Unitary Group Integration

This section details the integration of polynomial functions over the unitary group $U(d)$ with respect to the Haar measure. It covers the theoretical foundations based on Weingarten Calculus and provides practical examples using IntegrateUnitary.jl.

Overview

IntegrateUnitary.jl allows for evaluating integrals of the form:

\[\int_{U(d)} U_{i_1 j_1} \dots U_{i_n j_n} \bar{U}_{k_1 l_1} \dots \bar{U}_{k_n l_n} dU\]

where $U$ is a $d \times d$ unitary matrix ($U U^\dagger = I$). The integral is taken with respect to the Haar measure, which is the unique translation-invariant probability measure on the compact Lie group $U(d)$.

[!NOTE] Special Unitary Group $SU(d)$: For all currently supported "balanced" polynomial expressions (where the number of $U$ and $\bar{U}$ factors are equal), the integration over $SU(d)$ is equivalent to $U(d)$. Non-stable-range effects involving $\epsilon$-tensors for specific small $d$ are not currently covered. In the current backend, non-balanced queries are evaluated with the same phase-invariance rule as $U(d)$ and therefore return 0.

The result is expressed in terms of the dimension $d$ and Kronecker deltas matching the indices.

Theory: Weingarten Calculus

The integration relies on the Weingarten Calculus, a combinatorial method for evaluating integrals over compact groups.

The Integration Formula

The general formula for the integral is given by [Collins & Śniady, 2006]:

\[\int_{U(d)} U_{i_1 j_1} \dots U_{i_n j_n} \bar{U}_{k_1 l_1} \dots \bar{U}_{k_n l_n} dU = \sum_{\sigma, \tau \in S_n} \delta_{i, k_\sigma} \delta_{j, l_\tau} \text{Wg}(\sigma \tau^{-1}, d)\]

where:

  • $S_n$ is the symmetric group of permutations of $\{1, \dots, n\}$.
  • $\delta_{i, k_\sigma} = \prod_{m=1}^n \delta_{i_m, k_{\sigma(m)}}$ contracts row indices according to $\sigma$.
  • $\delta_{j, l_\tau} = \prod_{m=1}^n \delta_{j_m, l_{\tau(m)}}$ contracts column indices according to $\tau$.
  • $\text{Wg}(\pi, d)$ is the Weingarten function, which depends only on the cycle structure of the permutation $\pi$ and the dimension $d$.

The Weingarten Function

The Weingarten function $\text{Wg}(\pi, d)$ is a rational function of $d$. It is defined via the inverse of the Gram matrix of Schur functions or via character theory.

For small $n$, the values are:

  • n=1: $\text{Wg}([1], d) = \frac{1}{d}$
  • n=2:
    • Identity $\text{Wg}([1,1], d) = \frac{1}{d^2-1}$
    • Transposition $\text{Wg}([2], d) = -\frac{1}{d(d^2-1)}$
  • n=3:
    • $\text{Wg}([1,1,1], d) = \frac{d^2-2}{d(d^2-1)(d^2-4)}$
    • $\text{Wg}([2,1], d) = -\frac{1}{(d^2-1)(d^2-4)}$
    • $\text{Wg}([3], d) = \frac{2}{d(d^2-1)(d^2-4)}$

IntegrateUnitary.jl computes these values symbolically for any $n$, using the Murnaghan-Nakayama rule to evaluate characters of the symmetric group.

Implementation Details

IntegrateUnitary.jl automates the following steps:

  1. Index Identification: It parses the symbolic expression to identify which variables correspond to elements of $U$ and $\bar{U}$, extracting their indices ($i, j, k, l$).
  2. Degree Matching: It checks if the number of $U$ factors matches the number of $\bar{U}$ factors. If they differ ($n \neq m$), the integral vanishes (returns 0) due to phase invariance ($U \to e^{i\theta}U$).
  3. Symbolic Summation: It generates the sum over permutations $\sigma, \tau \in S_n$, computing the Kronecker delta products symbolically.
  4. Weingarten Evaluation: It computes the values of $\text{Wg}(\pi, d)$ using Combinatorics.jl characters.

Symbolic Dimension

A key feature of IntegrateUnitary.jl is the ability to leave the dimension $d$ as a symbolic variable. This is achieved through the SymbolicMatrix type, which represents a matrix of arbitrary (symbolic) size.

Unitary Designs

In many applications, full Haar integration is not required. Instead, one uses Unitary $t$-designs, which are ensembles that mimic the first $t$ moments of the Haar measure. A unitary $t$-design is a probability distribution $\nu$ such that for any polynomial $P$ of degree at most $t$ in $U$ and $\bar{U}$:

\[\mathbb{E}_{U \sim \nu} [P(U, \bar{U})] = \mathbb{E}_{U \sim \text{Haar}} [P(U, \bar{U})]\]

Usage: dDesign

Use the dDesign(d, t) measure to define a unitary $t$-design of dimension $d$ and order $t$.

using IntegrateUnitary, Symbolics
@variables d
# E[|U_11|^2] using a 2-design (degree 1 in U, 1 in U*)
@integrate abs(U[1,1])^2 dDesign(d, 2)
# Output: 1/d

Integration Behavior and Guards

IntegrateUnitary.jl enforces validity of integration over $t$-designs with the following behaviour:

  1. Balanced integrands (equal degree $q$ in $U$ and $U^\dagger$):

    • \[q \le t\]

      : returns the exact Haar-averaged result via Weingarten calculus.

    • \[q > t\]

      : throws an ArgumentError (e.g., Integrand degree (3) exceeds design order t=2), preventing accidental reliance on non-guaranteed values.
  2. Unbalanced integrands (different degree in $U$ and $U^\dagger$): returns zero, which is the Haar-correct value. This is guaranteed correct for total degree $\le t$; for higher-degree unbalanced monomials, a $t$-design does not guarantee this value, but no error is raised.

This guard mechanism helps ensure that physical simulations and protocol verifications (such as randomized benchmarking) remain mathematically rigorous for balanced polynomial integrands.

Examples

1. Basic Integration using @integrate

The @integrate macro provides a convenient way to integrate expressions without manually declaring variables. It uses heuristics to identify random unitaries (usually U) and dimensions.

using IntegrateUnitary, Symbolics
@variables d
# E[|U_11|^2]
@integrate abs(U[1, 1])^2 dU(d)
# Output: 1/d

# High-degree moments (Example A: different rows/cols)
res_a = @integrate abs(U[1, 1])^2 * abs(U[2, 2])^4 * abs(U[1, 3])^6 dU(d)

# High-degree moments (Example B: same row, Dirichlet behavior)
res_b = @integrate abs(U[1, 1])^2 * abs(U[1, 2])^4 * abs(U[1, 3])^6 dU(d)

# Example C: "2-design style" correlator identity
res_c = @integrate U[1, 1] * conj(U[1, 2]) * U[2, 2] * conj(U[2, 1]) dU(d)

2. Manual Integration

For more control, or when dealing with multiple matrices, you can declare symbols explicitly.

using IntegrateUnitary, Symbolics
@variables d
U = SymbolicMatrix(:U, :U)

# 1. Norm of a matrix element
# Integral of |U_{11}|^2
integrate(abs(U[1, 1])^2, dU(d))
# Output: 1/d

3. Higher Unitary Moments

using IntegrateUnitary, Symbolics
@variables d
U = SymbolicMatrix(:U, :U)

# 2. Fourth moment
# Integral of |U_{11}|^4
integrate(abs(U[1, 1])^4, dU(d))
# Output: 2 / (d*(d + 1))

4. Trace Moments

using IntegrateUnitary, Symbolics
@variables d
U = SymbolicMatrix(:U, :U)

# 3. Trace moments
# Integral of |tr(U)|^2
integrate(abs(tr(U))^2, dU(d))
# Output: 1

5. Matrix Integration

You can integrate matrix-valued expressions directly. For generic AbstractArray inputs, integrate applies element-wise integration. For direct matrix-valued integration of SymbolicMatrix and SymbolicMatrixProduct expressions, the output dimensions must be concrete integers.

using IntegrateUnitary

# Matrix-valued integration (concrete dimension required)
res = @integrate U * U' dU(2)
# Output: 2x2 identity matrix

For symbolic dimensions, use scalar contractions such as tr(...) instead of requesting a full matrix-valued result.

6. Complex Trace Powers

IntegrateUnitary.jl supports trace absolute values with integer-power workflows used in practice, in particular even powers $|tr(U)|^{2k}$. These are processed via the lazy trace engine. Odd or non-integer powers are currently unsupported and raise an IntegrationError. Also note that pure trace moments $|tr(U)|^{2k}$ with $k > 1$ require a concrete integer dimension (the result depends on $d$ as a step function, not a polynomial).

using IntegrateUnitary, Symbolics
U = SymbolicMatrix(:U, :U, 10)

# Integral of |tr(U)|^4 (requires concrete d)
integrate(abs(tr(U))^4, dU(10))
# Output: 2

7. HCIZ Integrals

IntegrateUnitary.jl provides direct support for Harish-Chandra-Itzykson-Zuber (HCIZ) integrals.

[!NOTE] hciz supports eigenvalue-vector and matrix interfaces. For SymbolicMatrix inputs, the dimension must be a concrete integer (symbolic d is unsupported in this path). For numeric matrix inputs with degenerate eigenvalues, IntegrateUnitary sorts both spectra and applies tiny independent perturbations to both before evaluating the HCIZ formula.

using IntegrateUnitary, LinearAlgebra

# Define matrices
A = diagm([1.0, 2.0])
B = diagm([0.5, 1.5])

# Compute ∫ dU exp(Tr(A U B U'))
res = hciz(A, B)
println(res)
# Output: 20.9329...

See the API Reference for more details.

Potential Pitfalls

[!IMPORTANT]

Symbolic (d) Pitfalls

  • Small Dimensions: For Haar-related measures (Unitary, Orthogonal, Circular), element-wise results are rational functions with poles at small $d$ (typically $d < n$ for degree $n$ moments). Pure trace moments $|\mathrm{tr}(U)|^{2k}$ are an exception: they depend on $d$ as a step function and require a concrete integer dimension.
  • Removable Singularities: Substituting numeric values can yield $0/0$ forms (e.g., at $d=1$ or $d=2$).
  • Automatic Handling: IntegrateUnitary.jl's evaluate function automatically simplifies expressions to resolve removable singularities when a denominator evaluates to zero.
  • Computational Complexity: The sum involves $(n!)^2$ terms. While optimized to group cycles, integrals with high degrees ($n > 6$) can become computationally expensive. The number of terms grows factorially.

References

  1. Collins, B. (2003). Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability. International Mathematics Research Notices, 2003(17), 953-982.
  2. Collins, B., & Śniady, P. (2006). Integration with respect to the Haar measure on unitary, orthogonal and symplectic groups. Communications in Mathematical Physics, 264(3), 773-795. arXiv:math-ph/0402073
  3. Puchala, Z., & Miszczak, J. A. (2017). Symbolic integration with respect to the Haar measure on the unitary groups. Bulletin of the Polish Academy of Sciences. Technical Sciences, 65(1), 21-27.

See Also