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arxiv: 2605.23830 · v2 · pith:MJWRNWHQnew · submitted 2026-05-22 · 🪐 quant-ph · cs.MS

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

{\L}ukasz Pawela , Zbigniew Pucha{\l}a This is my paper

Pith reviewed 2026-06-30 15:55 UTC · model grok-4.3

classification 🪐 quant-ph cs.MS
keywords Weingarten calculusHaar measuresymbolic integrationJulia packagequantum informationrandom matrix theoryWick contractionscompact groups
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The pith

A new Julia package computes exact symbolic integrals of polynomials over Haar measures on compact groups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents IntegrateUnitary.jl, which performs exact symbolic integration of entry-wise and trace polynomials over Haar measures on groups including U(d), O(d), Sp(d), and SU(d), along with various ensembles and designs. The implementation rests on Weingarten calculus and Wick contractions, with support for symbolic dimension d in many cases and selected workflows that require concrete integers. Such integrals arise repeatedly when averaging over random unitaries or states in quantum information and random matrix theory, so an open-source tool removes the need to derive each case by hand or rely on closed-source software.

Core claim

The package supplies a fully open-source realization of Weingarten calculus and Wick contractions that handles symbolic-d entry-wise and trace-polynomial integrals over U(d), O(d), Sp(d), SU(d) for balanced polynomials, circular and Gaussian ensembles, Ginibre ensembles, permutation groups, random pure states, and unitary t-designs, using the Murnaghan-Nakayama rule and symplectic-orthogonal duality together with a high-level trace interface and an ITensors.jl bridge.

What carries the argument

Weingarten calculus realized via the Murnaghan-Nakayama rule and symplectic-orthogonal duality, which reconstructs integration graphs from index-free expressions and performs the required contractions over the Haar measure.

Load-bearing premise

The concrete implementation of the Murnaghan-Nakayama rule, symplectic-orthogonal duality, and Wick contractions inside the package correctly reproduces the known integrals for all claimed groups, ensembles, and polynomial degrees.

What would settle it

Evaluating a standard integral such as the expectation of |tr(U)|^4 over U(d) for symbolic d and obtaining a result that differs from the established closed-form expression would show the implementation does not match known results.

Figures

Figures reproduced from arXiv: 2605.23830 by {\L}ukasz Pawela, Zbigniew Pucha{\l}a.

Figure 1
Figure 1. Figure 1: Architecture of IntegrateUnitary.jl. Scalar expressions follow the left path through normalization, substitution, expansion, and dispatch into the library lookup. Trace expressions are first analyzed by the symbolic trace logic, which reconstructs the Weingarten graph before entering the same dispatch stage. Common patterns are returned directly by the library lookup in O (1) time, while misses fall throug… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper presents IntegrateUnitary.jl, a Julia package implementing symbolic integration over Haar measures for compact groups (U(d), O(d), Sp(d), SU(d)) and ensembles (circular, Gaussian, Ginibre, permutation, random pure states, unitary t-designs). It realizes Weingarten calculus and Wick contractions with symbolic-d support for entry-wise and trace-polynomial integrals, includes the Murnaghan-Nakayama rule and symplectic-orthogonal duality, offers a high-level trace interface, automatic asymptotics, and an ITensors.jl bridge, while noting that some workflows require concrete integer dimensions.

Significance. If the implementation is verified correct, the package would supply a valuable open-source tool automating previously manual Weingarten and Wick calculations across a wide range of groups and polynomial degrees, with symbolic-d support and tensor-network integration; this directly addresses computational needs in quantum information and random matrix theory. The work ships reproducible code as a public Julia package implementing published algorithms, which is a concrete strength.

major comments (1)
  1. [Abstract] Abstract and manuscript body: the central claim that the package supplies a 'correct' implementation of Weingarten calculus, Wick contractions, Murnaghan-Nakayama rule, and symplectic-orthogonal duality for all listed groups, ensembles, and polynomial degrees is load-bearing, yet the text supplies no explicit verification examples (e.g., side-by-side comparison against known closed-form values of the Weingarten function for small partitions or the integral ∫|tr(U)|^{2k} dμ for k>1).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and constructive feedback. We address the major comment below and will revise the manuscript to incorporate explicit verification examples as suggested.

read point-by-point responses

  1. Referee: [Abstract] Abstract and manuscript body: the central claim that the package supplies a 'correct' implementation of Weingarten calculus, Wick contractions, Murnaghan-Nakayama rule, and symplectic-orthogonal duality for all listed groups, ensembles, and polynomial degrees is load-bearing, yet the text supplies no explicit verification examples (e.g., side-by-side comparison against known closed-form values of the Weingarten function for small partitions or the integral ∫|tr(U)|^{2k} dμ for k>1).

    Authors: We agree that the absence of explicit verification examples weakens the manuscript's support for the central correctness claims. In the revised version we will add a new subsection (likely in Section 3 or 4) containing side-by-side comparisons. These will include: (i) Weingarten function values Wg(λ,d) for small partitions such as λ=(2), (1,1), (3), (2,1) against the closed-form expressions tabulated in the literature (e.g., Collins et al.); (ii) explicit symbolic results for ∫|tr(U)|^{2k} dμ for k=2 and k=3 together with the corresponding known formulas or numerical cross-checks; and (iii) brief checks for the Murnaghan-Nakayama rule and symplectic-orthogonal duality on low-degree cases. The examples will be presented both symbolically and, where feasible, with concrete integer d to allow direct numerical verification. This addition directly substantiates the implementation claims without altering the package itself. revision: yes

Circularity Check

0 steps flagged

No circularity: paper describes software implementation of prior algorithms

full rationale

The manuscript is a software paper presenting IntegrateUnitary.jl. It implements Weingarten calculus, Wick contractions, Murnaghan-Nakayama rule, and symplectic-orthogonal duality for Haar integrals over listed groups and ensembles. No new mathematical derivation or prediction is offered; the text discusses high-level interfaces and algorithms drawn from existing literature. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claim to its own inputs appear. The package's correctness is an implementation detail outside the scope of derivation circularity analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a software-package description rather than a theoretical derivation; no free parameters, axioms, or invented entities are introduced beyond standard references to known combinatorial rules.

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