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

• URL: http://arxiv.org/abs/2005.12959v1
• Date: Tue, 26 May 2020 18:26:05 GMT
• Title: The decomposition of an arbitrary $2^w\times 2^w$ unitary matrix into signed permutation matrices
• Authors: Alexis De Vos and Stijn De Baerdemacker
• Abstract summary: Birkhoff's theorem tells that any doubly 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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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