Decomposition of Clifford Gates

TL;DR

This paper introduces a fast algorithm to decompose any Clifford gate into a minimal product of Clifford transvections, aiding in identifying commuting Pauli matrices using a novel symplectic group graphical method.

Contribution

It presents a new, efficient algorithm for minimal decomposition of Clifford gates and a graphical approach leveraging symplectic group structure.

Findings

Algorithm efficiently decomposes Clifford gates into minimal transvections.

Enables quick identification of Pauli matrices commuting with a given Clifford gate.

Uses a novel graphical method based on symplectic group properties.

Abstract

In fault-tolerant quantum computation and quantum error-correction one is interested on Pauli matrices that commute with a circuit/unitary. We provide a fast algorithm that decomposes any Clifford gate as a $\textit{minimal}$ product of Clifford transvections. The algorithm can be directly used for finding all Pauli matrices that commute with any given Clifford gate. To achieve this goal, we exploit the structure of the symplectic group with a novel graphical approach.