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Efficient Fault-Tolerant Syndrome Measurement of Quantum Error-Correcting Codes Based on "Flag"

Fault-tolerant syndrome measurement plays an important role in the process of quantum error correction, and considerable effort had been taken for reducing the physical overhead of syndrome measurement which including the ancilla qubits, the CNOT gates and the time-slots. The two extra qubits syndrome measurement technique, known as "flag"-style syndrome measurement, cuts down the number of extra qubits to the utmost. However, it works slowly because it measures only one syndrome at a time. We extend the technique to extract all syndromes of the distance-3 quantum error-correcting code at once. We propose a new method that increases the parallelism of the syndrome measurement circuit and reduces the time overhead by allocating and adjusting the order of CNOT gates for measuring data block reasonably, which we call dynamic time-slot allocation scheme, and which is applicable to both Hamming codes and color codes. For a CSS quantum error-correcting code with the number of stabilizers m and the maximum weight w, we only need 2m extra qubits and 2 × (w+2) time-slots for one-shot measurement of all syndromes.