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Quantum codes in the Lee metric

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

arXiv:2610.04834v1quant-phcs.ITmath.COmath.ITmath.MG

TL;DR

Qudit code analyses often treat all errors on a qudit equally, although realistic systems can have dominant error types and small-shift errors. The paper develops stabilizer coding over Zq using Lee metrics and applies it to code constructions, gate behavior, and memory models. It finds qubit codes with improved error-correction parameters over Hamming-metric designs and local quantum codes with self-correction for Z errors, while protection of both sectors remains unresolved.

  • Problem

    Existing qudit code analyses treat all q² − 1 independent errors as equally likely, although errors such as spin-j angular-momentum noise can be dominant and scaling single-qudit codes remains unresolved.

    Source: §1.1 · p. 3

  • Method

    The paper develops stabilizer-code formalism over Zq using joint and separate Lee metrics to measure X- and Z-shift errors.

    Source: §1.2 · p. 5

  • Results

    A [[9, 1, 5]] code corrects any two X or Z errors or any single Y error, while the shortest code correcting any two Hamming-metric errors has eleven qubits.

    Source: §1.2 · p. 5

  • Takeaways & Limitations

    Lee-distance designs can achieve the stated X- and Z-error correction with fewer physical qubits than the shortest code correcting two errors in the Hamming metric.

    Source: §1.2 · p. 5

    Whether quantum Lee-LDPC stabilizer codes can have arbitrarily high passive-memory thresholds through mechanisms beyond those protecting only Z errors remains unresolved and requires direct analysis of specified local thermal dynamics.

    Source: §8.3 · p. 63

Abstract

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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.

1 Introduction & summary

The paper develops Lee-metric quantum coding to characterize qudit errors whose shifts are small, addressing the assumption that all errors on a qudit are equally significant. It applies this framework to code constructions, Clifford gates, Lee-LDPC bounds, and passive-memory models.

1.1 Qudits, done right

  • Small-shift qudit errors motivate Lee-metric coding because conventional distance treats all errors on a qudit equally, despite realistic systems having dominant error types.The framework models modular X and Z shifts and counts their powers, not only the number of qudits affected.Source: §1.1 · pp. 3–4

1.2 Summary of results

  • A [[9, 1, 5]] code corrects any two X or Z errors or any single Y error, while the shortest code correcting any two Hamming-metric errors has eleven qubits.The joint Lee metric counts Y errors twice, unlike the conventional Hamming metric.Source: §1.2 · p. 5
  • The paper defines joint and separate Lee distances, derives sphere-packing bounds, and identifies perfect codes including families whose Lee distance grows with q at fixed n.The perfect-code results include one- and two-qudit families and CSS codes on n qudits of dimension 2n + 1 for n mod 3 ≠ 1.Source: §1.2 · p. 5
  • For composite q, CSS codes are described through integer-chain-complex homology, whose Smith normal form identifies logical-qudit dimensions dividing q and explicit conjugate logical operators.The logical quotient ker(HX)/im(HTZ) is expressed as homology.Source: §1.2 · p. 5
  • Shortened self-dual Hamming and Golay codes yield lifted CSS families whose Lee distances grow with q, alongside a Reed–Solomon example with Lee distance four and Hamming distance three.The lifted Hamming distance reaches seven by q = 16 and cannot exceed seven; lifted Golay examples include distances 10, 12, and 17.Source: §1.2 · p. 5
  • Clifford gates have a Lee-weight spread that bounds fault distance, and transversal gates are not automatically fault tolerant in this metric.Phase and SUM gates have spread two, whereas MUL2 has spread (q − 1)/2 for odd q; specific CSS codes retain fault-tolerant transversal gates.Source: §1.2 · p. 6
  • The Gray-map qubitization of Z4 codes can produce qubit Clifford stabilizer codes and depth-one circuits containing non-Clifford gates.The construction preserves distance under stated CSS conditions, and the single-block transversal Z4 Clifford gates become depth-one two-qubit circuits.Source: §1.2 · p. 6
  • Quantum CSS Lee-LDPC codes have Lee distance O(n) uniformly in q, while hypergraph products of helical codes have exponentially long memory for Z errors; protection of both sectors remains unresolved.The quantum constructions protect the Z-error sector in the examples studied, and whether both error sectors can be protected at arbitrarily high temperature remains open.Source: §1.2 · pp. 6–7

2 Motivating the Lee metric

The Lee metric measures shift magnitude on qudits rather than only the number of affected qudits, and its joint form also distinguishes qubit Y errors from X and Z errors. The paper motivates the metric with spin-noise expansions concentrated at low Lee weight and shows qubit codes that exploit it.

2.1 q-dimensional qudits with clock and shift errors

  • A Lee weight reflects the magnitude of each shift, so it can distinguish errors acting on the same number of qudits but with different shift sizes.The motivation is a setting where X is more likely than X2 and Z more likely than Z2.Source: §2.1 · p. 8

2.2 Warm-up with qubits: Y errors are rarer than X or Z errors

  • For qubits, the joint Lee metric counts X and Z as weight one and Y as weight two, matching independent bit- and phase-flip processes.The separate Lee metric instead records the X- and Z-type weights independently and is suited to CSS codes.Source: §2.2 · pp. 8–9
  • Two inequivalent [[6, 2, 3]] codes correct any single X or Z error and detect any single Y error, whereas Hamming distance three on six qubits allows only one logical qubit.Two logical qubits with Hamming distance three first occur at eight physical qubits in the cited classification.Source: §2.2 · p. 9
  • Using nine qubits, the code corrects every error of Lee weight at most two, while the shortest code correcting any two Hamming errors uses eleven qubits.The five-qubit code gains no distance from the Lee metric; its Lee distance remains three in every single-qubit Clifford frame.Source: §2.2 · p. 11

2.3 Spin noise expansion

  • The spin-noise test applies to spin-j qudits with q = 2j + 1 levels, where dephasing and transverse fluctuations are generated by Jz and Jx or Jy.The passage identifies dephasing as the primary error in nuclear and semiconductor spin qubits and treats the linear angular-momentum terms.Source: §2.3 · p. 12
  • For every spin, Lee weight one carries more than 6/π² (about 61%) of Jz’s squared norm; as q → ∞, the Lee-weight distribution tends to 6/(π²w²).For |b|L ≪ q, Jz’s Fourier coefficients decay inversely with Lee weight, with the slow decay caused by the sawtooth jump between m = j and m = −j.Source: §2.3 · p. 13
  • As q → ∞, the fraction of Jx at joint Lee weight one tends to 3π2/32, approximately 93%.For j = 9/2, Figure 1 reports 0.87 at weight one and 0.96 at weight at most two; the expansions have nonzero tails, so the Lee-noise approximation is not exact.Source: §2.3 · p. 13

3 Stabilizer codes

The paper extends stabilizer codes over Zq to joint and separate Lee distances, defines perfect codes through sphere-packing bounds, and relates Lee distance to Hamming distance. It gives perfect-code examples and shows that CSS Lee-LDPC distance has a q-independent linear-in-n upper bound in at least one sector.

3 Stabilizer codes

  • A code with joint Lee distance at least 2t + 1 corrects errors in a Lee ball of radius t, while separate distance at least 2t + 1 corrects errors whose X and Z parts each have Lee weight at most t.The joint ball lies inside the separate-metric square for the same t, so the separate-distance requirement is more demanding.Source: §3 · p. 13, Figure 2 · p. 14

3.1 Perfect codes

  • Perfect codes saturate the sphere-packing bound when the correctable errors have distinct syndromes; Table 2 lists joint and separate-metric examples, including CSS codes.The table caption states that every listed entry with q ≥ 4 is perfect over its listed parameter range.Source: §3.1 · p. 16, Table 2 · p. 17
  • The [[6, 4, 3⋄]]5 code is perfect and corrects any single X±1 or Z±1 error on its six qudits while encoding four qudits.Its radius-one Lee ball contains 25 errors with distinct syndromes, one for each syndrome value.Source: §3.1 · p. 16
  • For q = 3m with m ≥ 2, a perfect two-qudit family has joint Lee distance three and encodes a logical space of dimension m2.For m = 2, this is a [[2, 2 log6 2, 3⋄]]6 code storing a four-dimensional logical space in two six-dimensional qudits.Source: §3.1 · pp. 16–17

3.2 Relation to Hamming distance

  • The Lee and Hamming distances obey bounds linked to the logical operator's Hamming weight; for qubits, the Y-parity code has d⋄ = 2dH, while the five-qubit code attains dH = d⋄ = 3 and d□ = 2.The lower bound in (3.13b) is attained by the five-qubit code.Source: §3.2 · pp. 17–18
  • For CSS codes, at least one of dX and dZ is at most linear in n, uniformly in q, although Lee distance can grow unboundedly with q at fixed n.The passage also notes examples whose distance grows as √q at fixed K or m.Source: §3.2 · p. 18

4 CSS codes

This section develops CSS and stabilizer-code frameworks over Z_q and illustrates them with examples across several alphabets, including perfect and lifted codes. It also connects these codes to discrete and multimode GKP codes, relating logical classes and Lee weight to displacement weight.

4.1 Classical codes on Zq

  • Smith factors express a classical Zq code as cyclic factors determined by gcd(q,d_i), with free Zq factors and possible torsion factors.Torsion occurs when a factor Z_gcd(q,d_i) is neither trivial nor Zq.Source: §4.1 · p. 19

4.2 Quantum CSS codes

  • For composite q, the two quantum CSS formulas for k agree because |ker(H)| |im(H^T)| = q^n.This establishes that the encoded Hilbert-space dimension is well-defined even when k is not an integer.Source: §4.2 · p. 20
  • A single Smith normal form over Z describes general quantum CSS codes through an integer chain complex and yields logical-operator bases, including torsion logical qudits when q is composite.The resulting codespace can contain factors Z_d for divisors d of q; when q is prime, the codespace is isomorphic to Z_q^k.Source: §4.2 · pp. 21–22

4.3 Small examples

  • A one-qudit example has codespace Z_c′ and logical operators Z^(q/a′) and X^(a′/c′); it is perfect exactly when a′ = b′ and q/a′ = 2t + 1 is odd.Choosing a = b = K(2t + 1) and q = K(2t + 1)^2 gives perfect [[1, log_q K, 2t + 1]]_q codes.Source: §4.3 · p. 22
  • Two-dimensional Lee-ball tilings yield perfect two-qudit CSS codes, including [[2, 2 log_10 2, 3]]_10, which has Hamming distance one yet admits continuous transversal logical gates.This code corrects every error of Lee weight one despite its Hamming distance of one.Source: §4.3 · p. 23
  • Radius-one Lee tilings exist in every dimension and yield perfect quantum codes with parameters [[n, n - 2, 3]]_(2n+1); for n ≥ 3, both distances are exactly 3.The first members are [[3, 1, 3]]_7, [[5, 3, 3]]_11, and [[6, 4, 3]]_13.Source: §4.3 · p. 23
  • Shortening self-dual codes yields self-dual CSS codes; at q = 2, the extended Hamming and Golay codes give the Steane [[7, 1, 3]]_2 and quantum Golay [[23, 1, 7]]_2 codes.The extended Hamming and Golay families are self-dual for q powers of two, with Lee distances that grow with q.Source: §4.3 · pp. 23–24
  • The quantum heptacode is [[7, 1, 4]]4; this family’s distance never exceeds seven and saturates at q = 16.Shortening the extended code at two coordinates yields a [[6, 2, 3]]4 code.Source: §4.3 · p. 24
  • The lifted Golay family gives [[23, 1, 10]]4, [[23, 1, 12]]8, and [[23, 1, 17]]16 codes, with distance capped at 23.Shortening the extended code at two coordinates yields a [[22, 2, 8]]4 code.Source: §4.3 · p. 24
  • The quantum Reed–Solomon code [55] [56] [110] on Z5 is [[5, 1, 4]]5, with Lee distance four despite Hamming distance three.The passage states that no nondegenerate CSS code with these parameters has a larger Lee distance.Source: §4.3 · pp. 24–25

4.4 Realization of Zq stabilizer codes as multimode GKP codes

  • Every Z_q CSS code has a multimode GKP realization with corresponding logical classes and reduced weights proportional to Lee weight.One unit of Lee weight corresponds to one elementary displacement on the GKP grid.Source: §4.4 · pp. 25–27
  • The GKP construction extends to arbitrary Z_q stabilizer codes, including non-CSS codes, with the same proportionality between reduced weights.Source: §4.4 · p. 27

5 Fault tolerance

In the Lee metric, transversal gates can increase error weight, so transversality alone does not guarantee fault tolerance. The section defines Lee error spread, bounds its effect on fault tolerance, and applies the framework to Clifford gates and example codes.

5.1 Lee error spread

  • Phase and SUM gates have spread two, while MUL2 has spread (q −1)/2 for odd q.For odd q, the spread of MUL2 can therefore grow with q.Source: §5.1 · pp. 28–29
  • Lee spread equals the largest Lee weight of a symplectic-matrix column, and depth-D circuits with gate spread at most w have spread at most w^D.Spread is submultiplicative, while disjoint gate layers have the maximum spread of their gates.Source: §5.1 · p. 29
  • For a logical Clifford gate of spread wB, the bound certifies fault tolerance only when wB ≤ t and reduces the guaranteed fault distance to about t/wB.Here t is the code's correctable Lee-weight radius.Source: §5.1 · pp. 29–30
  • Spread-one gates are qudit permutations and single-qudit Fourier gates; they preserve joint Lee distance and define natural equivalence of Lee stabilizer codes.For CSS codes, the relevant subgroup also preserves separate Lee distances and maps CSS codes to CSS codes.Source: §5.1 · p. 30

5.2 Codes with Clifford gates

  • CSS codes with HX = HZ have transversal Fourier, phase, and SUM gates; their per-block spread is one, so they tolerate t faults.The joint-metric bound certifies only ⌊t/2⌋ faults for phase and SUM, a loss the text says is only in the bound for CSS codes.Source: §5.2 · pp. 30–31
  • On the Z5 Reed–Solomon code, transversal single-qudit Clifford gates realize the full logical Clifford group, isomorphic modulo phases and Paulis to the order-120 group 2I.The comparison code cited in the text is a qubit code, and the two codes are not directly comparable.Source: §5.2 · p. 31
  • For perfect two-qudit CSS codes at fixed m, t grows with q; spread-two logical phase gates tolerate at least one fault once q ≥ m(2w^2 + 2w + 1).The transversal Fourier and phase gates have fault distance t + 1 in this family.Source: §5.2 · p. 31
  • The perfect-code GKP realizations converge to two square GKP qudits mixed by a balanced 50:50 beam splitter as q grows at fixed m.In the limit, each Lee-weight-one fault corresponds to a grid displacement whose size tends to zero.Source: §5.2 · p. 32

6 Qubitizing codes with Gray maps

This section studies Gray-map qubitization of Z_4 stabilizer codes, including when code distance is retained and how quartrit Pauli, Clifford, and transversal operations translate into qubit operations. It also presents qubitized code examples and their parameters.

6.1 Quantum Gray maps

  • The Z4 Gray map preserves Lee weight and distance as Hamming weight and distance, though its image of a Z4-linear code is generally nonlinear.The image is linear exactly when 2u*v belongs to the code for all codewords u and v.Source: §6.1 · p. 33
  • Quantum Gray maps are restricted to q = 4: for q > 4, some qubit phase flip pulls back to an operator with Pauli terms of Lee weight at least two.For the reflected Gray code at q = 8, 4 has Hamming weight two but Lee weight four.Source: §6.1 · p. 33
  • Under the Z-basis quantum Gray map, quartrit Pauli operators map to qubit Clifford gates, and Z4 Clifford gates map into the third level of the qubit Clifford hierarchy.The image is a qubit Clifford gate if and only if the reduced symplectic matrix represents a permutation of the quartrits.Source: §6.1 · pp. 34, 36

6.2 Qubitizing Z4 stabilizer codes and transversal gates

  • A qubitized Z4 stabilizer code is generally a joint eigenspace of commuting qubit Clifford gates; even stabilizer generators suffice for its image to be an ordinary stabilizer code.Its joint Lee distance is bounded below by the input code’s joint Lee distance recomputed with a discount of two at sites where a logical operator is X±1Z2.Source: §6.2 · p. 37
  • The smallest quartrit code with joint Lee distance 2 qubitizes to a two-qubit Y-parity code with joint Lee distance 2 and Hamming distance 1.The quartrit code has separate Lee distance 1; its qubitized joint Lee distance is preserved, while Hamming distance loses a factor of two.Source: §6.2 · p. 38
  • For pure Z4 CSS codes, the Z-basis Gray map preserves distance min(dX, dZ) in both metrics, while µX < min(dX, dZ) can reduce the qubitized distance.More generally, the distance is min(dX, dZ) when every nonzero X-check vector has Lee weight at least min(dX, dZ).Source: §6.2 · p. 39
  • Semi-self-dual Z4 CSS codes qubitize without loss: their qubitized joint Lee and Hamming distances both equal min(dX, dZ).The same holds for the X-basis map when the Z-check module is contained in the X-check module.Source: §6.2 · p. 39
  • A Bell-basis map raises the seven-quartrit example’s qubit Hamming distance from 2 to 3 when dZ is the smaller CSS distance.The Z-basis map gives a ((14, 4, 2))H code, whereas the Bell-basis map gives a ((14, 4, 3))H code; an X-basis analogue serves when dX is smaller.Source: §6.2 · pp. 39–40
  • The Gray-mapped code is a qubit stabilizer code exactly under two conditions, including linearity of φ(C); when they hold, its CSS checks are generated by φ(D) and φ(C)⊥.For K ≥ 2, the resulting code has distance min(dX, dZ) in both metrics.Source: §6.2 · p. 40
  • Gray-map examples include [[30, 2, 6]]H and [[46, 2, 6]]H CSS codes obtained from [[15, 1, 6]]4 and [[23, 1, 6]]4 codes, respectively.The smallest instance is the [[4, 1, 2]]H Chuang–Leung–Yamamoto code [129], obtained by qubitizing a [[2, log4 2, 2]]4 code.Source: §6.2 · p. 40
  • Transversal Z4 Clifford gates and SUM translate under the Gray map into two-fold transversal qubit gates in the level-three hierarchy.The Table 6 codes need no Pauli correction for transversal S because their classical codes are Type II.Source: §6.2 · p. 41, Table 6 · p. 41

7 Lee-LDPC codes

This section develops the structure and limits of quantum Lee-LDPC codes, including bounds on logical operators, Lee distances, and energy barriers, and the role of integer commutation in controlling Lee-weight error propagation. It also examines hypergraph-product and helical-code constructions, their memory properties, and limits on the lifted-Hamming family’s Lee-LDPC status.

7 Lee-LDPC codes

  • Lee-LDPC checks bound Lee weight as well as density, because bounded Hamming-weight checks can still permit hook errors with Lee weight growing with q.Ordinary quantum LDPC codes limit hook-error weight through bounded-weight generators; the Lee-LDPC extension addresses the failure of Hamming weight alone to control Lee-weight propagation.Source: §7 · p. 42, §7.1 · p. 42

7.1 Quantum Lee-LDPC codes

  • When q > wC^2, lifting the checks of a (wB, wC)-Lee-LDPC CSS code to integers makes HXHZ^T = 0 over Z, enabling codespace decomposition from the separate Smith normal forms.The integer commutation condition is the basis for the logical decomposition stated in Proposition 7.3.Source: §7.1 · pp. 42–43
  • For torsion logical pairs, reduced Lee weight is at most wC n/2 for X or Z operators and at least q/αX or q/αZ for their respective duals.The X-side bounds use αX, and the Z-side bounds use αZ.Source: §7.1 · pp. 43–44

7.2 Bounds on logical operators and energy barriers

  • Every nontrivial quantum CSS Lee-LDPC code has a nontrivial logical operator with energy barrier at most O(n), and every noncommuting logical pair has a noncommuting witness of Lee weight at most O(n), uniformly in q.These are the section's main bounds on logical operators and energy barriers.Source: §7.2 · p. 45
  • Any integer codeword admits a single-site path whose syndrome Lee weight stays at most 2wB n, and reducing the path modulo any q ≥ 2 preserves that bound.The path is constructed by rounding a continuous interpolation and connecting successive configurations with single-coordinate updates.Source: §7.2 · pp. 45–46
  • Quantum CSS Lee-LDPC codes have a nontrivial logical operator in at least one error sector with energy barrier O(n), with a constant independent of q.The theorem bounds the barrier by 2w_B n when q > w_C^2.Source: §7.2 · pp. 46–47
  • Every nontrivial quantum Lee-LDPC CSS code has Lee distance O(n), uniformly in q, and the corresponding subsystem codes have dressed Lee distance O(n).The short noncommuting-witness bound is proved by relaxing an integer minimization and norms from ℓ1 to ℓ2, then applying generalized dual-lattice transference inequalities.Source: §7.2 · p. 47
  • The Lee-metric and Z_q Pauli-algebra assumptions are essential to the distance bound; a cited example outside those assumptions has distance q with bounded-weight stabilizers.The cited example is Example 8.7 in Section 8.3.Source: §7.2 · p. 47

7.3 Hypergraph products and the lifted-Hamming toric code

  • The lifted-Hamming construction is nontrivial exactly when 7 divides L, and for L=7m it encodes 18 qudits of dimension q; its translation-enriched order has an order-seven action.For q=4, coarse-graining by seven sites leaves nine decoupled toric codes; the same order-seven action for larger q=2^r is expected, while braiding was verified only at q=4.Source: §7.3 · pp. 49–50
  • For q=4 and L=7, the lifted-Hamming toric code is [[98,18,6]]_4 with Hamming distance four, versus [[98,2,7]]_4 for the Z_4 toric code on the same torus.Its Lee-to-Hamming distance ratio is 3/2; the reported Lee distance is attained by exhaustive enumeration of errors of Lee weight at most five.Source: §7.3 · p. 49
  • As q grows, the lifted-Hamming checks’ Lee weights grow at least as √q, so this family is Lee-LDPC only at fixed q.The check support remains eight edges, while string weights also grow without bound in q at fixed geometry.Source: §7.3 · p. 49

7.4 Helical repetition codes and their hypergraph products

  • For sufficiently large q, the helical repetition code has exponentially growing Lee distance, d > a^(n−1).The proposition also identifies the open-boundary codespace and states when periodic-boundary codewords coincide with it.Source: §7.4 · p. 50
  • Hypergraph products of horizontal and vertical helical repetition codes give local two-dimensional codes whose checks act on at most four neighboring qudits, and whose Z-error sectors inherit the classical codes’ long memory.Their check Lee weight is at most a_h + b_h + a_v + b_v, independent of n and q.Source: §7.4 · p. 51

8 High-temperature memory

This section examines how energy and entropy barriers in single-site Metropolis dynamics can produce long-lived helical-code memories, including fixed-temperature protection against Z errors under construction-dependent conditions. It also discusses counterexamples and explains why distance and energy-barrier bounds do not settle the unresolved existence of arbitrarily high-threshold quantum Lee-LDPC memories.

8 High-temperature memory

  • Memory time is the first time decoding failure reaches ε, and any lower bound on memory time also lower-bounds mixing time.For a nontrivial code, decoding failure under the Gibbs distribution is at least 1/2, yielding the memory-to-mixing comparison.Source: §8 · pp. 51–52

8.1 Memory from energy barriers

  • At any fixed β > 0, the periodic helical repetition code has memory time exponential in q as q grows exponentially with chain length n.Linear soundness creates a free-energy bottleneck between codeword sectors: the energetic contribution dominates shell entropy at finite temperature, producing Gibbs-measure condensation.Source: §8.1 · pp. 54, 57
  • In mixed-boundary helical HGP codes, the Z-error sector has exponentially large memory time at any fixed β > 0 along the specified parameter families.The mechanism uses an O(q) energy barrier and approximate linear confinement around codewords.Source: §8.1 · p. 58

8.2 Memory from “entropic” effects

  • For open-boundary helical repetition codes, memory time is exponential at every finite temperature when q grows superlinearly with n.The energy barrier is only O(n) and independent of q, but the energy-flat logical direction is exponentially long while transverse escape costs energy of order q; the classical code also has a glassy free-energy landscape.Source: §8 · p. 51, §8.2 · pp. 58–59
  • For fixed β > 0, endpoint Gibbs probabilities each decrease exponentially with n, yet any endpoint set holding a fixed positive Gibbs fraction must contain exponentially many values.The endpoint partition functions are far from uniform: the largest-to-average ratio grows exponentially with n.Source: §8.2 · p. 60
  • The endpoint-weight structure resembles spin-glass shattering but arises without disorder, without fragmentation, and without Gibbs weight concentrating on finitely many endpoint values.Neighboring endpoint values remain connected by local updates.Source: §8.2 · p. 60
  • Open-boundary helical hypergraph-product codes can have exponentially long Z-error Metropolis memory at any fixed inverse temperature, via an exponentially long bottleneck cylinder.The proof uses a generalized cylinder whose side is energetically suppressed; reduction modulo q preserves the bottleneck.Source: §8.2 · pp. 60–61

8.3 On the possibility of high-temperature memory

  • O(n) distance and energy-barrier bounds do not by themselves rule out arbitrarily high passive-memory thresholds; the transference bound also depends on the Zq Pauli algebra.The section argues that behavior must be analyzed for a specified stochastic dynamics rather than inferred from distance and barrier alone.Source: §8.3 · p. 61
  • For odd q, a bounded-weight-stabilizer code using a more general nice error basis has distance q and a memory temperature scale Tc ∼ q.Its elementary errors fall outside the Zq Pauli algebra assumed by the transference bound; for even q, they cannot connect different logical sectors.Source: §8.3 · pp. 61–62
  • A classical length-n repetition code has distance n and an O(1) energy barrier, yet its uncorrected memory time is Θ(n^2) under the described dynamics.For any fixed finite u, a global decoder recovers the initial codeword with probability approaching 1 after T = un steps.Source: §8.3 · pp. 62–63
  • The results neither establish nor rule out arbitrarily high passive-memory thresholds for quantum Lee-LDPC codes through mechanisms beyond those protecting the Z-error sector.Resolving this requires direct analysis of specified local thermal dynamics.Source: §8.3 · p. 63

9 Outlook

The outlook presents the Lee metric as a more faithful way to characterize robustness to small clock and shift errors. It also highlights both possible physical consequences and unresolved engineering and memory questions.

9 Outlook

  • The Lee metric captures robustness to small clock and shift errors more faithfully than the Hamming metric used in existing literature.Source: §9 · p. 63
  • Reduced error-graph connectivity could let codes protect against more errors at the same Hilbert-space dimension and, in some cases, support high-temperature self-correction in local one-dimensional models.Source: §9 · p. 63
  • The Lee-distance framework could extend to more general nice error bases, but a systematic account of what is possible remains future work.Source: §9 · p. 63
  • The analysis does not evaluate Lee-LDPC codes under circuit-level noise, including whether syndrome extraction converts small Lee-weight errors into larger errors.Source: §9 · p. 64

A Background and relation to previous work

The background connects Lee-metric coding to small-shift noise and several qudit settings, then positions the paper against existing work. It identifies the absence of a general Lee-metric stabilizer-code theory as the gap addressed.

A Background and relation to previous work

B Proof of Theorem 4.5

The proof lifts the code's chain complex over Zq to an integer chain complex, then relates the resulting homology to the original logical-operator spaces. Smith normal form identifies the torsion structure underlying the logical operators.

B Proof of Theorem 4.5

  • The integer lift of the chain complex is described as a mapping cone, and the proof uses it to relate integer homology to the code over Zq.Source: B Proof of Theorem 4.5 · p. 65
  • The reduction map induces an isomorphism from the integer-lift homology quotient to the corresponding kernel/image quotient over Zq.The proof establishes injectivity and surjectivity by lifting elements and using HXz = qt when HXz vanishes modulo q.Source: B Proof of Theorem 4.5 · p. 66
  • Smith normal form shows that nontrivial Z-logical operators arise from torsion in kerZ(D1) not contained in imZ(D2), with cyclic factors whose orders divide q.Source: B Proof of Theorem 4.5 · p. 67
  • The X-logical construction follows by exchanging X and Z roles, corresponding to the cochain complex.Source: B Proof of Theorem 4.5 · pp. 65, 67

C Proof of Proposition 7.3

The proof constructs logical-operator bases using integer representatives and Smith normal form, then establishes how the X- and Z-bases pair. Reduction modulo q yields dual logical bases.

C Proof of Proposition 7.3

  • Integer representatives are used in Zn during the proof, while codewords and logical operators are defined over Zq.Source: C Proof of Proposition 7.3 · p. 68
  • The logical Z-operator basis separates into free and torsion blocks, with torsion contributions determined by gcds involving the Smith-normal-form entries and q.Source: C Proof of Proposition 7.3 · p. 69
  • After reduction modulo q, the constructed X- and Z-operator bases are dual logical bases with the stated free- and torsion-block pairings.Source: C Proof of Proposition 7.3 · p. 70

D Proof of Theorem 7.8

The proof bounds noncommuting logical operators by separating torsion cases from a residual free-lattice case. In that residual case, lattice transference supplies a short noncommuting witness.

D Proof of Theorem 7.8

  • Torsion components can already yield a logical representative with Lee weight O(n), avoiding the residual lattice argument.The proof reduces stabilizer coefficients to [−1/2, 1/2] and bounds the resulting weight using the Lee weight of independent check rows.Source: D Proof of Theorem 7.8 · p. 71
  • For the residual case, projecting away torsion preserves the logical pairing and produces dual Euclidean lattices for the free logical operators.The proof then uses transference to find a short vector that pairs nontrivially with a logical operator.Source: D Proof of Theorem 7.8 · pp. 71–72

D.1 Proof of Lemma D.2

  • If a lattice vector is nontrivial modulo q, Lemma D.3 supplies a vector pairing nontrivially modulo q with squared Euclidean norm at most 4Dq(v).This converts a distance-to-multiples quantity into a short vector that preserves a nonzero pairing.Source: D Proof of Theorem 7.8 · pp. 72–73

E Short noncommuting witnesses in non-CSS codes

Appendix E extends the short-witness result to general stabilizer codes, beyond CSS codes. Its proof separates free and torsion logical components before treating noncommuting pairs.

E Short noncommuting witnesses in non-CSS codes

  • Appendix E extends Theorem 7.8 to general stabilizer codes, using a free/torsion decomposition of the logical group.The decomposition is built from Smith normal form and dual logical bases.Source: E Short noncommuting witnesses in… · pp. 75–76
  • For Lee-LDPC codes, each torsion pair has at least one logical class with a representative of reduced Lee weight O(n).Here “free” and “torsion” refer to the integer decomposition used in the proof.Source: E Short noncommuting witnesses in… · p. 76
  • Theorem E.4 treats nontrivial noncommuting logical pairs in quantum non-CSS stabilizer codes and establishes a short noncommuting witness.The proof projects the residual pair into the free part while preserving its nonzero symplectic pairing.Source: E Short noncommuting witnesses in… · pp. 79–80

F Proof of Proposition 8.4

The proof analyzes memory protection for Z errors in a mixed helical hypergraph-product code. It gauge-fixes errors so the leading syndrome reduces to independent horizontal helical-code syndromes.

F Proof of Proposition 8.4

  • The proof reads off the code-space and distance information from the Smith normal forms of the Z- and X-check matrices.The Z-check matrix has rank n(n−1) with nonzero Smith factors equal to 1; the X-check matrix has rank n^2 and one remaining factor Dh after n^2−1 factors of 1.Source: F Proof of Proposition 8.4 · p. 81
  • The q-scale bottleneck established here is specific to the Z-error sector and does not establish arbitrarily high-temperature memory protection for X errors.The proof’s lower-bound argument is consequently restricted to the Z-error side.Source: F Proof of Proposition 8.4 · p. 81
  • Gauge fixing bounds one block of qudit variables, leaving the leading X-syndrome as the syndromes of n independent horizontal periodic helical codes.The bounded variables contribute an O(n^2) correction to the confinement estimate.Source: F Proof of Proposition 8.4 · pp. 81, 83

G Proof of Proposition 8.5

The proof lower-bounds escape from a cylinder around a helical codeword orbit by separating cap and side exits. The cylinder is selected so its lifted representatives remain distinct after reduction modulo q.

G Proof of Proposition 8.5

  • When 2R + 1 < q/a, reduction modulo q maps the lifted cylinder one-to-one into the configuration set.Distinct affine lifts of the wrapped orbit are at least q/a apart in ℓ1 distance.Source: G Proof of Proposition 8.5 · p. 85
  • The cylinder boundary splits into side and cap exits, which the proof analyzes separately.For a cap exit, each single-site update changes the axial coordinate by at most Δ.Source: G Proof of Proposition 8.5 · pp. 85–86
  • The proof combines the cap-exit and side-exit bounds to control the probability of leaving the set while the decoder remains correct throughout it.The conclusion is made only after both exit routes are bounded.Source: G Proof of Proposition 8.5 · p. 88

H Free-energy landscape of the classical helical repetition code with open boundary conditions

This section examines the Gibbs-weight landscape of the classical helical repetition code with open boundary conditions, focusing on endpoint slices along its energy-flat direction. The weight is highly nonuniform, concentrated on exponentially many slices that form an exponentially small fraction of all fixed-endpoint slices, reflecting configurations with occasional low-density slips between codeword sectors.

Free-energy landscape

  • Any set of endpoint slices carrying a fixed positive fraction of the Gibbs weight must contain exponentially many slices, and every individual slice has exponentially small probability.The unrestricted partition function diverges along the flat direction, so the distribution is defined after quotienting by the primitive translation and studied on fixed-endpoint slices.Source: H Free-energy landscape of the… · pp. 89–90
  • For fixed β > 0, transfer-operator asymptotics give spectral radius 1 < ρa,b(β) < a, yielding exponentially decaying Gibbs probability for each fixed-endpoint slice.The Ruelle–Perron–Frobenius theorem gives the asymptotics; the row-sum bound and Zβ(r) ≤ Zβ(0) imply the slice-probability decay.Source: H Free-energy landscape of the… · pp. 91–94
  • The endpoint r is in bijection with an a-ary digit block B, whose probability under the independent-bond measure equals Zβ(r)/Ztot.The digits are generated by Euclidean division with carries, and the resulting block uniquely records the endpoint.Source: H Free-energy landscape of the… · pp. 94–95
  • The infinite carry construction yields stationary, ergodic digits B with entropy rate hβ < log a.The carries are independent of the bonds to their left; the stationary ergodic digit process is obtained as a measurable shift-equivariant function of the joint carry-and-bond process.Source: H Free-energy landscape of the… · pp. 96–97

H Free-energy landscape of the classical helical repetition code with open boundary conditions

  • For every fixed β > 0, 1 < ρ_a,b(β) < a, so the r = 0 slice has exponentially greater Gibbs weight than the average slice, while every individual slice has exponentially vanishing probability.Consequently, exponentially many slices are needed to carry any fixed positive fraction of the total Gibbs probability.Source: H Free-energy landscape of the… · p. 98
  • A set S_n−1 carries Gibbs probability tending to 1 while its size divided by a^n−1 tends to 0 exponentially fast.Thus the weight is concentrated on exponentially many slices, but they form an exponentially small fraction of all fixed-endpoint slices.Source: H Free-energy landscape of the… · p. 99
  • The dominant configurations locally resemble helical codewords but jump between codeword sectors; for a = 2 and b = 1, low-density slips yield relatively large slice weights.The passage illustrates a slip near a fixed position that makes a slice's partition function smaller than, but not much smaller than, Z_β(0).Source: H Free-energy landscape of the… · p. 99

I Proof of Proposition 8.6

The proof establishes a primitive integer logical Z generator, bounds the time to escape an integer cylinder under single-site Metropolis dynamics, then transfers that bound to the code over Zq. For fixed β > 0, the Z-error memory time is exponentially large in n when q grows sufficiently fast relative to the entropy scale.

Step 1: Logical generator

  • Every integer Z-logical operator is an integer multiple of the primitive generator ℓZ, modulo integer Z-stabilizers.The proof first lifts the check matrices and logical operators to the integer lattice.Source: I Proof of Proposition 8.6 · pp. 100–101

Step 2: Integer cylinder and memory-time estimate

  • The integer cylinder restricts syndrome energy and selects one fundamental interval along the logical direction, allowing a decoder to map its states to the zero codeword.The memory time is lower bounded by the time needed to reach the cylinder boundary.Source: I Proof of Proposition 8.6 · p. 102
  • The proof bounds escape through the cylinder’s caps and side; before the smaller of those escape times, cap exit is impossible and side-exit probability is at most 1/4.The side estimate uses the number and Gibbs weight of integer syndromes in an energy shell.Source: I Proof of Proposition 8.6 · pp. 102–103

Step 3: Reduction modulo q and the memory-time bound

  • For fixed β > 0, the Z-error Metropolis memory time is exponentially large in n when q grows sufficiently fast compared with the entropy scale n^2 log.The integer-cylinder bound transfers to Zq because reduction is injective there and preserves syndrome and Lee energy.Source: I Proof of Proposition 8.6 · pp. 99, 104
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