Majorana-XYZ subsystem code

Abstract

Building fault-tolerant quantum computers is a current priority for the quantum computing industry and academic counterparts. We present a quantum error correction code encoding a single logical qubit into $n=L^2$ physical qubits with distance $d = L$ as a subsystem code, and $n-O(\sqrt{n})$ logical qubits with distance $d=3$ as a standard stabiliser code. The logical single-qubit gates in the subsystem code are transversal. Under independent $X,Z$ noise we find a threshold of $1.1\% \pm 0.3\%$ for odd distance and $0.6\% \pm 0.1\%$ for even distance for the subsystem variant. The code detects every 1- and 2-qubit error, and additionally every error of weight 3 and higher (constrained by the distance) that is not a product of the 3-qubit check operations. These operations act on the gauge qubits leaving the subsystem code space invariant. The code can be implemented by qubits in a triangular configuration with the plaquettes as 3-qubit check operations.