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 Fpn 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 Ω(n). Here we show that the qudit analog of the n-qubit magic state has stabilizer rank Ω(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.