Between Shor and Steane: A unifying construction for measuring error syndromes

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 paper, 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 $L\times L$ and find that blocks of size $m\times m$ can be used to decode errors in $O(L/m)$ rounds of measurements. Our method can be applied to any Calderbank-Shor-Steane codes and presents a new direction for optimizing fault-tolerant quantum computation.