A general-purpose path-recording oracle is defined that perfectly simulates random elements of any closed subgroup of U(N) by storing t input-output pairs whose updates are governed by the commutant of the group's tensor-power representation.
Quantum Simulation of Random Unitaries from Clebsch-Gordan Transforms
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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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quant-ph 1years
2026 1verdicts
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Quantum Lazy Sampling and Path Recording for Any Group
A general-purpose path-recording oracle is defined that perfectly simulates random elements of any closed subgroup of U(N) by storing t input-output pairs whose updates are governed by the commutant of the group's tensor-power representation.