For quantum error correction (QEC) of logical qubits, it is necessary to perform error decoding, which estimates physical-qubit errors and measurement errors based on the syndrome information obtained with ancillary qubits. Since we need to solve a minimum-weight perfect matching (MWPM) problem for error decoding at a given throughput within an allowed latency, a scalable algorithm and its parallel hardware implementation are required especially for higher code-distance cases. In this talk, I will introduce our research project on QEC as the JST Moonshot Goal 6 project, where we develop an error correction algorithm for a surface code and an FPGA backend system the allows scalable execution of the algorithm, targeting superconducting qubits.