On Local Equivalence, Surface Code States and Matroids

TL;DR

This paper investigates the properties of stabilizer states related to local equivalence, proving the existence of infinitely many counterexamples to the LU-LC conjecture and characterizing surface code states using binary matroids.

Contribution

It demonstrates the existence of infinitely many stabilizer states violating the LU-LC conjecture and characterizes surface code states through binary matroids, linking quantum information and matroid theory.

Findings

Existence of infinitely many stabilizer states violating LU-LC conjecture for n ≥ 28

Surface code states do not have encoded non-Clifford transversal gates

Characterization of CSS surface code states via minor closed binary matroids

Abstract

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.