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Symbolic integration with respect to the Haar measure on the unitary group
Zbigniew Puchała, Jarosław Adam Miszczak
TL;DR
Polynomial integration over unitary groups with Haar measure is the subject of the paper. It presents IntU, a Mathematica package with special-case optimizations, and reports that calculation time grows rapidly for high-degree polynomials unless an optimization applies.
Problem
The paper addresses calculation of polynomial integrals over U(d) with respect to Haar measure.
Source: §1 · p. 1Method
IntU symbolically calculates polynomial Haar integrals and applies special-case formulas to reduce Weingarten-function computations for some classes.
Source: §1 · p. 1, §3.2 · p. 2, §3.4 · p. 3Results
For large-degree polynomials, calculation time grows rapidly, and calculation is possible only when one of the special-case optimizations applies.
Source: §6 · p. 6Takeaways & Limitations
IntU provides Mathematica calculations of polynomial Haar integrals, with examples including applications to quantum-state geometry.
Source: §6 · p. 6Calculation time depends strongly on integrand degree; large-degree calculations are possible only with a special-case optimization.
Source: §6 · p. 6
Abstract
from arXiv · showhide
We present IntU package for Mathematica computer algebra system. The presented package performs a symbolic integration of polynomial functions over the unitary group with respect to unique normalized Haar measure. We describe a number of special cases which can be used to optimize the calculation speed for some classes of integrals. We also provide some examples of usage of the presented package.
1. Introduction
Haar integration over unitary groups is studied across mathematical physics, random matrix theory, and algebraic combinatorics. This paper presents IntU, a Mathematica package for polynomial integrals, with special-case optimizations and usage examples.
1. Introduction
- Collins and Śniady proved a formula for calculating monomial integrals with respect to unitary-group Haar measure.Source: §1 · p. 1
- IntU calculates polynomial integrals over U(d) with respect to Haar measure and includes special cases to speed some classes of calculations.Examples include applications to the geometry of quantum states.Source: §1 · p. 1
- The paper introduces notation and mathematical background, presents faster special cases, describes IntU's functions, and gives usage examples before concluding.Source: §1 · p. 1
2. Mathematical background
The background defines unitary-group and permutation notation, then expresses Haar integrals through moments and the Weingarten function. Its representation-theoretic ingredients include Schur-polynomial dimensions and symmetric-group characters.
2.1. Basic concepts
- U(d) denotes the compact group of d × d unitary matrices, equipped with normalized Haar measure; random matrices drawn from it form the Circular Unitary Ensemble.Source: §2.1 · p. 1
2.2. Moments of the U(d)
- Polynomial integrals reduce by linearity to monomial integrals, called moments of U(d), evaluated using the Weingarten function; unequal multi-index lengths give zero.Source: §2.2 · p. 1
2.3. Weingarten function
- The Weingarten function is expressed as a sum over partitions of n with length at most d, using Schur polynomials and symmetric-group characters.Source: §2.3 · pp. 1–2
2.3.1. The dimension of irreducible representation of U(d)
- Evaluating the Schur polynomial at d ones gives the dimension of the corresponding irreducible representation of U(d).Source: §2.3.1 · p. 2
2.3.2. Irreducible character of Sn
- An irreducible character of S_n depends on permutation cycle type; nontrivial character values can be calculated with the Murnaghan–Nakayama rule.The Weingarten function likewise depends only on cycle type, so it is constant on each conjugacy class.Source: §2.3.2 · p. 2
3. Special cases
The section develops special cases that reduce the Weingarten-function work needed for selected Haar integrals. The savings vary by integral class, from minor for low-degree moments to substantial for row/column and cycle cases.
3. Special cases
- These special cases replace direct use of formula (3), which processes ∏_i k_i!∏_j l_j! permutations for the relevant multi-indices.The implemented optimizations aim to increase package efficiency by minimizing Weingarten-function calculations.Source: §3 · p. 2
3.1. First two moments of U(d)
- Formulas for the first two moments allow polynomial integrals of degree less than 5 to be calculated without formula (3), but yield only a minor efficiency improvement.Direct Weingarten-function calculation is already very fast for these cases.Source: §3.1 · p. 2
3.2. Elements from one row (column)
- For elements from one row or column, simplex-based integration and grouped permutation sums reduce Weingarten evaluations; one optimization avoids (∑pi)!∏pi! executions.The row or column's squared absolute values are uniformly distributed on the standard d-simplex; the exact time gain also depends on conjugacy-class cardinalities.Source: §3.2 · p. 2
- For general nonnegative p_j, Proposition 2 groups permutations within blocks of sizes p_j, avoiding at least ∏p_j! Weingarten-function executions versus direct use of formula (3).The exact time efficiency also depends on conjugacy-class cardinalities for partitions of p_1,...,p_d.Source: §3.2 · p. 3
3.4. Cycle permutations
- For cycle-permutation integrals, the formula avoids more than (k!)^(2m−1) Weingarten-function executions compared with direct use of formula (3).This case assumes positive integer k, m ≤ d, and σ is a cycle in S_m.Source: §3.4 · p. 3
4. Package description
IntU provides Mathematica functions for integrating polynomial expressions or index tuples over U(d) with Haar measure. Supporting functions calculate Weingarten values and provide partition, character, and beta-function utilities.
4. Package description
- The package groups its functions into main functionality, Weingarten-related calculations, and helpers.The main functions use supporting functions to find integral values.Source: §4 · p. 3, §4.2 · p. 4
4.1. Main functionality
- IntU's main functions integrate polynomial expressions directly or calculate the integral for supplied four-tuples of indices.The package also supports multiple integrals with separate variable and dimension specifications.Source: §4.1 · pp. 3–4
4.2. Weingarten
- Weingarten[type,dim] returns the Weingarten function for a partition cycle type and unitary-group dimension.Supporting routines calculate symmetric-group characters, Schur-polynomial values, and permutation cycle types.Source: §4.2 · p. 4, §4.3 · p. 4
4.3. Helper functions
- Helper functions include multinomial Beta and conjugacy-class cardinality calculations used by the optimizations in Sections 3.2 and 3.3.Other helpers provide conjugate and binary partition representations.Source: §4.3 · p. 4
5. Usage examples
The examples demonstrate IntU on elementary integrals, matrix expressions, local-unitary orbit averages, and moments of a maximally entangled numerical shadow. The reported calculations reproduce known covariance and moment results in the cases checked.
5. Usage examples
- The section presents a series of examples to illustrate the package's main features.Source: §5 · p. 4
5.1. Elementary integrals
- The elementary examples show integrations of unitary-entry expressions and an index-based integral call.One index example specifies d = 6; another example simplifies a product of entries and conjugates before integration.Source: §5.1 · pp. 4–5
5.3. Matrix expressions
- The matrix-expression example applies IntegrateUnitaryHaar to an integrand built from U ⊗ U and its conjugate.It defines U ∈ U(d) and U2 = U ⊗ U ∈ U(d²).Source: §5.3 · p. 5
5.4. Mean value of local unitary orbit
- For the local-unitary orbit example, the mean value is calculated for independent Haar-distributed unitary matrices U and V of size d; at the fixed dimension, IntegrateUnitaryHaar’s integral agrees with calculations in.The mean-value calculation uses d = 3.Source: §5.4 · p. 5
5.5. Moments of maximally entangled numerical shadow
- For the maximally entangled numerical shadow, the computed first and second moments agree with Eqs. (38) and (39) after algebraic manipulation.The example forms the state by vectorizing a Haar-distributed unitary matrix and evaluates the moments using IntU.Source: §5.5 · pp. 5–6
6. Summary
IntU is a Mathematica package for polynomial integrals over U(d) with respect to Haar measure. Special cases can improve calculation speed, while runtime grows rapidly with integrand degree and high-degree calculations require an optimization.
6. Summary
- IntU calculates polynomial integrals over U(d) with respect to Haar measure and includes special cases to optimize speed.The package is also used in examples concerning the geometry of quantum states.Source: §6 · p. 6
- For large-degree polynomials, calculation time grows rapidly and calculation is possible only when a special-case optimization is used.For small-degree polynomials, the package can use the direct formula (3).Source: §6 · p. 6
- The package may be useful in investigations involving the Circular Unitary Ensemble, quantum-state geometry, and quantum entanglement.Source: §6 · p. 6