The 27-qubit Counterexample to the LU-LC Conjecture is Minimal

TL;DR

This paper proves that the smallest known counterexample to the LU-LC conjecture involves 27 qubits, and for fewer qubits, LU and LC equivalences coincide, establishing minimality.

Contribution

It demonstrates that the 27-qubit counterexample is minimal by showing equivalence for all smaller graph states, using advanced code and graph transformation techniques.

Findings

Counterexample to LU-LC conjecture is minimal at 27 qubits

LU and LC equivalence coincide for graph states up to 26 qubits

Connection established with triorthogonal and Reed-Muller codes

Abstract

It was once conjectured that two graph states are local unitary (LU) equivalent if and only if they are local Clifford (LC) equivalent. This so-called LU-LC conjecture was disproved in 2007, as a pair of 27-qubit graph states that are LU-equivalent, but not LC-equivalent, was discovered. We prove that this counterexample to the LU-LC conjecture is minimal. In other words, for graph states on up to 26 qubits, the notions of LU-equivalence and LC-equivalence coincide. This result is obtained by studying the structure of 2-local complementation, a special case of the recently introduced r-local complementation, and a generalization of the well-known local complementation. We make use of a connection with triorthogonal codes and Reed-Muller codes.