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

• URL: http://arxiv.org/abs/2603.25219v1
• Date: Thu, 26 Mar 2026 09:18:10 GMT
• Title: The 27-qubit Counterexample to the LU-LC Conjecture is Minimal
• Abstract summary: We prove that a pair of 27-qubit graph states that are LU-equivalent, but not LC-equivalent, was discovered.<n>For graph states on up to 26 qubits, the notions of LU-equivalence and LC-equivalence coincide.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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