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.
Highlights
The Gottesman-Knill Theorem [9, 18] states that any quantum circuit consisting of Clifford gates can be efficiently classically simulated
This means that circuits consisting only of Clifford gates cannot provide computational advantage over classical computers
It is widely believed that universal quantum computers cannot be efficiently simulated by classical computers: state-of-the-art simulators using modern day supercomputers are only able to simulate a few dozens of qubits [6, 12, 19, 22]
Summary
Accepted in Quantum 2022-02-02, click title to verify. Published under CC-BY 4.0. It turns out that stabilizer states correspond to functions in this broader class This establishes a surprising link between higher-order Fourier analysis and quantum information theory. It was shown in [7] (see [2]) that stabilizer states are quadratic forms taking values in Z8 defined on affine subspaces. We think that the techniques used here might pave the way to super-linear lower bounds for decompositions in terms of stabilizer states defined on the full space Fnp. Higher-order Fourier analysis and quantum information theory. In analogy with analysis of Boolean functions, we hope that higher-order Fourier analysis proves useful in quantum information theory
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