High-threshold stabilizer code on the honeycomb lattice

We consider a topological stabilizer code on the honeycomb lattice, the XYZ² code. The code is inspired by the Kitaev honeycomb model and fits in the framework of “matching codes” discussed by Wootton [J. Phys. A: Math. Theor. 48 215302 (2015)]. It utilizes weight-six (XYZXYZ) and weight-two (XX, ZZ or YY) parity checks on a planar hexagonal grid composed of 2d² qubits, with weight-three checks at the boundaries, stabilizing one logical qubit. We study the properties of the code assuming perfect stabilizer measurements using maximum likelihood decoding. For pure X, Y, or Z biased noise we can solve for the logical failure rate analytically, giving a threshold of 50%. In contrast to the rotated surface code and the XZZX code, which have code distance d² only for pure Y noise, here the code distance is 2d² for both pure Z and pure Y noise (for XX parity checks). The code also possesses distinctive properties in a biased noise error model which give rise to unidirectional syndromes in three directions depending on the bias.