On Local Equivalence, Surface Code States and Matroids - Quantum PhysicsReport as inadecuate

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Abstract: Recently, Ji et al disproved the LU-LC conjecture and showed that the localunitary and local Clifford equivalence classes of the stabilizer states are notalways the same. Despite the fact this settles the LU-LC conjecture, asufficient condition for stabilizer states that violate the LU-LC conjecture ismissing. In this paper, we investigate further the properties of stabilizerstates with respect to local equivalence. Our first result shows that thereexist infinitely many stabilizer states which violate the LU-LC conjecture. Inparticular, we show that for all numbers of qubits $n\geq 28$, there existdistance two stabilizer states which are counterexamples to the LU-LCconjecture. We prove that for all odd $n\geq 195$, there exist stabilizerstates with distance greater than two which are LU equivalent but not LCequivalent. Two important classes of stabilizer states that are of greatinterest in quantum computation are the cluster states and stabilizer states ofthe surface codes. To date, the status of these states with respect to theLU-LC conjecture was not studied. We show that, under some minimalrestrictions, both these classes of states preclude any counterexamples. Inthis context, we also show that the associated surface codes do not have anyencoded non-Clifford transversal gates. We characterize the CSS surface codestates in terms of a class of minor closed binary matroids. In addition tomaking connection with an important open problem in binary matroid theory, thischaracterization does in some cases provide an efficient test for CSS statesthat are not counterexamples.

Author: Pradeep Sarvepalli, Robert Raussendorf

Source: https://arxiv.org/

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