High-Rate Concatenated Quantum Error-Correcting Codes for Qudits

@misc{nishi2026highrate,
  title = {{High-Rate Concatenated Quantum Error-Correcting Codes for Qudits}},
  author = {Takanori Nishi and Hayato Goto},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11225},
  eprint = {2610.11225},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11225},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - High-Rate Concatenated Quantum Error-Correcting Codes for Qudits
AU  - Takanori Nishi
AU  - Hayato Goto
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11225
UR  - https://arxiv.org/abs/2610.11225
AB  - Qudits provide a larger local Hilbert space than qubits and have recently attracted interest as building blocks for efficient fault-tolerant quantum computation (FTQC). In this work, we extend many-hypercube (MHC) codes, high-rate concatenated quantum codes, from qubits to qudits of prime dimension q. We construct the level-2 and level-3 MHC codes with parameters [ [6^2,4^2,2^2] ]_q and [ [6^3,4^3,2^3] ]_q, respectively, and generalize the level-by-level minimum-distance (LLMD) decoder originally proposed for the qubit MHC codes. In a qudit X-error model, we evaluate the code-capacity performance of the qudit MHC codes for dimensions up to q=13. We find that the error threshold increases monotonically with the local dimension q, reaching 10.0% for q=13, nearly double the value of 5.1% for qubits. Furthermore, for sufficiently large local dimensions, we observe a pronounced waterfall regime in which the logical error rate decreases substantially faster than expected from conventional distance-based estimates. To understand the origin of the observed qudit advantage, we analyze the decoding performance in terms of the entropy of the underlying q-ary noise channel. This analysis reveals a competition between the increased syndrome information available at larger q and the simultaneous growth of the typical physical error weight at fixed entropy. These results demonstrate that higher-dimensional quantum systems can substantially improve the performance of high-rate concatenated quantum codes and motivate further development of qudit-based FTQC architectures.
ER  -