Quantum codes in the Lee metric

AI summary

Overview

• Research area: Quantum error correction, qudit stabilizer codes, and fault-tolerant quantum memories.
• Technical level: Intermediate.
• Scope: The paper develops quantum coding methods based on the Lee metric, which measures the size of shifts and phase shifts rather than simply counting how many qudits an error touches.

What This Paper Is About

Most quantum codes treat all non-identity errors on a qudit as equally costly. This paper studies a more selective model: small shifts and phase shifts are treated as more likely than large ones, and errors are measured using the Lee metric. The authors build a theory for codes over (\mathbb{Z}_q), investigate how these codes behave under gates and noise, and examine whether large-(q) codes can support long-lived quantum memories.

Key Contributions

1. A quantum Lee-metric framework. The paper defines joint and separate Lee distances for stabilizer codes over (\mathbb{Z}_q), including composite as well as prime values of (q). The joint metric counts the size of both the (X)- and (Z)-shifts; for qubits, this makes a (Y) error count twice.

2. New code constructions and small-code results. The authors identify qubit codes whose Lee distance exceeds the best Hamming distance for the same numbers of physical and logical qubits. They also construct and analyze qudit codes, including lifted Hamming and Golay codes, perfect Lee codes, and codes that can be converted to qubit systems using the Gray map.

3. Results on gates, fault tolerance, and limits of sparse checks. The paper quantifies how Clifford gates can increase Lee weight and shows why transversal gates are not automatically fault tolerant under this error model. It also proves that quantum CSS Lee-LDPC codes have Lee distance at most (O(n)), uniformly in (q).

4. Long-lived classical and quantum memories at large (q). The authors introduce helical repetition codes with local checks and show that certain versions have exponentially long memory times under local Metropolis dynamics. Hypergraph products of these codes yield local two-dimensional quantum codes that protect the (Z)-error sector, while protection against both error sectors remains unresolved.

Main Findings

• Two ways to measure quantum errors: The separate Lee metric tracks (X)- and (Z)-error weights individually, which fits CSS codes. The joint Lee metric adds them together, fitting situations where bit and phase errors arise independently.

• Lee distance can improve small qubit codes: Among codes on at most nine qubits, the authors find codes with larger joint Lee distance than the maximum Hamming distance for the same code parameters. For example, a ([[9,1,5]]) code corrects any two (X) or (Z) errors, or any single (Y) error; the paper contrasts this with the eleven-qubit code needed to correct any two errors in the Hamming metric.

• Large (q) can increase distance at fixed code size, but not for sparse CSS families without limit: For general codes, Lee distance can grow with (q) even at fixed (n). In contrast, the paper proves that a nontrivial quantum CSS Lee-LDPC code has distance (O(n)), with a bound that does not grow with (q).

• Some qudit gates strongly amplify errors: The paper defines a gate’s Lee-weight spread and shows that it determines how much a Clifford gate can increase the weight of an error. The phase and SUM gates have spread two, while the spread of (\mathrm{MUL}_2) is ((q-1)/2) for odd (q).

• The Gray map connects qudit codes to qubit circuits: The (\mathbb{Z}_4) Gray map preserves Lee distance as Hamming distance on pairs of bits. Under this mapping, some qudit Clifford gates become gates beyond the qubit Clifford group, including the non-Clifford gate ((T\otimes T)\mathrm{CS}).

• Some large-(q) memories are long-lived, but the quantum result is partial: Helical repetition codes can have exponentially long memory times at fixed temperature under the conditions described in the paper. Their hypergraph products protect against (Z) errors in the examples studied; whether a quantum Lee-LDPC code can protect both error sectors at arbitrarily high temperature is open.

Methodology in Plain English

The researchers first define how to assign a cost to each shift and phase shift: small powers of the qudit operators have low Lee weight, while larger powers have higher weight. They use this to adapt the stabilizer-code framework and derive bounds on how many errors codes can detect or correct.

They then apply the framework in several ways. For qubits, they examine a database of small stabilizer codes and compare their best Lee distances with their Hamming distances. For qudits, they construct code families from classical codes and use algebraic tools to describe logical operators, including when (q) is composite. They also analyze how Clifford gates transform errors, and use the Gray map to translate certain (\mathbb{Z}_4) codes into qubit systems.

Finally, they study sparse-check codes and memory dynamics. Their proofs of limits on CSS Lee-LDPC codes use geometry-of-numbers results. Their memory-time analysis considers local Metropolis updates, where one site changes by a small shift at a time, and compares energetic barriers with the number of paths through the state space.

Why This Matters

The paper offers a way to judge a code by how errors are expected to occur on the hardware, rather than by treating all non-identity errors as equally likely. This could help guide code design for systems where small shifts or phase changes dominate.

Potential applications include:

• Qudit quantum processors: Designing error-correcting codes for hardware that uses more than two levels per physical system.
• Spin and nuclear-spin systems: Modeling noise dominated by spin operators and small changes in spin states, as discussed in the paper.
• Oscillator and rotor systems: Connecting Lee stabilizer codes to GKP codes, where logical errors can be viewed as phase-space displacements.
• Passive quantum memories: Exploring whether local interactions and large internal state spaces can keep stored information stable for long times.

Industry relevance: The work is theoretical, not a demonstration of a deployed device or a measured hardware improvement. Its relevance is in giving hardware and error-correction researchers a framework to assess when qudit encodings may be useful, what error propagation they must control, and which memory designs merit further testing.

Future Directions

• Determine whether quantum Lee-LDPC codes can protect both (X)- and (Z)-error sectors at arbitrarily high temperature.
• Develop practical decoders and evaluate whether the theoretical Lee-distance advantages translate into improved performance under realistic noise.
• Study fault-tolerant syndrome-extraction circuits: the paper highlights the need to understand whether circuits turn low-Lee-weight errors into larger errors.
• Explore code designs tailored to experimentally measured noise, including how the methods extend beyond the particular shift-and-phase error model studied here.

Target Audience

This paper is most useful to researchers and graduate students in quantum error correction, stabilizer codes, qudit computing, and quantum memory design. It may also interest experimentalists working with spin, atomic, trapped-ion, superconducting, oscillator, or rotor systems who want to connect their hardware’s error structure to code choice. Readers new to quantum coding can follow the motivation and main conclusions, though the mathematical development assumes familiarity with basic quantum errors and stabilizer codes.

Authors’ 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.