QECTOR Decoder v3: Syndrome-Faithful Decoding - Foundations, Algorithms, and Architecture of a Fifteen-Backend Quantum Error Correction Engine

Abstract

QECTOR Decoder v3: Syndrome Faithful Decoding, Foundations, Algorithms, and Architecture of a Fifteen Backend Quantum Error Correction Engine QECTOR Decoder v3 is a quantum error correction decoding engine written in Rust with PyO3 Python bindings. It provides fifteen specialized decoding backends unified by a single invariant enforced on every call: syndrome faithfulness, the requirement that H c = s (mod 2) for any correction the engine returns. Scope of this manual This reference manual documents implementation version 1.0.0 of qector decoder v3. Every mathematical statement in it is anchored to source modules and regression tests in the frozen 1.0.0 workspace. Hardware dependent measurements such as latency, throughput, and memory are intentionally excluded from the manual; the methodology for producing and reporting such figures is specified instead, so any future claim is reproducible and auditable. Backend families covered Exact minimum weight perfect matching using Edmonds' blossom algorithmEvent driven Sparse Blossom using radix heap region growthZero allocation Union Find (UF 01)Belief propagation with ordered statistics (BP OSD) for quantum LDPC codesAmbiguity clustering with learned predecodersSpace time and streaming decoding across multiple roundsTwo stage CSS decoding for correlated X and Z noiseCUDA and OpenCL batch kernels, proven bit identical to the CPU reference on tested configurationsA native routing and orchestration layer with a seven tier fallback chain What the manual includes The manual develops the binary field mathematics and stabilizer formalism from first principles, proves the core correctness theorems shared by every backend including syndrome faithfulness, the logical error criterion, and path flipping faithfulness of MWPM, and walks through worked numerical examples including the Steane [[7,1,3]] code, the rotated surface code at distance 3, and a Wilson confidence interval computed by hand. It documents the public API surface, both stable and provisional, the detector error model pipeline, the licensing model and tier enforcement, ecosystem integrations such as a PyMatching compatible shim, Sinter decoders, and a Qiskit plugin, and the testing and validation strategy across correctness, property based, cross decoder equivalence, and GPU bit identity layers. Reproducibility policy Claims in this manual are explicitly scoped: exactness, bit identity, and accuracy statements are attached only to the workloads and configurations on which they were established. No universal performance claim is made anywhere in the document, and prior benchmark artifacts that did not survive a core fingerprint change were formally withdrawn. Appendix D provides the exact reproduction commands, and Appendix C maps every claim to its evidence anchor in the source tree. Keywords quantum error correction, QEC, decoder, MWPM, Union Find, BP OSD, LDPC, qLDPC, surface code, Stim, Sinter, PyMatching, Rust, Python, PyO3, CUDA, batch decoding, reproducible research, benchmarking Author Guillaume Lessard, iD01t Productions, ORCID 0009 0000 3465 3753 Companion document (ground-truth verification campaign, 2026-09-27). PROOFS.pdf, attached to this version, gives the complete re-derived proofs for the decoder theorems of this manual (syndrome fidelity Hc = s, MWPM path-flipping, Union-Find parity algebra and peeling, OSD, AmbiguityCluster, TwoStage, LP optimality certificate), with explicit scope limits and a separate section listing excluded conjectures. See the record errata manifest for retractions applied across the corpus.