Quantum Higher Order Fourier Analysis and the Clifford Hierarchy

TL;DR

This paper introduces a quantum version of higher-order Fourier analysis that characterizes the Clifford hierarchy, providing new tools for understanding quantum complexity.

Contribution

It develops a mathematical framework for quantum higher-order Fourier analysis and links it to the Clifford hierarchy in quantum information.

Findings

Defines quantum measures generalizing classical uniformity norms

Characterizes the Clifford hierarchy using quantum Fourier analysis

Provides an analytic condition for Clifford hierarchy membership

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.