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arXiv · 2005.12959

The decomposition of an arbitrary $2^w\times 2^w$ unitary matrix into signed permutation matrices

Alexis De Vos·Stijn De Baerdemacker
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Abstract

Birkhoff's theorem tells that any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. A similar theorem reveals that any unitary matrix can be decomposed as a weighted sum of complex permutation matrices. Unitary matrices of dimension equal to a power of~2 (say $2^w$) deserve special attention, as they represent quantum qubit circuits. We investigate which subgroup of the signed permutation matrices suffices to decompose an arbitrary such matrix. It turns out to be a matrix group isomorphic to the extraspecial group {\bf E}$_{2^{2w+1}}^+$ of order $2^{2w+1}$. An associated projective group of order $2^{2w}$ equally suffices.

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Alexis De Vos, Stijn De Baerdemacker. 2020-05-26. The decomposition of an arbitrary $2^w\times 2^w$ unitary matrix into signed permutation matrices. https://doi.org/10.1016/j.laa.2020.07.017

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