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

TL;DR

This paper explores how any unitary matrix of size 2^w can be decomposed into a specific subgroup of signed permutation matrices, revealing a connection to a particular extraspecial group relevant for quantum circuits.

Contribution

It identifies a subgroup of signed permutation matrices, isomorphic to an extraspecial group, that can decompose any 2^w-sized unitary matrix, advancing quantum circuit analysis.

Findings

Decomposition uses a subgroup isomorphic to an extraspecial group.

The associated projective group of order 2^{2w} also suffices.

Provides a group-theoretic framework for unitary matrix decomposition.

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.