Sp(d), and SU(d)SU(d)SU(d) for balanced polynomials), circular and Gaussian ensembles, Ginibre ensembles, permutation groups, random pure states, and unitary ttt-designs. The package provides a fully open-source implementation of the Weingarten calculus and Wick contractions with broad symbolic-ddd support for entry-wise and trace-polynomial integrals, while selected workflows currently require concrete integer dimensions (including higher pure trace moments ∣tr(U)∣2k|\mathrm{tr}(U)|^{2k}∣tr(U)∣2k for k>1k > 1k>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.
Abstract The randomized quantum marginal problem asks about the joint distribution of the partial traces (“marginals”) of a uniform random Hermitian operator with fixed spectrum acting on a space of tensors. We introduce a new approach to this problem based on studying the mixed moments of the entries of the marginals. For randomized quantum marginal problems that describe systems of distinguishable particles, bosons, or fermions, we prove formulae for these mixed moments, which determine the joint distribution of the marginals completely. Our main tool is Weingarten calculus, which provides a method for computing integrals of polynomial functions with respect to Haar measure on the unitary group. As an application, in the case of two distinguishable particles, we prove some results on the asymptotic behavior of the marginals as the dimension of one or both Hilbert spaces goes to infinity.
Rimu...jl is a Julia package for solving many-body quantum problems. The core of the package is a matrix-free implementation of Hamiltonians and other operators and compact representation of Fock states, which together allow for efficient methods suitable for high-performance computing. Rimu...jl includes a Julia implementation of the full configuration interaction quantum Monte Carlo (FCIQMC) algorithm which is a type of projector QMC algorithm for stochastically solving the time-independent Schrödinger equation. It also includes many well-known model Hamiltonians, and an interface for exact diagonalisation based on external eigenvalue solvers. Both the stochastic and exact diagonalisation methods are accessed with a CommonSolve...jl interface. We describe the FCIQMC algorithm and how to obtain estimators of observables as well as the key features of the implementation.
We are upgrading the Python-version of RTNI, which symbolically integrates tensor networks over the Haar-distributed unitary matrices. Now, PyRTNI2 can treat the Haar-distributed orthogonal matrices and the real and complex normal Gaussian tensors as well. Moreover, it can export tensor networks in the format of TensorNetwork so that one can make further calculations with concrete tensors, even for low dimensions, where the Weingarten functions differ from the ones for high dimensions. The tutorial notebooks are found at GitHub: https://github.com/MotohisaFukuda/PyRTNI2. In this paper, we explain maths behind the program and show what kind of tensor network calculations can be made with it. For the former, we interpret the element-wise moment calculus of the above random matrices and tensors in terms of tensor network diagrams, and argue that the view is natural, relating delta functions in the calculus to edges in tensor network diagrams.
GroupFunctions.jl is a Julia library for computing individual matrix elements of irreducible representations of U(d). These matrix elements, called group functions, can be evaluated symbolically or numerically. For SU(2), they reduce to the Wigner D-functions. The library computes these matrix elements in a carrier-space basis enumerated by Gelfand-Tsetlin patterns. It can also compute entire representation operators, construct input unitaries from parameterisations common in quantum optics, translate Gelfand-Tsetlin patterns into occupation-number kets, and compute the associated Schur functions. Results can be exported in a form compatible with Mathematica.
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