Quantum Simulation of Random Unitaries from Clebsch-Gordan Transforms

Understand

We construct exact compressed oracles for Haar-random group actions associated with an arbitrary finite-dimensional unitary representation of a compact group.

• The construction is a representation-theoretic version of Zhandry's compressed-oracle technique: the memory of the oracle is stored in the Fourier basis, and each update is implemented by Clebsch-Gordan transforms.
• This framework naturally gives forward, conjugate, transpose, and inverse compressed oracles.
• For the unitary group with the defining representation, we present efficient implementation based on high-dimensional Clebsch-Gordan transforms.