Phase-Aware Syndrome Decoding

Decoder inference that retains Pauli phases and degeneracy when estimating logical error classes. The main concepts are symplectic phases, posterior cosets, biased noise. Each must be defined in the context of a concrete physical model, mathematical representation, or experimental task. Terminology alone does not establish the behavior of the system.

Syndrome extraction measures commuting stabilizer eigenvalues, but the syndrome does not determine a unique physical error. Errors differing by stabilizers share their logical action, and a maximum-likelihood decoder should rank logical equivalence classes rather than individual Pauli strings. Coherent error rotations also possess relative phases that a simple Pauli-twirled stochastic noise model can discard. A genuinely phase-aware model must distinguish physically observable effects from arbitrary Pauli representation phases.

Compare a complete small-code channel calculation against Pauli-twirled decoding, keeping the same initial state, syndrome circuit, readout noise and logical observable. Evaluate process fidelity or logical diamond error as appropriate, not only syndrome success. Test stabilizer degeneracy and bias extremes to find cases where the ranking of corrections changes. State which coherent correlations the decoder can actually exploit.