Flag fault-tolerant error correction with arbitrary distance codes

摘要

In this paper we introduce a general fault-tolerant quantum error correctionprotocol using flag circuits for measuring stabilizers of arbitrary distancecodes. Flag circuits use extra ancilla qubits to signal when errors resultingfrom $v$ faults in the circuit have weight greater than $v$. The flag errorcorrection protocol is applicable to stabilizer codes of arbitrary distancewhich satisfy a set of conditions and uses fewer qubits than other schemes suchas Shor, Steane and Knill error correction. We give examples of infinite codefamilies which satisfy these conditions and analyze the behaviour ofdistance-three and -five examples numerically. Requiring fewer resources thanShor error correction, flag error correction could potentially be used inlow-overhead fault-tolerant error correction protocols using low density paritycheck quantum codes of large code length.