  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. In
particular, we show that for all numbers of qubits $n\geq 28$, there exist
distance two stabilizer states which are counterexamples to the LU-LC
conjecture. We prove that for all odd $n\geq 195$, there exist stabilizer
states with distance greater than two which are LU equivalent but not LC
equivalent. Two important classes of stabilizer states that are of great
interest in quantum computation are the cluster states and stabilizer states of
the surface codes. To date, the status of these states with respect to the
LU-LC conjecture was not studied. We show that, under some minimal
restrictions, both these classes of states preclude any counterexamples. In
this context, we also show that the associated surface codes do not have any
encoded non-Clifford transversal gates. We characterize the CSS surface code
states in terms of a class of minor closed binary matroids. In addition to
making connection with an important open problem in binary matroid theory, this
characterization does in some cases provide an efficient test for CSS states
that are not counterexamples.
