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PreprintCC BY 4.0arXiv

The Quantum Hamming Bound in Arbitrary Local Dimension

  • Yu-Xuan Zhang
  • Jing-Ling Chen
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Abstract

The quantum Hamming bound arises from a direct dimension estimate for exact quantum error-correcting codes. For nondegenerate codes, distinct correctable errors map the code space into orthogonal subspaces, so dimension counting directly gives the bound; degeneracy allows these subspaces to overlap and invalidates the argument. Although linear-programming, asymptotic, and structural approaches have advanced the problem from several directions, the general local-dimension case has remained open. Here we prove the quantum Hamming bound for every local dimension q≥3. Building on the exact two-center Fourier reduction, we develop new overlap estimates for nonbinary Hamming spaces. For q≥4, Jacobi estimates and monotonicity give a uniform gap; qutrits form the boundary case where the gap vanishes and are handled by a filtered correction. This completes the finite-length nonbinary case; together with the existing qubit result, it establishes the quantum Hamming bound for arbitrary local dimension.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026quantum,
  title = {{The Quantum Hamming Bound in Arbitrary Local Dimension}},
  author = {Yu-Xuan Zhang and Jing-Ling Chen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.22538},
  eprint = {2606.22538},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.22538},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Quantum Hamming Bound in Arbitrary Local Dimension
AU  - Yu-Xuan Zhang
AU  - Jing-Ling Chen
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2606.22538
UR  - https://arxiv.org/abs/2606.22538
AB  - The quantum Hamming bound arises from a direct dimension estimate for exact quantum error-correcting codes. For nondegenerate codes, distinct correctable errors map the code space into orthogonal subspaces, so dimension counting directly gives the bound; degeneracy allows these subspaces to overlap and invalidates the argument. Although linear-programming, asymptotic, and structural approaches have advanced the problem from several directions, the general local-dimension case has remained open. Here we prove the quantum Hamming bound for every local dimension q≥3. Building on the exact two-center Fourier reduction, we develop new overlap estimates for nonbinary Hamming spaces. For q≥4, Jacobi estimates and monotonicity give a uniform gap; qutrits form the boundary case where the gap vanishes and are handled by a filtered correction. This completes the finite-length nonbinary case; together with the existing qubit result, it establishes the quantum Hamming bound for arbitrary local dimension.
ER  -