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arXiv · 2509.26623

Quantum Simulation of Random Unitaries from Clebsch-Gordan Transforms

Dmitry Grinko·Satoshi Yoshida
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Abstract

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. We also explain how Ma-Huang's approximate path-recording oracle compares to our exact construction. For general compact groups, we describe the corresponding path-recording bases, achieved via generalized Schur transforms. These results clarify the relation between exact representation-theoretic compressed oracles and the path-recording bases used in algorithmic and cryptographic compressed-oracle arguments.

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Dmitry Grinko, Satoshi Yoshida. 2025-09-30. Quantum Simulation of Random Unitaries from Clebsch-Gordan Transforms. https://arxiv.org/abs/2509.26623

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