Quantum codes in the Lee metric
Abstract
We introduce a quantum coding framework for discrete small-shift noise, in which errors on qudits are modeled as low-weight X- and Z-type Pauli shifts, analogous to small phase-space displacements in continuous-variable systems. This structure is approximately respected by nuclear-spin noise and captured by the Lee metric, motivating a quantum extension of classical Lee-metric coding theory. We develop such a formalism for stabilizer codes over ℤq for arbitrary q, using joint and separate Lee metrics for the X- and Z-components of errors. For qubits, the joint metric counts Y errors twice and can yield codes that detect and correct X and Z errors with fewer physical qubits than codes designed for the conventional Hamming metric. We discuss Lee-weight spreading under Clifford gates and qubitize codes over ℤ4 via the Gray map, finding codes with two-fold transversal non-qubit-Clifford gates. For quantum CSS Lee-LDPC codes on n qudits, we prove that the Lee distance cannot exceed O(n), uniformly in q, demonstrating an unexpected obstruction to using the large internal Hilbert space of a large-q qudit to make high-Lee-distance codes. Under local Metropolis dynamics, certain classical q-ary ``helical repetition codes'' have exponentially long memory times at fixed temperature for sufficiently large q (that grows with system length), which can be understood as spontaneous symmetry breaking at finite temperature, even in local one-dimensional models. Hypergraph products of these helical repetition codes provide local two-dimensional quantum codes that inherit self-correction for Z errors, but self-correction for X errors remains an open question.
Related
- Construction of Quantum Rank-Metric Codes Using Hermitian Orthogonality
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BibTeX
@misc{guo2026quantum,
title = {{Quantum codes in the Lee metric}},
author = {Jinkang Guo and Yiqiu Han and Aranya Chakraborty and Shubham P. Jain and Tianhao Liu and Victor V. Albert and Andrew Lucas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04834},
eprint = {2610.04834},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04834},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum codes in the Lee metric
AU - Jinkang Guo
AU - Yiqiu Han
AU - Aranya Chakraborty
AU - Shubham P. Jain
AU - Tianhao Liu
AU - Victor V. Albert
AU - Andrew Lucas
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04834
UR - https://arxiv.org/abs/2610.04834
AB - We introduce a quantum coding framework for discrete small-shift noise, in which errors on qudits are modeled as low-weight X- and Z-type Pauli shifts, analogous to small phase-space displacements in continuous-variable systems. This structure is approximately respected by nuclear-spin noise and captured by the Lee metric, motivating a quantum extension of classical Lee-metric coding theory. We develop such a formalism for stabilizer codes over ℤ_q for arbitrary q, using joint and separate Lee metrics for the X- and Z-components of errors. For qubits, the joint metric counts Y errors twice and can yield codes that detect and correct X and Z errors with fewer physical qubits than codes designed for the conventional Hamming metric. We discuss Lee-weight spreading under Clifford gates and qubitize codes over ℤ_4 via the Gray map, finding codes with two-fold transversal non-qubit-Clifford gates. For quantum CSS Lee-LDPC codes on n qudits, we prove that the Lee distance cannot exceed O(n), uniformly in q, demonstrating an unexpected obstruction to using the large internal Hilbert space of a large-q qudit to make high-Lee-distance codes. Under local Metropolis dynamics, certain classical q-ary ``helical repetition codes'' have exponentially long memory times at fixed temperature for sufficiently large q (that grows with system length), which can be understood as spontaneous symmetry breaking at finite temperature, even in local one-dimensional models. Hypergraph products of these helical repetition codes provide local two-dimensional quantum codes that inherit self-correction for Z errors, but self-correction for X errors remains an open question.
ER -