Subsystem many-hypercube codes

We propose high-rate quantum error-correcting codes with low-weight syndrome measurements based on the concatenated [[4,2,2]] error-detecting codes, which are a family of the many-hypercube codes [1]. Concatenated codes in general contain syndrome measurements of weights that grow exponentially with respect to the concatenation level. We resolve this problem by making the concatenated codes into subsystem codes [2]. The resulting codes have a constant syndrome-measurement weight of 4, while keeping high encoding rates. We develop the block-MAP and neural-network decoders and show that they demonstrate better performance than the BP-OSD and bounded-distance decoder.

[1] H. Goto, Science Advances 10, eadp6388 (2024).

[2] R. Nakai and H. Goto, arXiv:2510.04526.