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arXiv:2607.10867

GroupFunctions.jl: computing individual entries of the irreducible representations of the unitary group U(d)

David Amaro-Alcalá, Konrad Szymański

quant-phcs.MSSubmitted Jul 12, 2026• Announced Jul 14, 2026
arXivHTMLPDF

Abstract

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.

Journal reference
J. Open Source Software, 11(126), 11169 (2026)
DOI
10.21105/joss.11169
Comments
4 pages, to be submitted to JOSS

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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.
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.
Numerical investigations are an important research tool in quantum information theory. There already exists a wide range of computational tools implemented in various programming languages. However there is little effort in implementing this kind of tools in the Julia language. Julia is a modern numerical computation library with excellent support for vector and matrix algebra, extended type system that allows for implementation of elegant application interfaces and support for parallel and distributed computing. QI.jl is a new quantum information theory library implemented in Julia that provides functions for creating and analyzing quantum states, and for creating quantum operations in various representations. An additional feature of the library is a collection of functions for sampling random quantum states and operations such as unitary operations and generic quantum channels.
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