We study a class of gauge fixings of the Bacon-Shor code at the circuit level, which includes a subfamily of generalized surface codes. We show that for these codes, fault tolerance can be achieved by direct measurements of the stabilizers. By simulating our fault-tolerant scheme under biased noise, we show the possibility of optimizing the performance of the surface code by stretching the bulk stabilizer geometry. To decode the syndrome efficiently and accurately, we generalize the union-find decoder to biased noise models. Our decoder obtains a $0.83\%$ threshold value for the surface code in quadratic time complexity.

• Received 16 December 2019
• Accepted 10 February 2020

DOI:https://doi.org/10.1103/PhysRevA.101.042312