  Efficient decoding to estimate error locations from outcomes of syndrome
measurement is the prerequisite for quantum error correction. Decoding in
presence of circuit-level noise including measurement errors should be
considered in case of actual quantum computing devices. In this work, 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 the circuitlevel 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.
