On Local Equivalence, Surface Code States and Matroids

Understand

Recently, Ji et al disproved the LU-LC conjecture and showed that the local unitary and local Clifford equivalence classes of the stabilizer states are not always the same.

• Despite the fact this settles the LU-LC conjecture, a sufficient condition for stabilizer states that violate the LU-LC conjecture is missing.
• In this paper, we investigate further the properties of stabilizer states with respect to local equivalence.
• Our first result shows that there exist infinitely many stabilizer states which violate the LU-LC conjecture.