Type: Article
Publication Date: 2022-02-09
Citations: 6
DOI: 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 \cite{gowers1998new}. We observe that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-qudit stabilizer states are so-called nonclassical quadratic phase functions (defined on affine subspaces of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">F</mml:mi></mml:mrow><mml:mi>p</mml:mi><mml:mi>n</mml:mi></mml:msubsup></mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-qubit magic state has stabilizer rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">&#x03A9;</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. Here we show that the qudit analog of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-qubit magic state has stabilizer rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">&#x03A9;</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, 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.