Distributed Symmetry Inference and Anomaly-Resilient Quantum Error Correction: An Icosahedral Realization

We present a two-layer quantum error correction architecture based on Distributed Symmetry Inference (DSI), realized through the binary icosahedral group 2I ∼= SL(2, 5) and its binary tetrahedral subgroup 2T. The storage layer uses the standard [[5, 1, 3]] stabilizer code; the DSI resource layer uses a non-stabilizer weighted graph state K5(π/3) whose symmetry properties classify errors by conjugacy class sector: center (order 2), face (order 3,6), edge (order 4), and vertex (order 5,10).Code-switching to the ((7, 2, 3)) computation code achieves non-Clifford universality consistent with the Eastin–Knill theorem. Hardware experiments on dual independent IBM Heron processors confirm the predicted sector accessibility hierarchy at > 5σ significance, validated by ideal baselines, axis-scrambled controls, and three independent measurement protocols. We prove a frame indistinguishability theorem establishing that no single-qubit measurement can identify the active decryptionframe—connecting the QECC construction to encrypted quantum cloning. Short cycles in the constraint structure of 2I create a likely performance degradation for belief-propagation LDPC decoding in group-derived constructions, motivating code-switching as a natural complement to the group-theoretic architecture. Numerical simulations demonstrate coherence lifetime enhancements of 2–3× under correlated, non-Markovian noise, with the gain mechanism requiring structured noise (no advantage under Markovian conditions).