The 27-qubit Counterexample to the LU-LC Conjecture is Minimal
This resolves a foundational question in quantum information theory about the minimal size of counterexamples to the LU-LC conjecture, which is important for understanding quantum entanglement and equivalence classes of graph states.
The paper proves that the previously discovered 27-qubit counterexample to the LU-LC conjecture is minimal, showing that for graph states on up to 26 qubits, local unitary equivalence and local Clifford equivalence coincide.
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.