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arXiv:2610.04834

Quantum codes in the Lee metric

Jinkang Guo, Yiqiu Han, Aranya Chakraborty, Shubham P. Jain, Tianhao Liu, Victor V. Albert, Andrew Lucas

quant-phcs.ITmath.COmath.MGSubmitted Oct 4, 2026• Announced Oct 6, 2026• 0 citations
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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.

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Finite-Size Decoding Guarantees for Stabilizer Quantum Error-Correcting Codes

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Abstract We develop a general framework for bounding logical error probabilities at finite code distance in stabilizer quantum error-correcting codes. Rather than relying on asymptotic threshold argu ments, the analysis isolates decoder locality and control over syndrome connectivity as sufficient conditions for exponential suppression of logical errors at finite size. Under a stochastic Pauli noise model, we show that component-local decoders admit an explicit finite-size bound determined by the probability of large connected syndrome structures. The framework is formulated for stabilizer codes defined on bounded-degree syndrome graphs and is independent of any specific decoding algorithm. We prove a general finite-size decoding theorem and illustrate its application to the two-dimensional surface code, where the bound scales exponentially with the code distance. We further discuss the extension of the framework to quantum low-density parity-check and hypergraph-product codes. A minimal numerical illustration confirms the qualitative finite-size scaling predicted by the theory. These results provide a complementary perspective to fault-tolerance threshold theorems by mak ing explicit the mechanisms governing logical error suppression at finite code distance, with direct relevance to near- and intermediate-scale quantum devices.

•Dec 19, 2025•0 citations
We introduce a quantum coding framework for discrete small-shift noise, in which errors on qudits are modeled as low-weight XXX- and ZZZ-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\mathbb{Z}_qZq​ for arbitrary qqq, using joint and separate Lee metrics for the XXX- and ZZZ-components of errors. For qubits, the joint metric counts YYY errors twice and can yield codes that detect and correct XXX and ZZZ 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\mathbb{Z}_4Z4​ via the Gray map, finding codes with two-fold transversal non-qubit-Clifford gates. For quantum CSS Lee-LDPC codes on nnn qudits, we prove that the Lee distance cannot exceed O(n)O(n)O(n), uniformly in qqq, demonstrating an unexpected obstruction to using the large internal Hilbert space of a large-qqq qudit to make high-Lee-distance codes. Under local Metropolis dynamics, certain classical qqq-ary ``helical repetition codes'' have exponentially long memory times at fixed temperature for sufficiently large qqq (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 ZZZ errors, but self-correction for XXX errors remains an open question.
We develop a theory of stabilizer codes whose stabilizer groups consist of commuting qudit Paulis, oscillator displacements, and/or planar-rotor displacements. We consider codes whose stabilizer groups form generalized lattices in quantum phase space, a condition that guarantees a finite logical dimension. We construct oscillator-rotor, rotor-qudit, and oscillator-rotor-qubit codes that cannot be decomposed into separate subsystems by generalized Clifford transformations. We also introduce an oscillator-qubit code arising from a partially lifted Golay code, where Construction A is applied to half of the Golay code coordinates. We derive analytical forms of logical operators from the symplectic dual lattice, determine Clifford-Gaussian gates from lattice symmetries, show that the code dimension equals the covolume of the stabilizer lattice, and organize the syndrome subspaces into a vector bundle. Our technique builds on earlier non-commutative geometric results by Rieffel, but it can often be interpreted simply as concatenating a given stabilizer code with a Gottesman-Kitaev-Preskill (GKP) code, applying conventional lattice-theoretic results, and unconcatenating. We formulate distances, optimal decoding, and quantum weight enumerators, recovering the Pauli, qudit, and GKP versions together with known MacWilliams identities. We construct finite-energy codewords whose encoding is an isometry up to an error exponentially small in the inverse damping parameter.
We construct self-correcting 3D surface codes over Zq\mathbb Z_qZq​ with local dimension qqq growing with system size. The model preserves the local geometry and stabilizer support of the ordinary 3D surface code and remains translation invariant in the bulk, while modifying only the stabilizer Pauli powers. We use the Lee metric to characterize error weight and consider local Metropolis dynamics with single-qudit X±1X^{\pm1}X±1 and Z±1Z^{\pm1}Z±1 updates, each changing the energy by at most a constant independent of qqq and system size NNN. We prove that, below the critical temperature, when the local dimension scales as q=Θ(N)q=Θ(N)q=Θ(N), the memory times in both error sectors grow at least as exp⁡[Ω(N1/3)]\exp\left[Ω(N^{1/3})\right]exp[Ω(N1/3)]. Our numerical results further support these analytical results. Together, these results establish growing local Hilbert-space dimension as a new resource for designing self-correcting quantum memories while retaining simple local geometry and local thermal dynamics.
We introduce an adaptive simulated annealing algorithm to search for moderately-sized quantum error-correcting codes with high encoding rates and large distances. Our search targets two classes of stabilizer codes: (1) CSS codes, and (2) a subclass of CSS codes that we call ``self-dual with equivalent logicals'' (SWEL) codes, the latter of which which admit transversal implementations of logical Hadamard and phase gates that can be leveraged to construct fault-tolerant gate sets. The search is guided by an energy function that acts as a surrogate for the logical error rate in a code-capacity noise model, combining code distance with a count of minimum-weight logical operators to resolve the discrete plateaux that impede naïve distance optimization. For block lengths of up to 505050 physical qubits, our search finds state-of-the-art CSS and SWEL codes whose distances frequently meet or exceed the variants of the quantum Gilbert-Varshamov bound. In addition to providing favorable seed codes for fault-tolerant architectures based on code concatenation, the codes found in this work are promising candidates for high-rate code demonstrations on near-term quantum computing hardware.
We introduce an intrinsic formulation of quantum error correction based on representation theory, in which error-protection structure is encoded directly in a unitary group representation, rather than being tied to a particular embedding into a larger Hilbert space. In this framework, error models are classified according to the isotypic decomposition of the conjugation action on the operator algebra. Our main result, the \emph{Schur bootstrap}, shows that if an intrinsic code satisfies the Knill--Laflamme conditions on a given symmetry sector, then the same error-protection relations hold for every extrinsic realization obtained from a group-equivariant isometric embedding into a larger Hilbert space. Thus a single intrinsic verification certifies the corresponding symmetry-resolved error-correction conditions across an entire family of physical realizations. We further introduce an intrinsic notion of distance, called depth, defined via adjoint order. For standard multi-qudit systems this coincides with conventional code distance, while for more general representations it refines the usual weight-based notion. We also prove an intrinsic Eastin--Knill theorem: any intrinsic code of depth at least two has a discrete logical symmetry group, with the obstruction to continuous covariant gates arising from the representation-theoretic structure of the adjoint action. We illustrate the framework with several examples, including a minimal SU(2)\mathrm{SU}(2)SU(2) construction that unifies permutation-invariant qubit codes and bosonic codes, and higher-dimensional constructions exhibiting transversal Clifford symmetries and realizations beyond qubit systems.
We study the structure and transversal logical capabilities of stabilizer quantum error correcting codes. Among our results, we identify universal lower bounds on circuit depth to generate a full logical Clifford algebra, and develop novel constructions of logical transversal gates including a new depth-one transversal phase S‾\mathrm{\overline{S}}S gate in the rotated surface code and a depth-one intra-block CZ‾\mathrm{\overline{CZ}}CZ gate in the 2D-toric code that generalizes to all odd distances and all lengths L≥3L\ge3L≥3, respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters [[n,n,Θ(nβ)]][[n,\sqrt{n},Θ({n^β})]][[n,n​,Θ(nβ)]] where β≈0.2823β\approx 0.2823β≈0.2823 in one demonstrated case, that provably possesses a constant-depth complete 2-local transversal logical Clifford basis instruction set architecture (ISA) composed of all individually targeted S‾\mathrm{\overline{S}}S, SHS‾=X\mathrm{\overline{SHS}} = \sqrt{X}SHS=X​, and CZ‾\mathrm{\overline{CZ}}CZ gates. This ISA is depth-one for certain subfamilies that we design and generally constant-depth under certain conditions. The code family is built from a small code with parameters [[n0,2,d0]][[n_0, 2, d_0]][[n0​,2,d0​]], and is tunable in the standard way: it tiles out to form utility-scale logical qubit counts, and it scales up through concatenation to achieve higher distances and error suppression. We show that this construction preserves the depth-one complete transversal logical Clifford basis ISA when composed with these commuting construction actions, inheriting structure from the core codes so that at scale the complete logical Clifford basis ISA remains depth-one up to depth-two addressable operations between tiled cores. We call these Quantum Logic Codes.
Abstract We develop a general framework for bounding logical error probabilities at finite code distance in stabilizer quantum error-correcting codes. Rather than relying on asymptotic threshold argu ments, the analysis isolates decoder locality and control over syndrome connectivity as sufficient conditions for exponential suppression of logical errors at finite size. Under a stochastic Pauli noise model, we show that component-local decoders admit an explicit finite-size bound determined by the probability of large connected syndrome structures. The framework is formulated for stabilizer codes defined on bounded-degree syndrome graphs and is independent of any specific decoding algorithm. We prove a general finite-size decoding theorem and illustrate its application to the two-dimensional surface code, where the bound scales exponentially with the code distance. We further discuss the extension of the framework to quantum low-density parity-check and hypergraph-product codes. A minimal numerical illustration confirms the qualitative finite-size scaling predicted by the theory. These results provide a complementary perspective to fault-tolerance threshold theorems by mak ing explicit the mechanisms governing logical error suppression at finite code distance, with direct relevance to near- and intermediate-scale quantum devices.