# Quantum higher-order Fourier analysis and the Clifford hierarchy

## TL;DR

This paper introduces quantum higher-order Fourier analysis, a new mathematical framework that helps quantify the complexity of quantum computing operations.

## Contribution

The paper introduces quantum higher-order Fourier analysis and shows how it characterizes the Clifford hierarchy in quantum computation.

## Key findings

- Quantum higher-order Fourier analysis generalizes classical higher-order Fourier analysis to quantum settings.
- The framework provides a way to quantify the complexity of unitary gates in quantum computation.
- A necessary and sufficient analytic condition for a unitary to belong to a specific level of the Clifford hierarchy is established.

## Abstract

Quantum Fourier analysis is a powerful tool in mathematical physics. Here, we introduce quantum, higher-order Fourier analysis (q-HOFA). We explore some mathematical properties of this theory and show that it provides a way to quantify the complexity of unitary gates in quantum computation. This should provide a natural starting point for a more general study of q-HOFA.

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 recent advances in number theory and combinatorics. We define a family of “quantum measures” on linear transformations on a Hilbert space, that reduce in the case of diagonal matrices to the uniformity norms introduced by Timothy Gowers. 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 computation. In particular, we give a necessary and sufficient analytic condition that a unitary is an element of the kth level of the Clifford hierarchy.