Stabilizer rank and higher-order Fourier analysis

Abstract

We establish a link between stabilizer states, stabilizer rank and higherorder Fourier analysis|a still-developing area of mathematics that grew out of Gowers's celebrated Fourier-analytic proof of Szemerfiedi's theorem [10]. We observe that n-qudit stabilizer states are so-called nonclassical quadratic phase functions (defined on afine subspaces of Fnp 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 (n). Here we show that the qudit analogue of the n-qubit magic state has stabilizer rank (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.