Surface code states are an important class of stabilizer states that play a prominent role in quantum information processing. In this paper we show that these states do not contain any counterexamples to the recently disproved LU-LC conjecture. In the process we show that surface codes do not have any encoded non-Clifford transversal gates. We also prove some interesting structural properties of the CSS surface code states. We show that these states can be characterized as a class of minor closed binary matroids. This characterization could be of independent interest in that it makes a connection with the theory of binary matroids.
展开▼
