Metachecks in Bivariate Bicycle Codes: Syndrome Distance, Measurement Faults, and Repair Limits

I. INTRODUCTION

• Redundant stabilizer checks provide parity constraints on valid syndromes, complementing repeated extraction, which adds latency and enlarges the decoding problem.The paper asks when these constraints distinguish measurement faults and how ambiguity can leave a residual error for the data decoder.Source: §I · p. 1

A. Relation to Prior Work and Scope

• The work extends prior BB syndrome analysis beyond coprime periods and adds translation-orbit and quotient-algebra accounts of measurement ambiguity, finite-code costs, partial repetition, and decoder comparisons.It also uses an annihilator–colon decomposition and certified component minima established in [20].Source: A. Relation to Prior Work and Scope · p. 1

B. Main Results

• The 108-qubit example’s colon component contains a weight-10 logical, although the component with obvious one-block witnesses first appears at weight 12.Both logical components must be checked; a one-block search alone would report the wrong distance.Source: B. Main Results · pp. 1–2
• Translation classes identify syndrome locations with identical single-fault metacheck information, characterizing collisions and the checks that must be remeasured under a static-data assumption.The same analysis bounds distinguishable fault locations when encoded dimension stays bounded as the lattice grows.Source: B. Main Results · p. 2
• Section V assigns each leading ambiguity the minimum data-error weight that produces its residual syndrome, separating repair-ambiguity frequency from the difficulty of the remaining data error.Finite-code and decoder experiments compare these predictions with minimum-weight repair, joint decoding, and repeated space-time decoding.Source: B. Main Results · p. 2

II. NOTATION AND BB SYNDROME STRUCTURE

• The notation sets N = lm and models the two N-qubit blocks using translations on Z_l × Z_m and multiplication matrices over a binary group ring.A 3 × 3 torus supplies the running example for the later definitions and calculations.Source: §II · p. 2, Figure 2 · p. 3

A. CSS convention and selected error sector

• Decoding X errors makes Z-type checks produce the measured syndrome; the selected BB check matrices have equal rank under the antipode permutation.For the BB(3, 3) example, H has rank five, so four of nine syndrome bits are redundant parity relations.Source: A. CSS convention and selected error… · p. 3

B. Syndrome code and metachecks

• The running example’s metacheck basis comes from four independent translates of c = (1 + x)(1 + y), with the resulting matrix satisfying MH = 0.The paper frames the design question as how well metacheck columns separate measurement faults and what data errors accompany remaining collisions.Source: B. Syndrome code and metachecks · p. 3
• The metacheck space consists of parity constraints on valid syndromes, and its matrix cancels the data contribution from a noisy measurement.For the running BB(3, 3) example, four independent metachecks distinguish all nine single measurement faults.Source: B. Syndrome code and metachecks · pp. 3–4

C. Matrix and group-ring descriptions

• Table II serves as a dictionary between the ordinary binary-matrix and group-ring descriptions of the main objects.The matrix calculations are used alongside the group-ring form, which exposes annihilators, ideals, and translations.Source: C. Matrix and group-ring descriptions · p. 4, Table II · p. 5

III. LOGICAL DECOMPOSITION AND DISTANCE OBSTRUCTIONS

• For the same running example, k = 8; Proposition 1 gives four logical dimensions in the annihilator kernel and four in the colon quotient, at the level of dimensions.Source: §III · p. 5
• The running BB(3, 3) example has k = 8 and a weight-two annihilator logical, so it is useful for notation and syndrome redundancy but not as a good-distance code.Its exact sequence divides the eight logical dimensions into four annihilator dimensions and four colon-quotient dimensions.Source: §III · p. 5
• A component label does not fix a representative's support: adding a stabilizer can populate the other block and lower its weight.Thus a left-block witness gives an upper bound on the annihilator minimum, not necessarily its value.Source: §III · p. 5
• In the 108-qubit example, the colon-component logical has weight 10, while the one-sided search finds a weight-12 witness.The decomposition explains why the true distance is ten rather than merely showing a lighter representative that occupies both blocks.Source: §III · p. 5
• When the generators are equal and the code encodes a logical qubit, paired single-qubit errors have identical syndromes and yield a nontrivial weight-two logical, so dX = dZ = 2.The paired-error argument also supplies a decoder-independent lower bound under independent X errors and a fast rejection rule.Source: §III · p. 5

Examples

• For BB(3, 3), KM = {1} and dS = 3, so all nine single-fault metasyndromes are distinct; BB(6, 3) instead has 18 locations in nine indistinguishable pairs.The BB(6, 3) pairs are {h, x3h}.Source: §IV · p. 6

Distance-two characterization and orbit factorization

• A nontrivial translation g ∈ KM makes the valid syndrome 1 + g weight two, characterizing the leading single-fault ambiguity when k > 0.The orbit theorem also factors metasyndrome formation through orbit parities and then a smaller matrix.Source: §IV · p. 6, Figure 6 · p. 6
• For a translation subgroup KM of size κ, single faults at h and gh have identical metasyndromes exactly when g ∈ KM, partitioning syndrome locations into indistinguishable orbits.The metasyndrome retains each orbit’s fault parity, not the fault’s location within the orbit.Source: §IV · p. 6

Finite-rate obstruction for metasyndrome-only repair

• When dS = 2 and κ > 1, metasyndrome-only repair fails on at least a fraction 1 − 1/κ of uniformly located single faults; minimum-weight lookup attains this bound.The independent-flip bound applies to metasyndrome-only exact reconstruction, not joint decoders that retain the raw syndrome and a data prior.Source: §IV · p. 6

Partial repetition

• Under a static-data assumption, removing every single-fault ambiguity requires remeasuring all but one check in each KM-orbit.For Gross, 72 checks form 36 pairs, so remeasuring one check per pair uses 36 additional measurements; intervening data faults require a space–time decoder.Source: §IV · p. 7

Running example

• For BB(3, 3), the quotient algebra has 16 elements, nine of them units, and its nine translations map bijectively to those units, so all nine single faults have distinct metasyndromes.Here dim A = k/2 = 4.Source: A. Syndrome quotient and unit-group… · p. 7

Unit-group bound

• The number of distinct single-fault metasyndromes is bounded by |A×|, and dS ≥ 3 exactly when G embeds in A×; if N > |A×|, then dS = 2.The quotient-algebra bound can be sharper than the coarse bound from the number of distinct nonzero columns of a full-rank metacheck matrix.Source: A. Syndrome quotient and unit-group… · p. 7

Finite-code calculations

• For the five standard codes in Table III, |A×| is 36, 9, 9, 36, and 36; surjectivity holds in the listed trinomial examples but fails in all three symmetric rows.For BB(4, 4), BB(8, 8), and BB(6, 6), |G/KM| is 4, 8, and 36 while |A×| is 8, 128, and 288, respectively.Source: A. Syndrome quotient and unit-group… · p. 7

Algebraic interpretation

• The algebraic unit-group bound refines the translation-class count, while repeated-root cases use local rather than finite-field factors in the algebra decomposition.The listed trinomial examples attain the bound, but surjectivity is not universal across BB codes.Source: A. Syndrome quotient and unit-group… · p. 7

Bounded-dimension families

• For any BB family with bounded k and N growing, metasyndrome-only exact-reconstruction failure tends to one at every fixed pM ∈ (0, 1/2].The bound concerns metasyndrome-only repair, not joint decoding.Source: A. Syndrome quotient and unit-group… · p. 7

• wamb is the lightest data preimage of the orbit-pair syndrome ambiguity that produces the linear exact-repair term, not a full soundness measure.Translation symmetry makes the reference syndrome position irrelevant when searching for this minimum.Source: §V · p. 8
• For Gross and BB(6, 3), the same 50% single-fault ambiguity has minimum and average data costs of six and two, respectively.Both codes have |KM| = 2, so an optimal metasyndrome-only exact-reconstruction rule must misidentify half of single measurement faults.Source: §V · p. 8
• The average minimum data weight over nontrivial within-orbit displacements captures typical ambiguity severity beyond the best-case wamb.For the running example, dS = 3, so no weight-two orbit ambiguity occurs and these severity metrics are unnecessary.Source: §V · p. 8
• For [[108, 8, 10]], wamb is 4 while the average minimum data cost is 5.2, so the lightest ambiguity understates typical cost.When every column of H has odd weight, the syndrome parity matches the data-error weight and µ(g) is even.Source: §V · p. 8

• The logical component distances in the table are certified in [20, Table VII], while syndrome-side quantities are recomputed in this work.For [[288, 12, 18]], the exact certification in [20, Thm. 4, Table IV] matches the previously reported weight-18 upper bound.Source: §VI · p. 8
• In [[72, 12, 6]], dS = 3, so minimum-weight repair corrects every single measurement fault, and both logical components have distance six.In [[108, 8, 10]], dcol = 10 < dann = 12; the minimum-distance logical is identified by its colon class.Source: §VI · p. 8
• BB(3, 3) is non-coprime but semisimple, while BB(6, 6) has repeated-root factors because the characteristic divides both periods.For BB(6, 3), direct enumeration gives dX = 4, and a weight-preserving CSS-sector permutation gives dZ = 4.Source: §VI · p. 8

• Ranks, translation symmetries, quotient-algebra data, and low-weight ambiguity searches are computed exactly, while sustained logical-failure rates are estimated by Monte Carlo.Logical component distances are certified values from [20, Table VII] and serve as a precondition for interpreting metacheck results.Source: §VII · pp. 8–9
• Table III covers standard BB codes and reports component distances, unavoidable single-fault mis-reconstructions, and unit-group sizes.Its component distances are certified in [20, Table VII].Source: Table III · p. 9
• Tables IV–VI provide additional rows for syndrome behavior, finite-code checks and ranking quantities, and the decoders used in sustained-memory experiments.Source: Table IV · p. 9, Table V · p. 9

A. Exact finite-code calculations

• Exact calculations cross-check metacheck subgroup and orbit identities, compute algebraic dimensions by row reduction, and obtain ambiguity costs by an exact meet-in-the-middle search through weight six.For every code, the calculations verify dS = 2 iff |KM| > 1, the orbit factorization and counts, and rM = k/2.Source: A. Exact finite-code calculations · p. 9

B. Metacheck graph used by BP+OSD

• The BP+OSD experiments use a deterministic minimum-total-weight basis of the metacheck space, since alternative bases can change the Tanner graph seen by belief propagation.The second sorting key fixes ties reproducibly; the row-weight profiles are implementation data, not code invariants.Source: B. Metacheck graph used by BP+OSD · pp. 9–10

C. Exact low-weight decoder enumeration

• Every single and double measurement fault is exhaustively decoded by minimum-weight lookup and BP+OSD, separating information limits from decoder failures.A failure is defined as a reconstructed fault bξ differing from ξ.Source: C. Exact low-weight decoder enumeration · p. 10

D. Sustained phenomenological memory

• Sustained-memory trials use ten noisy rounds followed by an ideal round, balanced data and measurement error rates, and a final logical-failure test.Five decoder approaches include separated repair, a raw-syndrome reference, joint data–measurement decoding, and repeated space–time decoding; sampling stops at 100 failures or 2 × 10^4 trials, with anytime-valid confidence sequences.Source: D. Sustained phenomenological memory · p. 10

E. Decoder settings and reproducibility

• BP+OSD uses minimum-sum belief propagation with parallel scheduling, a 100-iteration cap, and order-two OSD-CS.The decoder uses the ldpc package version 2.4.1; its input is a syndrome vector and the channel prior for each variable is the corresponding physical rate.Source: E. Decoder settings and reproducibility · pp. 10–11

A. Structural predictions

• BP+OSD fails on 9 of 36 single faults in [[72, 12, 6]], although minimum-weight repair corrects every single fault.BP already returns a metasyndrome-consistent but incorrect vector in those cases, so OSD is not invoked; raising its order to seven leaves the count unchanged.Source: A. Structural predictions · p. 11
• The [[108, 8, 10]] code has dcol = 10 < dann = 12, placing its minimum-distance logicals in the colon component.The component minima are equal for [[72, 12, 6]], [[90, 8, 10]], Gross, and [[288, 12, 18]].Source: A. Structural predictions · p. 11

B. Exact syndrome-repair curves

• Gross has 36 distinct single-fault labels for 72 measurement locations, while all 36 locations of [[72, 12, 6]] are distinct.At pM = 10^-2, the orbit bound is 0.3036 versus exact repair failure 0.3373 for Gross, and 0.0865 versus 0.0891 for BB(6, 3).Source: B. Exact syndrome-repair curves · p. 11

C. Sustained-memory decoder comparison

• The joint [H | IN] decoder has the lowest central estimate among one-round decoders at every reported operating point, while space–time decoding can outperform it.For the more ambiguous rows, separated repair shows little benefit or a penalty relative to the raw heuristic at sampled points; this is suggestive, not a ranking law.Source: C. Sustained-memory decoder comparison · p. 12

IX. DESIGN LESSONS AND LIMITATIONS

• When k stays bounded as block length grows, available metasyndromes cannot keep pace with single-fault locations; under static data, the orbit count gives the minimum partial remeasurement that removes all single-fault collisions.The finite-rate bound applies after syndrome compression to a metasyndrome, not to a joint decoder retaining the syndrome and a data prior.Source: §IX · p. 12
• The ambiguity costs (wamb, μ̄1) are finite-length quantities, not a confinement theorem.They distinguish codes with the same single-fault collision rate but different data consequences.Source: §IX · p. 13
• On [[72, 12, 6]], BP converges to a different metasyndrome-consistent vector for 9 of 36 single faults, and order-seven OSD does not change those cases.OSD post-processing is not invoked after BP has already satisfied the metasyndrome.Source: §IX · p. 13
• The sustained-memory experiments are phenomenological, not threshold estimates, and omit circuit faults, leakage, hook errors, and schedule-dependent correlations.The joint-decoder comparison does not isolate the contribution of redundant rows; a rank-reduced-H ablation would test that contribution.Source: §IX · p. 13

X. CONCLUSION

• The logical decomposition has k/2-dimensional annihilator and colon components, and the selected distance is the smaller of their minima.In [[108, 8, 10]], the colon component reaches weight 10 before the annihilator component reaches weight 12; the logical certifications are taken from [20].Source: §IX · p. 13, §X · p. 14
• Gross has 36 unavoidable single-fault mis-reconstructions among 72 locations, while [[72, 12, 6]] minimum-weight repair corrects every single fault.The sustained experiments report that joint data–measurement decoding performs better than separated repair on the more ambiguous rows.Source: §X · p. 14

APPENDIX A PROOFS OF THE STRUCTURAL STATEMENTS

• A weight-two valid-syndrome word occurs exactly when its translation lies in KM, equivalently when the corresponding metacheck columns are equal.Thus dS ≥ 3 exactly when KM is trivial, or equivalently when the translation map is injective.Source: Appendix A · p. 14
• Within each KM orbit, metacheck columns are equal, so summing coordinates orbitwise compresses M to M = MΠKM.Conditioned on one fault, a κ-location orbit has only one recoverable location for a metasyndrome-only rule; the independent-fault argument yields success at most (1 − pM)^u1 for pM ≤ 1/2.Source: Appendix A · p. 14

APPENDIX B ALGORITHMS FOR THE EXACT CALCULATIONS

• Exact calculations use Gaussian elimination over F2, translation tests on a basis of CM, and breadth-first search over the 2^rM metasyndrome states.The exact component-distance certifications are taken from [20, Table VII], while the ambiguity search is exhaustive through data weight six.Source: Appendix B · p. 14