A Birkhoff Connection Between Quantum Circuits and Linear Classical Reversible Circuits

Abstract

Birkhoff's theorem tells how any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. Similar theorems on unitary matrices reveal a connection between quantum circuits and linear classical reversible circuits. It triggers the question whether a quantum computer can be regarded as a superposition of classical reversible computers.

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

“…However, if we loosen the requirement of a decomposition in strictly permutation matrices, we can lift the restriction on the equal linesum of the unitary matrix. In [5] [6] it is demonstrated that an arbitrary U(n) matrix can be decomposed as a weighted sum of complex permutation matrices and, in partcular, of signed permutation matrices if n is equal to a power of 2, say 2 w . Because it was demonstrated before by us [7] that prime-powers hold interesting properties, in the present paper, we will focus on the special case of n = 2 w .…”