Logical-Class Diagnostic Layer for Stabilizer Codes and Decoders via Supercharge Structures

Abstract

We develop a decoder-agnostic logical-class diagnostic layer for abelian Pauli stabilizer codes, motivated by fault-tolerant quantum computing. An ancilla fermionic register defines a nilpotent supercharge whose clean degree-zero harmonic sector is the code space. Projected-return blocks derived from graded commutators organize the recovery test of whether a decoder-selected correction preserves every encoded input for every error in a declared same-syndrome family; the passing syndromes form the certified faithful window. Demoting the supercharge to an exact binary diagnostic formulation classifies residuals modulo stabilizers, separating a check violation, a harmless stabilizer residual, and a syndrome-silent nontrivial logical action, with a two-sector form for CSS codes. Because the unknown fault cancels, the relative logical class of two record-compatible candidates is computable online. We verify the diagnostic formalisms on the Steane code and specify a static--circuit bridge audit protocol for the Gross bivariate-bicycle quantum low-density parity-check (qLDPC) code. The method supplements existing decoders without additional physical qubits for software use.