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.

翻译：曾有人猜想：两个图态在局部酉变换(LU)下等价当且仅当它们在局部克利福德变换(LC)下等价。这一被称为LU-LC猜想的命题在2007年被证伪——研究者发现了一对27量子比特的图态，它们LU等价但非LC等价。我们证明该反例具有最小性。换言之，对于不超过26量子比特的图态，LU等价性与LC等价性完全一致。这一结论通过研究2-局部补图（近日提出的r-局部补图的特例，也是经典局部补图的推广）的结构获得。我们利用了与三正交码及里德-穆勒码之间的关联。