Efficient decoding to estimate error locations from outcomes of syndrome measurements is the prerequisite for quantum error correction. Decoding in the presence of circuit-level noise including measurement errors should be considered in case of actual quantum computing devices. In this paper, we develop a decoder for circuit-level noise that solves the error estimation problems as Ising-type optimization problems. We confirm that the threshold theorem in the surface code under circuit-level noise is reproduced with an error threshold of approximately 0.4%. We also demonstrate the advantage of the decoder through which the $Y$ error detection rate can be improved compared with other matching-based decoders. Our results reveal that a lower logical error rate can be obtained using our algorithm compared with that of the minimum-weight perfect matching algorithm.

• Received 28 July 2023
• Accepted 19 October 2023
• Corrected 7 February 2024

DOI:https://doi.org/10.1103/PhysRevResearch.5.043261

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Published by the American Physical Society