Stabilizer rank and higher-order Fourier analysis

Quantum 6, 645 (2022).

https://doi.org/10.22331/q-2022-02-09-645

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 [10]. 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 [20] 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.