Global stabilizer quantum error correction with combinatorial arrays

Stabilizer codes are a fundamental class of error-correcting codes for quantum information that allow for syndrome decoding in the quantum domain. One of the substantial challenges in quantum error correction is that the quantum gates that perform error correction themselves are faulty in practice, which makes fault-tolerant implementation a vital component. Recently, a coding theoretic technique is proposed that makes it possible for syndrome decoding to help correct imperfectly extracted syndromes instead of fully relying on an external fault-tolerant mechanism. In particular, it was proved that any single-error-correcting stabilizer code can be made robust against single errors on either a data qubit or a syndrome bit by using at most one more stabilizer operator than necessary for standard syndrome decoding, while analogous overhead for making double-error-correcting stabilizer codes robust against double errors involving data qubits and/or syndrome bits was shown to be at most logarithmic. We generalize this result to t-error-correcting stabilizer codes and show that the overhead for achieving analogously defined global t-error correction for data qubits and/or syndrome bits is also at most logarithmic. The proof exploits combinatorial arrays that may be seen as parity-check matrices that detect errors of even weight but may overlook errors of odd weight.