Conjectured $DXZ$ decompositions of a unitary matrix

TL;DR

This paper explores a conjecture that all divisors of a unitary matrix's dimension admit specific ZXZ-like decompositions, extending known cases, and proposes an iterative algorithm to approximate such decompositions.

Contribution

It introduces a conjecture generalizing existing ZXZ decompositions for all divisors of matrix dimension and provides an iterative algorithm to approximate these decompositions.

Findings

The conjecture extends known decompositions to all divisors of the matrix dimension.

An iterative Sinkhorn algorithm is proposed for numerical approximation.

Preliminary results suggest the feasibility of the conjectured decompositions.

Abstract

For any unitary matrix there exists a ZXZ decomposition, according to a theorem by Idel and Wolf. For any even-dimensional unitary matrix there exists a block-ZXZ decomposition, according to a theorem by F\"uhr and Rzeszotnik. We conjecture that these two decompositions are merely special cases of a set of decompositions, one for every divisor of the matrix dimension. For lack of a proof, we provide an iterative Sinkhorn algorithm to find an approximate numerical decomposition.