Stabilizer rank and higher-order Fourier analysis

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 Szemerédi'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.

16 pages

References in corpus (2)

• Qudit versions of the qubit "pi-over-eight" gate
• Lower Bounds on Stabilizer Rank

Cited by in corpus (1)

• Lower Bounds on Stabilizer Rank