The decoding of surface codes with twists

The standard surface code consists of weight-4 and weight-2 stabilizers, and can be efficiently decoded by a Minimum Weight Perfect Matching (MWPM) decoder. Some variants of surface code contain a twist, a stabilizer (usually weight-5) with a Pauli-Y measurement. The inclusion of twists allow for easy implementation of Clifford gates. However, the resulting codes are non-CSS, giving rise to a challenge in decoding. We study how adding a twist to surface code affects the decoding, with a focus on Litinski’s edge-tracked code.

We found that the MWPM decoder, after an adaptation to take in non-CSS codes, can decode the edge-tracked surface code up to full distance. We also analyzed the cases where MWPM fails to decode errors in the triangle code (and stellated surface codes in general), and attribute the failure to the tight-packing of the stellated codes. Finally, we calculated the thresholds of the edge-tracked surface code in phenomenological error models without measurement errors.