  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.
