Stabilizer rank and higher-order Fourier analysis

Farrokh Labib

CWI, QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands

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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.

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Cited by

[1] Shir Peleg, Amir Shpilka, and Ben Lee Volk, "Lower Bounds on Stabilizer Rank", arXiv:2106.03214, Quantum 6, 652 (2022).

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