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Deterministic Rational Certificates for Causal Logical Decoding of CSS Codes

Deivison Ramos do Rosario

Published Oct 28, 2026• 0 citations
Publisher

Abstract

Deterministic Rational Certificates for Causal Logical Decoding of CSS Codes This work investigates the deterministic certification of logical decoding decisions in Calderbank–Shor–Steane (CSS) quantum error-correcting codes under independent phase errors and noisy intermediate syndrome measurements. The proposed framework combines causal Fourier–Walsh filtering, tensor-train approximations, exact rational arithmetic, and rigorous residual-based error bounds. In addition to global and spectrally refined estimates, the study develops a goal-oriented adjoint certification procedure that establishes whether an approximate logical-class probability contrast has a mathematically guaranteed sign. For the unrotated [[13,1,3]] surface code, the framework is evaluated across 300 correlated experimental configurations involving 25 simulated histories, four temporal horizons, and three tensor-train bond dimensions. The baseline and spectrally refined methods certify 184 and 192 decisions, respectively, while adjoint certification increases coverage to 272 decisions without modifying the frozen tensor candidates. All 300 certified intervals contain their corresponding exact physical-model contrasts, with no incorrect certified signs observed. An additional rationally verified [[41,1,5]] code instance demonstrates finite-size applicability beyond the distance-three benchmark. The accompanying reproducibility archive includes the manuscript, LaTeX sources, Python verification programs, frozen numerical data, rational certificates, adversarial tests, and SHA-256 integrity manifests. The results establish conditional mathematical guarantees under the specified stochastic model. They do not establish universal decoding thresholds, asymptotic scalability, or fault-tolerant hardware performance. Author: Deivison Ramos do Rosario Affiliation: Independent Researcher, Bengui, Belém, Pará, Brazil ORCID: https://orcid.org/0009-0002-7768-5047 License: Creative Commons Attribution 4.0 International (CC BY 4.0)

DOI
10.5281/zenodo.23749900

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Deterministic Rational Certificates for Causal Logical Decoding of CSS Codes This work investigates the deterministic certification of logical decoding decisions in Calderbank–Shor–Steane (CSS) quantum error-correcting codes under independent phase errors and noisy intermediate syndrome measurements. The proposed framework combines causal Fourier–Walsh filtering, tensor-train approximations, exact rational arithmetic, and rigorous residual-based error bounds. In addition to global and spectrally refined estimates, the study develops a goal-oriented adjoint certification procedure that establishes whether an approximate logical-class probability contrast has a mathematically guaranteed sign. For the unrotated [[13,1,3]] surface code, the framework is evaluated across 300 correlated experimental configurations involving 25 simulated histories, four temporal horizons, and three tensor-train bond dimensions. The baseline and spectrally refined methods certify 184 and 192 decisions, respectively, while adjoint certification increases coverage to 272 decisions without modifying the frozen tensor candidates. All 300 certified intervals contain their corresponding exact physical-model contrasts, with no incorrect certified signs observed. An additional rationally verified [[41,1,5]] code instance demonstrates finite-size applicability beyond the distance-three benchmark. The accompanying reproducibility archive includes the manuscript, LaTeX sources, Python verification programs, frozen numerical data, rational certificates, adversarial tests, and SHA-256 integrity manifests. The results establish conditional mathematical guarantees under the specified stochastic model. They do not establish universal decoding thresholds, asymptotic scalability, or fault-tolerant hardware performance. Author: Deivison Ramos do Rosario Affiliation: Independent Researcher, Bengui, Belém, Pará, Brazil ORCID: https://orcid.org/0009-0002-7768-5047 License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Quantum low-density parity-check (qLDPC) codes reduce the qubit overhead of fault-tolerant quantum computation by an order of magnitude, but their decoding is harder than its classical counterpart: because many physical errors are equivalent up to stabilizers, the degenerate maximum-likelihood (ML) decoder must compare the probabilities of entire equivalence classes of errors, that is, partition functions, rather than single errors. The workhorse decoder BP+OSD sidesteps degeneracy heuristically and offers no guarantees. We treat degenerate decoding as probabilistic inference in an undirected graphical model: the probability of each logical class is the partition function of an unconstrained, strictly positive Markov random field over the code's check variables, a construction that generalizes the random-bond Ising mapping of the surface code to arbitrary CSS codes and to spacetime decoding with measurement errors and circuit-level noise. On this model we build two decoders. The first estimates all class partition functions by annealed importance sampling with common random numbers and attaches to every decision a certificate of optimality: a paired bootstrap test, or, composed with constant-factor estimators such as WISH, an exact optimality proof. The second is region-based: the Bethe free energy, whose bias cancels between classes, reproduces exact ML decoding on every tested surface-code instance at millisecond cost, and enlarging the regions to elimination clusters makes exact degenerate ML decoding of the [[72,12,6]] bivariate bicycle code feasible. Across surface codes and the bivariate bicycle codes [[72,12,6]] and [[144,12,12]], under code-capacity, phenomenological, and circuit-level noise, the sampling decoder matches or exceeds BP+OSD while certifying the bulk of its decisions, and the certificate flags exactly the syndromes on which any fast decoder should be distrusted.
Abstract Quantum error correction (QEC) is the core mechanism enabling scalable and fault-tolerant quantum computation. Among various QEC schemes, Calderbank–Shor–Steane (CSS) codes play a central role. Efficiently verifying whether a quantum state remains within the target CSS code subspace is therefore essential both for understanding the theoretical performance of these codes and for assessing the error-correcting capability of practical quantum platforms. In this work, we develop an efficient framework for verifying whether a quantum system remains within the encoded subspace of a CSS code. We first show that the stabilizer structure of any CSS code can be mapped to a two-colorable graph, enabling the design of a hybrid verification strategy based solely on local Pauli measurements. We further analytically optimize the measurement probabilities and derive the resulting sample complexities for representative CSS codes. Specifically, for the Toric code, the efficiency of our hybrid strategy increases with lattice size and asymptotically approaches near-optimal performance. These results establish a unified and scalable approach to CSS code subspace verification, providing a foundation for certifiable and fault-tolerant quantum information processing.
This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal ZZZ-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal TTT realizes the logical transversal S†S^{\dagger}S†; (b) show that the condition C2∗C1⊆C1⊥C_2 \ast C_1 \subseteq C_1^{\perp}C2​∗C1​⊆C1⊥​ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal TTT implements the logical identity and the logical transversal TTT, respectively.
For an erasure pattern in a binary CSS code, the logical ambiguity supported on the erased coordinates is a finite-dimensional quotient. We give an explicit account of how its dimension becomes a decoder output, an exact conditional success probability, and a verifiable combinato- rial certificate. A restricted-rank formula yields the complement identity dZ (E) + dX (Ec) = k for every CSS code. For a cellulation of a connected closed orientable surface, the same index is computed by two graph component counts. We couple these counts to forest peeling, specify independently checkable component witnesses, and derive finite ambiguity-enumerator identi- ties. Complete enumeration on the square tori of sizes two and three gives exact sector and joint logical-success statistics. A register-transfer implementation is assessed through 1,462,400 recorded reference comparisons, scoped formal contracts, 72,000 specified single-run force injec- tions, and a separate persistent-fault experiment. The latter exhibits the distinction between an algebraic identity and an effective runtime audit: complementary indices derived from shared corrupted counters can still sum to the correct constant. Independent recounts and modular residues detect specified fault classes, while correlated value-and-residue changes remain un- detected. Mathematical exactness, computational checking, and empirical fault coverage are stated separately throughout.
We study hard-erasure belief-propagation (BP), understood as the sum-product algorithm on erasure messages, for a punctured MacKay-Neal/Hsu-Anastasopoulos (MN/HA) representation of Calderbank-Shor-Steane (CSS) codes. For every integer degree triple 2≤jZ<jX<k≤602\leq j_Z<j_X<k\leq602≤jZ​<jX​<k≤60, we prove that the constituent potential thresholds are jZ/kj_Z/kjZ​/k and 1−jX/k1-j_X/k1−jX​/k. The proof combines classical HA-MN duality with a completed exact rational certificate for all 1711 MN degree pairs in this range. The certificate establishes strict positivity on every physical nontrivial fixed-point branch, at every erasure probability in [0,1][0,1][0,1], and has an independent integer-arithmetic verifier. A reduction of the Z-side recursion and a fixed-channel vector-potential argument then prove seeded density-evolution convergence below the smaller constituent threshold for sufficiently large coupling width and an ideal seed interval at least that wide. Under jZ+jX=kj_Z+j_X=kjZ​+jX​=k, this threshold equals the quantum-erasure hashing-bound parameter determined by the CSS design rate. In particular, the degree triple (4,8,12)(4,8,12)(4,8,12) has potential threshold 1/31/31/3 without a positivity hypothesis. The numerical example at erasure probability 0.33250.33250.3325 reproduces the resulting decoding waves. The theorem uses an ideal auxiliary-message seed; its finite-code realization and logical block-error convergence are separate questions.
Stabilizer codes lie at the heart of modern quantum-error-correcting codes (QECCs). Of particular importance is a class called Calderbank-Shor-Steane (CSS) codes, which includes many important examples such as toric codes, color codes, and fractons. Recent studies have revealed that the decoding transition for these QECCs could be intrinsically captured by calculating information-theoretic quantities from the mixed state. Here we perform a simple analytic calculation of the coherent information for general CSS codes under local incoherent Pauli errors via diagonalization of the density matrices and mapping to classical statistical mechanical (SM) models. Our result establishes a rigorous connection between the decoding transition of the quantum code and the phase transition in the random classical SM model. It is also directly confirmed for CSS codes that exact error correction is possible if and only if the maximum-likelihood decoder always succeeds in the thermodynamic limit. Thus, the fundamental threshold is saturated by the optimal decoder.