Surface codes, Ising Models, and Topological Defects

Abstract

Surface codes are currently among the most promising schemes of quantum architectures that have a high threshold of noise within which reliable computations can be performed. While the major drawback of surface codes is their large overhead, recent implementations using topological twist defects on the code have been shown to significantly reduce the required quantum resources, but no theoretical estimates have been made for the thresholds of these methods. To attempt to answer this question, here we use the mapping to random bond Ising models (RBIMs) to compare twists to the topological Ising spin-flip and duality defects. We derive the mapping in the presence twists for an independent and identical X and Z Pauli error model and find that in the case of unbiased noise the threshold of the code is unchanged. For unbiased noise, we show that there exist, cousins of the twist defects, that when mapped to the Ising model, satisfy a modified version of topological invariance; one where shifting the entire defect line leaves the partition function invariant. We obtained a trivial solution for these defects, which we refer to as sink defects, and found that it leaves the threshold unchanged. However, the cases we investigated were restrictive and the problem of deriving the threshold of surface codes in the presence of twist defects remains broadly unanswered.