Unitary Designs

IntU support integration over Unitary $t$-designs.

Theory

A unitary $t$-design is a probability distribution (ensemble) $\nu$ over unitary matrices $U \in U(d)$ such that for any polynomial $P(U, U^\dagger)$ of degree at most $t$ in the entries of $U$ and at most $t$ in the entries of $U^\dagger$:

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

In other words, a $t$-design perfectly simulates the Haar measure for the first $t$ moments. This concept is crucial in Quantum Information for efficient randomization, benchmarking, and decoupling.

Usage

Use the dDesign(dim, t) function to define a measure representing a unitary $t$-design.

dDesign(dim, t)

1. Basic Integration using @integrate

The @integrate macro automatically identifies U as the random matrix.

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

1. Manual Integration

using IntU, Symbolics

# Define dimension and matrix
@variables d
U = SymbolicMatrix(:U, :U, d)

# Create a 2-design measure
design = dDesign(d, 2)
integrate(abs(U[1,1])^2, design)

Integration Behavior

When integrating with integrate(expr, design):

1. Degree Check: The integrator explicitly calculates the degree of the integrand in $U$ and $U^\dagger$.
2. Valid Degrees: If both degrees are $\le t$, the integral is computed using the standard unitary Weingarten calculus (same result as Haar measure).
3. Invalid Degrees: If the degree exceeds $t$, an error is thrown to indicate that the design does not support this moment. This prevents accidental reliance on non-guaranteed values.

Example

# 1. 2-design supports 4th moments (degree 2 in U, 2 in U*)
expr2 = abs(U[1,1] * U[2,2])^2
res2 = integrate(expr2, design)
println(res2)
# Output: 1 / (d^2 - 1) * ... (Matches Haar)

# 2. 2-design DOES NOT support 6th moments (degree 3)
try
    expr3 = abs(U[1,1])^6
    integrate(expr3, design)
catch e
    println(e)
    # Error: Integrand degree (3, 3) exceeds design order t=2
end

References

1. Dankert, C., Cleve, R., Emerson, J., & Livine, E. (2009). Exact and approximate unitary 2-designs and their application to fidelity estimation. Physical Review A, 80(1), 012304.
2. Gross, D., Audenaert, K., & Eisert, J. (2007). Evenly distributed unitaries: On the structure of unitary designs. Journal of Mathematical Physics, 48(5), 052104.
3. Ambainis, A., & Emerson, J. (2007). Quantum t-designs: t-wise independence in the quantum world. Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).