For the bivariate bicycle code [[288,12,18]] ("two-gross"), with the depth-8 syndrome-extraction cycle of Bravyi et al. (arXiv:2308.07915), this deposit gives explicit sets of 16 circuit faults that trigger no detector and apply a nontrivial logical operator. They exist for Z-type and X-type faults and under both readings of the CNOT term labels. One of them lies entirely inside the noise model and memory layout of Beverland et al., "Fail fast" (arXiv:2511.15177: Z-type noise, 18 noisy cycles and one fault-free cycle), whose search reported a lowest weight of 18. So the circuit distance of this cycle is at most 16. No lower bound is claimed, and no claim is made about logical error rates. Contents. A two-page note; the four fault sets listed gate by gate (cycle, timestep, gate, qubit coordinates, Pauli) so they can be checked by hand; a standalone checker (Python 3 + numpy) that builds the code and cycle from the polynomials and replays each set; tamper controls; logs. Checks. Two separately written circuit builds (one written from the papers only) reproduce the matrix sizes in Fail fast, Table 1, and agree on all four sets at cycles 1, 9 and 18; the gate order and indexing were checked against the text of the public BivariateBicycleCodes repository (inspected, not executed). A third propagator, written from the fault table and the papers only, is included. Removing any single fault makes detectors fire in every case tested. Use of AI. The author is not a professional researcher. The search, checks and drafting were carried out by AI models (Anthropic's Claude via Claude Code; OpenAI's ChatGPT as a reviewer) under the author's direction. The author is responsible for the content. No human expert has reviewed the work to date. Corrections are welcome.
For the bivariate bicycle code [[288,12,18]] ("two-gross"), with the depth-8 syndrome-extraction cycle of Bravyi et al. (arXiv:2308.07915), this deposit gives explicit sets of 16 circuit faults that trigger no detector and apply a nontrivial logical operator. They exist for Z-type and X-type faults and under both readings of the CNOT term labels. One of them lies entirely inside the noise model and memory layout of Beverland et al., "Fail fast" (arXiv:2511.15177: Z-type noise, 18 noisy cycles and one fault-free cycle), whose search reported a lowest weight of 18. So the circuit distance of this cycle is at most 16. No lower bound is claimed, and no claim is made about logical error rates. Contents. A two-page note; the four fault sets listed gate by gate (cycle, timestep, gate, qubit coordinates, Pauli) so they can be checked by hand; a standalone checker (Python 3 + numpy) that builds the code and cycle from the polynomials and replays each set; tamper controls; logs. Checks. Two separately written circuit builds (one written from the papers only) reproduce the matrix sizes in Fail fast, Table 1, and agree on all four sets at cycles 1, 9 and 18; the gate order and indexing were checked against the text of the public BivariateBicycleCodes repository (inspected, not executed). A third propagator, written from the fault table and the papers only, is included. Removing any single fault makes detectors fire in every case tested. Use of AI. The author is not a professional researcher. The search, checks and drafting were carried out by AI models (Anthropic's Claude via Claude Code; OpenAI's ChatGPT as a reviewer) under the author's direction. The author is responsible for the content. No human expert has reviewed the work to date. Corrections are welcome.
Bravyi et al. conjectured that their depth-8 syndrome cycle for the [[144,12,12]] gross code has circuit distance 10. We prove it. In their noise model and for every number of noisy cycles, no set of nine or fewer faults causes an undetected logical error of either Pauli type, and we give explicit ten-fault logical errors of both types. When we released our data on 15 October 2026, the best reported bounds for X-type faults were 9 ≤ d_circ ≤ 12, at 12 rounds; since then, Crocker and Pan have reported the value 10 in both bases, for 12 noisy cycles and for every number of cycles up to 12, respectively. For the same family of circuits we show that the [[90,8,10]] and [[108,8,10]] cycles have circuit distance 8 for every number of cycles, extending values reported for fixed numbers of rounds; for [[72,12,6]] we confirm the known value 6. We also find that, for three of these codes, Google's Tesseract test circuits have the same detector-error-model columns as the depth-8 cycle once restricted to the detectors of the memory basis, for every number of rounds we checked, so circuit-distance results carry over between the two at those round counts. The lower bounds combine 607 SAT refutations, each with a DRAT proof, with exhaustive enumerations of extended-code logicals that carry no certificate. Finally, we independently confirm two results that others reported first: the [[360,12]] bivariate bicycle code has distance 24, and the left-right gross-code circuit of Strikis, Browne and Beverland, as reconstructed from their figures, has circuit distance 11. All code, formulas, proofs and fault sets are available at doi:10.5281/zenodo.23394575.
We construct three new weight-6 bivariate-bicycle (BB) codes—two [[336,20,12]] codes, inequivalent under the BB presentation symmetries, over Z₂ × Z₈₄ and a [[360,24,12]] code over Z₆ × Z₃₀—and certify exact distance 12 in both the X and Z sectors. They encode 25% and 50% more logical qubits than the best distance-12, weight-6 code reported by IBM's evolutionary search (k = 16) [Cruz-Benito et al., arXiv:2606.02418]. They are also the only ones: a symmetry-reduced enumeration of every connected weight-6 BB code with three-term polynomials, n ≤ 360 and k ≥ 20 eliminates 1,448 of the 1,584 candidates with checked logical operators of weight at most 11, and the remaining 136 presentations fall into three classes under group automorphisms, renormalisation and A ↔ B, represented by the new codes; within the polynomial ansatz of [Bravyi et al., Nature 627, 778 (2024)], every one of the 432 such candidates has distance at most 10. The same certificates settle the distances of the IBM ladder [Bravyi et al., Nature 627, 778 (2024)]—[[72,12,6]], [[90,8,10]], [[108,8,10]], the gross code [[144,12,12]] and the two-gross code [[288,12,18]], both sectors—and certify 19 ≤ d ≤ 24 for [[360,12]], whose distance we establish to be exactly 24 by exhaustive enumeration, resolving the upper bound reported in [Bravyi et al., Nature 627, 778 (2024)]. For the depth-8 syndrome-extraction circuit they certify circuit-level distance exactly 6 for [[72,12,6]] in both memory bases and at least 6 for every other code tested. Certified lower bounds are DRAT refutations for n ≤ 144 and replayable search trees above, whose single pruning lemma is proved in Lean 4; a reader replays any certificate with a standard-library Python verifier plus drat-trim or a 200-line C replayer, without the software that produced it. In a 1.1-million-shot circuit-level benchmark under one frozen BP-OSD decoder, the new codes' block failure probabilities at p = 0.004 are 3.6–6.2 times lower than the gross code's and 3.4–5.6 times higher than the two-gross code's, in both memory bases. Every computation—search, certification, replay and benchmark—ran on one laptop (Apple M3 Pro, 18 GB) with no GPU and no cluster. Contents of the release paper/ — the manuscript (LaTeX source, bibliography and PDF). certificates/ — one manifest per code and sector (<code_id>_<sector>.json), its standard-library Python verifier (verify_qec_<code_id>_<sector>.py), the proofs in proofs/ (DRAT proofs and gzipped proof trees), the problem files and C source of the tree replayer, the Lean 4 minimality lemma (qec_minimality.lean), the weight-6 envelope witnesses with their checkers, the distance table, FORMAT.md (the certificate format) and REPLAY.md (a clean replay of every certificate: 44 of 44 pass). simulation/ — the projected detector error models, the BP-OSD and BP-LSD evaluation scripts, every 2,000-shot chunk record with its seed, the decoder-tuning grid and the joint-model controls. SHA256SUMS — the SHA-256 of every file. Verification In certificates/: python3 verify_qec_<code_id>_<sector>.py rebuilds the code from its polynomials, checks the logical basis by rank, regenerates every CNF or problem file byte for byte and replays the proof; its last line is the verdict. DRAT rows need drat-trim (set DRAT_TRIM); tree rows need a C compiler. The Lean lemma checks with lake env lean qec_minimality.lean in a project with the pinned Lean 4 v4.33.1 and Mathlib 0df444a3 (exit status 0). Integrity: shasum -a 256 -c SHA256SUMS. Hardware Every computation — search, certificate production, replay and the benchmark — ran on one MacBook Pro (Apple M3 Pro, 11 CPU cores, 18 GB memory, macOS 15.5), with no GPU and no cluster. License: Licensed by SmartLedger Solutions under CC BY-NC 4.0 (LICENSE.md). Commercial use requires a written license from SmartLedger Solutions (labs@smartledger.solutions). The certificate producer is proprietary and not distributed; verification does not need it.
The ultimate goal of quantum error correction is to create logical qubits with very low error rates (e.g. 1e-12) and assemble them into large-scale quantum computers capable of performing many (e.g. billions) of logical gates on many (e.g. thousands) of logical qubits. However, it is necessarily difficult to directly assess the performance of such high-quality logical qubits using standard Monte Carlo sampling because logical failure events become very rare. Building on existing approaches to this problem, we develop three complementary techniques to characterize the rare-event regime for general quantum low-density parity-check (qLDPC) codes under circuit noise. (I) We propose a well-motivated, low-parameter ansatz for the failure spectrum (the fraction of fault sets of each size that fail) that empirically fits all the QEC systems we studied and predicts logical error rates at all physical error rates. (II) We find min-weight logical operators of syndrome measurement circuits and exactly compute the number of min-weight failing configurations. (III) We generalize the splitting method to qLDPC codes using multi-seeded Metropolis sampling to improve convergence for systems with many inequivalent logical operators. We apply these tools to distance-6, -12, and -18 bivariate bicycle codes under circuit noise, observing strong low-error-rate performance with the recently proposed Relay decoder but also considerable scope for further improvement.
Faulty syndrome measurements can corrupt an otherwise correct quantum-error correction step. Bivariate bicycle (BB) codes contain dependent stabilizer checks, so every valid syndrome obeys additional parity constraints, or metachecks. We study how far this built-in redundancy can identify measurement faults and when the remaining ambiguity is unavoidable, while separately checking the code's logical structure. A logical decomposition is used as a preliminary safety check: it identifies a k/2-dimensional annihilator subspace and a k/2-dimensional colon quotient, and the minimum-weight logical need not be visible from the annihilator side alone. On the measurement side, translation symmetry partitions syndrome locations into classes that carry identical metacheck information. This gives an exact characterization of the leading single-fault ambiguity, a bound on how many fault locations can be distinguished, and a family-level repair limit when the encoded dimension stays bounded while the block length grows. Under a static-data assumption, the same calculation gives the minimum number of checks that must be remeasured to remove every single-fault ambiguity. Exact finite-code calculations illustrate both regimes: all single measurement faults are distinguishable in a 72-qubit BB code, whereas the 144-qubit Gross code merges its 72 syndrome locations into 36 indistinguishable pairs. In a 108-qubit example, one logical component first appears at weight 12 while the other contains a weight-10 logical. Sustained phenomenological experiments show that joint data--measurement decoding is more robust than a separated repair stage on the more ambiguous codes. The resulting tests apply to general two-block BB codes, including non-coprime periods and repeated-root cases.
We present new upper and lower bounds on the minimum distance of certain generalized bicycle (GB) codes beyond the reach of techniques for classical codes capable of even capturing the true minimum distance for some cases. These bounds are then applied to illustrate the existence and analyze two highly degenerate GB code families with parameters [[d2+1,2,d]] for odd d≥3 and [[d2,2,d]] for even d≥4, both having the property that each check qubit is connected to exactly four data qubits similar to surface codes. For the odd-distance family, we analyze the structure of low-weight logical Pauli operators and demonstrate the existence of a fault-tolerant logical CNOT gate between the two logical qubits, achievable through a simple relabeling of data qubits. We further construct a syndrome extraction pattern for both families that does not imply minimum distance reduction arising from extraction circuit faults that propagate from the check qubits to the data qubits. Finally, we numerically evaluate their logical error rates under a code capacity depolarizing noise model using the belief propagation ordered statistics decoding (BP-OSD) and minimum-weight perfect-matching (MWPM) decoders, yielding thresholds of approximately 14−16% for the odd and even families, very similar to those of rotated surface codes.
High-rate bivariate bicycle (BB) codes are promising low-overhead quantum memories, but their stabilizer checks typically have weight 6 or higher, making syndrome extraction challenging. We introduce subsystem bivariate bicycle (SBB) codes, a translation-invariant CSS subsystem construction that realizes BB-code logical structure using local weight-4 gauge measurements. Their stabilizer syndromes are inferred by multiplying the corresponding gauge outcomes. We further show that nonlocal stabilizers in translation-invariant CSS subsystem codes can be detected using a determinantal-ideal criterion based on the gauge-operator commutation matrix. When this criterion excludes nonlocal stabilizers, a finite-depth Clifford circuit decouples gauge qubits and identifies the protected subsystem with a corresponding BB stabilizer code. An SBB code is topological, meaning that it has no nontrivial local logical operators, if and only if the corresponding BB code is topological. A finite search yields low-overhead examples including [[27,6,3]], [[75,10,5]], and [[108,12,6]]; the latter encodes six times more logical qubits than a subsystem surface code at the same block length and distance. These results show how gauge degrees of freedom can make high-rate BB logical structure compatible with local weight-4 syndrome extraction.