Deciding Local Unitary Equivalence of Graph States in Quasi-Polynomial Time

TL;DR

This paper presents a quasi-polynomial time algorithm for determining local unitary equivalence of graph states, improving efficiency and bounds for LU- and LC-equivalence in quantum graph states.

Contribution

It introduces a quasi-polynomial time algorithm for LU-equivalence, extending graphical characterisation methods and refining bounds on graph state equivalences.

Findings

Algorithm runs in $n^{ ext{log}_2(n)+O(1)}$ time.

LU- and LC-equivalence coincide for graph states up to 19 qubits.

Improves bounds on size of LU- but not LC-equivalent graph states.

Abstract

We describe an algorithm with quasi-polynomial runtime $n^{\log_2(n)+O(1)}$ for deciding local unitary (LU) equivalence of graph states. The algorithm builds on a recent graphical characterisation of LU-equivalence via generalised local complementation. By first transforming the corresponding graphs into a standard form using usual local complementations, LU-equivalence reduces to the existence of a single generalised local complementation that maps one graph to the other. We crucially demonstrate that this reduces to solving a system of quasi-polynomially many linear equations, avoiding an exponential blow-up. As a byproduct, we generalise Bouchet's algorithm for deciding local Clifford (LC) equivalence of graph states by allowing the addition of arbitrary linear constraints. We also improve existing bounds on the size of graph states that are LU- but not LC-equivalent. While the smallest known examples involve 27 qubits, and it is established that no such examples exist for up to 8 qubits, we refine this bound by proving that LU- and LC-equivalence coincide for graph states involving up to 19 qubits.