Many-hypercube codes: High-rate quantum error-correcting codes for high-performance fault-tolerant quantum computing

Standard approaches to quantum error correction for fault-tolerant quantum computing are based on encoding a single logical qubit into many physical ones, resulting in asymptotically zero encoding rates and therefore huge resource overheads. To overcome this issue, high-rate quantum codes, such as quantum low-density parity-check codes, have been studied over the past decade. In this case, however, it is difficult to perform logical gates in parallel while maintaining low overheads. Here we propose concatenated high-rate small-size quantum error-detecting codes as a new family of high-rate quantum codes. Their simple structure allows for a geometrical interpretation using hypercubes corresponding to logical qubits. We thus call them many-hypercube codes. They can realize both high rates, e.g., 30% (64 logical qubits are encoded into 216 physical ones), and parallelizability of logical gates. Developing dedicated decoder and encoders, we achieve high error thresholds even in a circuit-level noise model. Thus, the many-hypercube codes will pave the way to high-performance fault-tolerant quantum computing.