Stabilizer rank and higher-order Fourier analysis

TL;DR

This paper connects stabilizer states in quantum computing with higher-order Fourier analysis, using mathematical tools to analyze stabilizer rank and generalize known results from qubits to qudits of any prime dimension.

Contribution

It establishes a novel link between stabilizer states and higher-order Fourier analysis, enabling new methods to analyze stabilizer rank for qudits of arbitrary prime dimension.

Findings

Stabilizer states are nonclassical quadratic phase functions in higher-order Fourier analysis.

The stabilizer rank of the qudit magic state is proven to be (n), generalizing previous qubit results.

Tools from higher-order Fourier analysis are effectively used to analyze quantum state complexity.

Abstract

We establish a link between stabilizer states, stabilizer rank, and higher-order Fourier analysis -- a still-developing area of mathematics that grew out of Gowers's celebrated Fourier-analytic proof of Szemer\'edi's theorem \cite{gowers1998new}. We observe that $n$-qudit stabilizer states are so-called nonclassical quadratic phase functions (defined on affine subspaces of $\mathbb{F}_p^n$ where $p$ is the dimension of the qudit) which are fundamental objects in higher-order Fourier analysis. This allows us to import tools from this theory to analyze the stabilizer rank of quantum states. Quite recently, in \cite{peleg2021lower} it was shown that the $n$-qubit magic state has stabilizer rank $\Omega(n)$. Here we show that the qudit analog of the $n$-qubit magic state has stabilizer rank $\Omega(n)$, generalizing their result to qudits of any prime dimension. Our proof techniques use explicitly tools from higher-order Fourier analysis. We believe this example motivates the further exploration of applications of higher-order Fourier analysis in quantum information theory.