Quantum codes in the Lee metric

2026-10-04

AI Summary

This study addresses the challenge of efficient coding against discrete small-shift noise in quantum error correction by constructing a quantum coding framework based on the Lee metric. Methodologically, it leverages stabilizer codes over Z_q, CSS-LDPC techniques, and Gray mapping, combined with hypergraph product constructions and Metropolis dynamics simulations to optimize error-correction efficiency. The primary contributions include proposing a joint Lee metric that reduces physical qubit overhead, enabling X/Z error correction with fewer qubits. Furthermore, this work establishes distance upper bounds for high-dimensional qudits and reveals the finite-temperature self-correcting mechanism of spiral repetition codes. It also demonstrates that two-dimensional quantum codes can inherit Z-error self-correction properties while identifying fundamental encoding barriers in the large-q regime.

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 $\mathbb{Z}_q$ 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 $\mathbb{Z}_4$ 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.

Problem

• Quantum error correction
• Lee metric
• Discrete small-shift noise
• Stabilizer codes
• Self-correcting quantum codes

Innovation

• Lee metric
• quantum error correction
• stabilizer codes
• LDPC codes
• self-correction