Deterministic Rational Certificates for Causal Logical Decoding of CSS Codes

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)