The decoder graphs of the toric code: the syndrome bits are vertices, the data errors are horizontal edges, and the type-I measurement errors are vertical edges. The ancilla blocks when aligned (a) lead to timelike correlations between directed repeated measurements. By offsetting the ancilla blocks (b), the timelike correlations require spacelike errors in order to correlate defects from top to bottom.
Fault-tolerant quantum error correction requires the measurement of error syndromes in a way that minimizes correlated errors on the quantum data. Steane and Shor ancilla are two well-known methods for fault-tolerant syndrome extraction. In this Letter, we find a unifying construction that generates a family of ancilla blocks that interpolate between Shor and Steane. This family increases the complexity of ancilla construction in exchange for reducing the rounds of measurement required to fault tolerantly measure the error. We then apply this construction to the toric code of size L×L and find that blocks of size m×m can be used to decode errors in O(L/m) rounds of measurements. Our method can be applied to any Calderbank-Shor-Steane code and presents a new direction for optimizing fault-tolerant quantum computation. https://doi.org/10.1103/PhysRevLett.127.090505
Shilin Huang and Kenneth R. Brown
