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Steane's seven-qubit quantum code is a natural choice for fault-toleranceexperiments because it is small and just two extra qubits are enough to correcterrors. However, the two-qubit error-correction technique, known as "flagged"syndrome extraction, works slowly, measuring only one syndrome at a time. Thisis a disadvantage in experiments with high qubit rest error rates. We extendthe technique to extract multiple syndromes at once, without needing morequbits. Qubits for different syndromes can flag errors in each other. Thisgives equally fast and more qubit-efficient alternatives to Steane'serror-correction method, and also conforms to planar geometry constraints.  We further show that Steane's code and some others can be error-correctedwith no extra qubits, provided there are at least two code blocks. The roughidea is that two seven-qubit codewords can be temporarily joined into atwelve-qubit code, freeing two qubits for flagged syndrome measurement.

Fault-tolerant quantum error correction for Steane's seven-qubit color code with few or no extra qubits