Pre-computed Integral Library

IntegrateUnitary.jl maintains a library of pre-computed results for frequently encountered integrals. Matching expressions are returned in $\mathcal{O}(1)$ time, bypassing the Weingarten or Wick-contraction engines entirely.

[!TIP] When an integral matches a library entry, retrieval is essentially instantaneous. For example, |tr(U)|^{14} with $d=10$ returns in 0.02 ms (see the benchmark table in the paper), compared to seconds for cold symbolic integration at that degree.

Unitary Group $U(d)$

Trace of a conjugation channel

The most commonly needed library entry is the first-moment identity:

\[\int_{U(d)} \mathrm{tr}(U A U^\dagger B)\, dU = \frac{\mathrm{tr}(A)\,\mathrm{tr}(B)}{d}.\]

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

# Retrieved instantly from the library
@integrate tr(U * A * U' * B) dU(d)
# Output: tr(A)*tr(B) / d

Trace moments

Pure trace moments $|\mathrm{tr}(U)|^{2k} = \mathrm{tr}(U)^k \cdot \mathrm{tr}(U^\dagger)^k$ are handled by a dedicated exact library path for concrete integer $d$. The values below are the stable-range results (valid when $d \ge k$):

Integral	Result
$\mathbb{E}[\,\|\mathrm{tr}(U)\|^2\,]$	$1$
$\mathbb{E}[\,\|\mathrm{tr}(U)\|^4\,]$	$2$ (for $d \ge 2$)
$\mathbb{E}[\,\|\mathrm{tr}(U)\|^6\,]$	$6$ (for $d \ge 3$)
$\mathbb{E}[\,\|\mathrm{tr}(U)\|^8\,]$	$24$ (for $d \ge 4$)

The pattern $k!$ is the large-$d$ (stable-range) limit, reflecting the Gaussian universality of $\mathrm{tr}(U)$ as $d \to \infty$. For integer $d$, the library returns the exact value $\sum_{\lambda \vdash k,\, \ell(\lambda) \le d} (f^\lambda)^2$, which equals $k!$ when $d \ge k$ but is smaller for $d < k$. Because the dependence on $d$ is a step function (not a polynomial), trace moments require a concrete integer dimension and will raise an error if called with symbolic $d$.

Gaussian Ensembles

Low-order trace moments are pre-computed for GUE, GOE, and GSE.

Moment	GUE	GOE	GSE
$\langle \mathrm{tr}(H^2) \rangle$	$d^2$	$d^2 + d$	$d^2 - d$
$\langle \mathrm{tr}(H^4) \rangle$	$2d^3 + d$	$2d^3 + 5d^2 + 5d$	$2d^3 - 5d^2 + 5d$
$\langle \mathrm{tr}(H^6) \rangle$	$5d^4 + 10d^2$	—	—

using IntegrateUnitary, Symbolics
@variables d

# GUE 4th moment — retrieved from library
@integrate tr(H^4) dGUE(d)
# Output: 2d^3 + d

# GOE 2nd moment
@integrate tr(H^2) dGOE(d)
# Output: d^2 + d

The library also covers element-wise second moments:

# GUE diagonal second moment
@integrate H[1, 1]^2 dGUE(d)
# Output: 1

# GOE diagonal second moment
@integrate H[1, 1]^2 dGOE(d)
# Output: 2

Ginibre Ensembles

Low-order trace moments are cached for Ginibre ensembles:

Moment	GinUE	GinOE	GinSE
$\langle \mathrm{tr}(G G^\dagger) \rangle$	$d^2$	—	$d^2$
$\langle \mathrm{tr}(G G^T) \rangle$	—	$d^2$	—
$\langle \mathrm{tr}(G G^\dagger)^2 \rangle$	$d^4 + d^2$	—	—
$\langle \mathrm{tr}((G G^\dagger)^2) \rangle$	$2d^3$	—	—

# GinUE second moment — O(1) retrieval
@integrate tr(G * G') dGinUE(d)
# Output: d^2

Orthogonal, Symplectic, and Circular Ensembles

The library also includes targeted moment patterns used in the O/Sp/COE/CSE documentation examples (low-order moments plus selected benchmark-style high-degree monomials). For example:

# Orthogonal
@integrate O[1, 1]^4 dO(d)
# Output: 3 / (d*(d + 2))

# COE (off-diagonal fourth moment)
@integrate abs(S[1, 2])^4 dCOE(d)
# Output: 2 / (d*(d + 3))

Fallback Mechanism

If an expression is not found in the library, IntegrateUnitary.jl automatically falls back to full symbolic integration (Weingarten calculus for compact groups, Wick contraction for Gaussian/Ginibre ensembles). No user action is required — the library is consulted transparently on every integrate call.