1 Introduction
Quantum computing is an emerging computing paradigm for potential practical use since it enables us to use properties of quantum matters, such as superpositions and entanglements, for faster pro-cessing in several applications [1], [8], [1] [8]. On the other hand, since the lifetime of computational elements, called a quantum bit (qubit), is limited up to hundreds of microseconds [2] due to heating and calibration noise [12], the reliability of current quantum devices is not enough for their practical applications [13]. To cope with this problem, we can construct a logical qubit with noisy physical qubits using quantum error-correcting (QEC) codes. By checking the error parities with a sufficiently faster period than the lifetime of qubits repetitively and by increasing the code distance, i.e., using more physical qubits in the code, the occurred errors can be reliably estimated and corrected, and error rates of logical qubits can be suppressed to an arbitrarily small value [7], [17]. This error-estimation task is called a decoding process. Currently, surface codes [6] are believed to be the most promising QEC codes, since they can be implemented with nearest-neighboring interactions on two-dimensional grids and show high error-correction performance. While the surface codes still require decoders to solve complicated graph-matching problems within a strict time restriction, several hardware implementations that satisfy the restriction at a large code distance have been proposed.