Quantum Simulation of Random Unitaries from Clebsch-Gordan Transforms

We present a general method for simulating an action of $t$ copies of a Haar random unitary for arbitrary compact groups. This construction can be viewed as a representation-theoretic generalization of Zhandry's compressed function oracle technique. It is conceptually simple, exact and utilizes Clebsch-Gordan transforms as main building blocks. In particular, for the unitary group, our method is efficient in space and time. Finally, our general oracle for forward queries can be easily modified into oracles for conjugate, transpose, and inverse queries, thus unifying all four query types.