Related papers: Quantum Higher Order Fourier Analysis and the Clifford Hierarchy

Quantum Higher Order Fourier Analysis and the Clifford Hierarchy

URL: http://arxiv.org/abs/2508.15908v2
Date: Sat, 06 Sep 2025 03:36:07 GMT
Title: Quantum Higher Order Fourier Analysis and the Clifford Hierarchy
Authors: Kaifeng Bu, Weichen Gu, Arthur Jaffe,
Abstract summary: We show that our quantum measures and our related theory of quantum higher-order Fourier analysis characterize the Clifford hierarchy.<n>In particular, we give a necessary and sufficient analytic condition that a unitary is an element of the k-th level of the Clifford hierarchy.
Score: 3.8233569758620054
License: http://creativecommons.org/licenses/by/4.0/
Abstract: We propose a mathematical framework that we call quantum, higher-order Fourier analysis. This generalizes the classical theory of higher-order Fourier analysis, which led to many advances in number theory and combinatorics. We define a family of quantum measures on a Hilbert space, that reduce in the case of diagonal matrices to the classical uniformity norms. We show that our quantum measures and our related theory of quantum higher-order Fourier analysis characterize the Clifford hierarchy, an important notion of complexity in quantum information. In particular, we give a necessary and sufficient analytic condition that a unitary is an element of the k-th level of the Clifford hierarchy.

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