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In this paper we introduce a general fault-tolerant quantum error correctionprotocol using flag circuits for measuring stabilizers of arbitrary distancecodes. In addition to extending flag error correction beyond distance-threecodes for the first time, our protocol also applies to a broader class ofdistance-three codes than was previously known. Flag circuits use extra ancillaqubits to signal when errors resulting from $v$ faults in the circuit haveweight greater than $v$. The flag error correction protocol is applicable tostabilizer codes of arbitrary distance which satisfy a set of conditions anduses fewer qubits than other schemes such as Shor, Steane and Knill errorcorrection. We give examples of infinite code families which satisfy theseconditions and analyze the behaviour of distance-three and -five examplesnumerically. Requiring fewer resources than Shor error correction, flag errorcorrection could potentially be used in low-overhead fault-tolerant errorcorrection protocols using low density parity check quantum codes of large codelength.

Flag fault-tolerant error correction with arbitrary distance codes