Fault-tolerant syndrome extraction using bare syndrome qubits on a compass code. On the left, each red rectangle represents a Z-type stabilizer. cnot gates are applied from data qubits to ancilla qubits in the order specified by the dashed arrows. If a Z-type error occurs on the ancilla qubits after each cnot gate represented by a dashed red line, those propagate to the three groups of data qubit errors circled in the middle. The right-hand picture represents the decoder graph, with the three corresponding correlated errors highlighted in red.
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. doi: 10.1103/PhysRevA.101.042312
Shilin Huang, and Kenneth R. Brown
