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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 2026-05-25 04:14 UTC · model grok-4.3

classification 🪐 quant-ph cs.MS
keywords Haar measureWeingarten calculussymbolic integrationJulia packagecompact groupsquantum informationrandom matrix theoryunitary designs
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The pith

IntegrateUnitary.jl 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, a Julia package for computing exact expectations of polynomial functions over the Haar measure on groups including U(d), O(d), Sp(d), and SU(d) for balanced polynomials. It extends to circular and Gaussian ensembles, Ginibre ensembles, permutation groups, random pure states, and unitary t-designs. The implementation relies on Weingarten calculus and Wick contractions with symbolic support for dimension d in entry-wise and trace-polynomial cases. These integrals form a core computational step in quantum information and random matrix theory, where exact closed forms replace numerical sampling or approximations.

Core claim

The package supplies a fully open-source realization of Weingarten calculus together with Wick contractions, equipped with broad symbolic-d support, that evaluates 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; selected higher-trace and matrix-valued workflows accept concrete integer dimensions while automatic asymptotic expansions and a symbolic trace interface are also provided.

What carries the argument

Weingarten calculus and Wick contractions, which convert polynomial expectations over Haar measure into sums over permutations or partitions evaluated via the Murnaghan-Nakayama rule and symplectic-orthogonal duality.

If this is right

  • High-degree moments of traces become available in closed symbolic form for both symbolic and concrete d.
  • Quantum information quantities that require averaging polynomials over random unitaries can be obtained exactly.
  • Tensor-network expectations over random unitaries are directly accessible through the ITensors.jl interface.
  • Automatic large-d expansions are generated for any supported polynomial integral.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Routine access to these integrals may allow researchers to derive new closed-form relations among quantum metrics without manual case-by-case calculation.
  • The same symbolic engine could be applied to verify conjectures about higher moments in random matrix ensembles.
  • Integration with computer-algebra systems outside Julia would let users combine these results with other symbolic manipulations.

Load-bearing premise

The algorithms including the Murnaghan-Nakayama rule and symplectic-orthogonal duality are correctly implemented to return the claimed exact symbolic results.

What would settle it

Evaluating the known integral of |tr(U)|^2 over U(d) and obtaining any value other than 1.

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

2 major / 2 minor

Summary. The manuscript presents IntegrateUnitary.jl, a Julia package implementing Weingarten calculus, Wick contractions, and related combinatorial algorithms (including the Murnaghan-Nakayama rule and symplectic-orthogonal duality) to compute exact symbolic expectations of polynomial functions over Haar measures on U(d), O(d), Sp(d), SU(d) (balanced case), circular/Gaussian/Ginibre ensembles, permutation groups, random pure states, and unitary t-designs, with support for symbolic d in entry-wise and trace-polynomial integrals plus selected workflows for concrete d.

Significance. A verified, open-source implementation with broad symbolic-d support would be a useful addition to the quantum-information and random-matrix-theory toolkit, enabling reproducible exact computations of high-degree moments and metrics that are currently handled case-by-case in the literature.

major comments (2)
  1. [Abstract and algorithm sections] Abstract and algorithm-description sections: the central claim that the package produces exact symbolic results rests on the correctness of the Murnaghan-Nakayama implementation and the symplectic-orthogonal duality; the manuscript describes these algorithms but supplies no explicit cross-checks against independently known closed forms (e.g., the exact expression for E[|tr(U)|^{2k}] for small k or the HCIZ integral at low order).
  2. [Demonstration sections] Demonstration and results sections: while the text states that the package 'efficiently handles high-degree moments,' no concrete symbolic outputs, error checks, or comparisons with existing Weingarten-function tables or other implementations are reported, so the load-bearing assumption that the graph-reconstruction and character-evaluation routines are free of off-by-one or sign errors for symbolic d remains untested in the manuscript.
minor comments (2)
  1. [Abstract] Clarify which workflows require concrete integer d versus fully symbolic d, and list the precise polynomial classes supported for each group.
  2. [Results] Add a short table or appendix entry showing at least one low-order symbolic result together with the corresponding known closed form.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We agree that the manuscript would be strengthened by the addition of explicit verification examples and concrete outputs. We will revise accordingly to address both major comments.

read point-by-point responses

  1. Referee: [Abstract and algorithm sections] Abstract and algorithm-description sections: the central claim that the package produces exact symbolic results rests on the correctness of the Murnaghan-Nakayama implementation and the symplectic-orthogonal duality; the manuscript describes these algorithms but supplies no explicit cross-checks against independently known closed forms (e.g., the exact expression for E[|tr(U)|^{2k}] for small k or the HCIZ integral at low order).

    Authors: We agree that explicit cross-checks are needed to support the claim of exact symbolic results. In the revised manuscript we will add a dedicated verification subsection that computes E[|tr(U)|^{2k}] for small k (k=1,2,3) and matches the outputs against the known closed-form expressions from the literature. We will likewise include low-order HCIZ integrals and compare them with independently tabulated results. These additions will directly test the Murnaghan-Nakayama rule and the symplectic-orthogonal duality implementations. revision: yes

  2. Referee: [Demonstration sections] Demonstration and results sections: while the text states that the package 'efficiently handles high-degree moments,' no concrete symbolic outputs, error checks, or comparisons with existing Weingarten-function tables or other implementations are reported, so the load-bearing assumption that the graph-reconstruction and character-evaluation routines are free of off-by-one or sign errors for symbolic d remains untested in the manuscript.

    Authors: We accept that the current demonstration sections do not contain the requested concrete outputs or comparisons. The revised version will include explicit symbolic results for selected high-degree moments, direct numerical comparisons against published Weingarten-function tables for both integer and symbolic d, and side-by-side checks with at least one other open implementation where available. These additions will confirm the correctness of the graph-reconstruction and character-evaluation routines. revision: yes

Circularity Check

0 steps flagged

No circularity: package description implements external algorithms

full rationale

The manuscript is a software package description presenting IntegrateUnitary.jl. It implements known methods (Weingarten calculus, Murnaghan-Nakayama rule, symplectic-orthogonal duality) drawn from prior literature rather than deriving new results. No predictions, fitted parameters, or self-referential derivations appear; the load-bearing assumption is correct implementation of externally documented combinatorial algorithms. This is self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The package's value rests on the assumption that standard domain algorithms (Weingarten calculus, Murnaghan-Nakayama rule) are correctly coded; no new free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Weingarten calculus and the Murnaghan-Nakayama rule correctly compute the integrals over the listed Haar measures
    The package implements these established techniques from prior literature rather than deriving them.

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