Algebraic characterisation of one-way patterns

TL;DR

This paper provides a comprehensive algebraic framework for understanding the structure of one-way quantum computation patterns, focusing on the positive branch and its relation to unitary matrices and measurement angles.

Contribution

It introduces a complete structural characterization of the positive branch of one-way patterns using phase map decomposition and matrix analysis.

Findings

Connected the structure of unitary matrices to measurement angles

Analyzed the primary structure of the phase map matrix

Progressed towards full characterization of efficient one-way implementations

Abstract

We give a complete structural characterisation of the map the positive branch of a one-way pattern implements. We start with the representation of the positive branch in terms of the phase map decomposition, which is then further analysed to obtain the primary structure of the matrix M, representing the phase map decomposition in the computational basis. Using this approach we obtain some preliminary results on the connection between the columns structure of a given unitary and the angles of measurements in a pattern that implements it. We believe this work is a step forward towards a full characterisation of those unitaries with an efficient one-way model implementation.