Quantum error correcting code can diagnose potential errors and correct them based on measured outcomes by leveraging syndrome measurement. However, mid-circuit measurement has been technically challenging for early fault-tolerant quantum computers and the readout-induced noise acts as a main contributor to the logical infidelity. We present a different method for syndrome extraction, namely generalized syndrome measurement, that requires only a single-shot measurement on a single ancilla, while the canonical syndrome measurement requires multiple measurements to extract the eigenvalue for each stabilizer generator. As such, we can detect the error in the logical state with minimized readout-induced noise. By adopting our method as a precheck routine for quantum error correcting cycles, we can significantly reduce the readout overhead, the idling time, and the logical error rate during syndrome measurement. We numerically analyze the performance of our protocol using Iceberg code and Steane code under realistic noise parameters based on superconducting hardware and demonstrate the advantage of our protocol in the near-term scenario. As mid-circuit measurements are still error-prone for near-term quantum hardware, our method could boost the applications of early fault-tolerant quantum computing.

Recently we became aware of an important reference that was published during the preparations of our manuscript, which we failed to cite in the original paper. In Ref. [1], the authors propose a similar scheme for the generation of multiple entangled pairs between qubit registers using a high-dimensional photonic qudit and cavity-mediated spin-photon gates. Contrary to Ref. [1], we show that such photonic qudit-mediated entanglement generation schemes have similar distribution rates as standard (parallel) qubit approaches but the memory requirements are significantly relaxed for the qudit schemes.

The generation of multiple entangled qubit pairs between distributed nodes is a prerequisite for a future quantum Internet. To achieve a practicable generation rate, standard protocols based on photonic qubits require multiple long-term quantum memories, which remains a significant experimental challenge. In this paper, we propose a novel protocol based on 2m-dimensional time-bin photonic qudits that allows for the simultaneous generation of multiple (m) entangled pairs between two distributed qubit registers and we outline a specific implementation of the protocol based on cavity-mediated spin-photon interactions. By adopting the qudit protocol, the required qubit memory time is independent of the transmission loss between the nodes, in contrast to standard qubit approaches. As such, our protocol can significantly boost the performance of near-term quantum networks.