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

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.

4th paper in a series of Birkhoff decompositions for unitary matrices [(1) arXiv:1509.08626; (2) arXiv:1606.08642; (3) arXiv:1812.08833]

References in corpus (1)

• The Birkhoff theorem for unitary matrices of arbitrary dimensions

Cited by in corpus (1)

• H2ZIXY: Pauli spin matrix decomposition of real symmetric matrices