Topologically Robust Syndrome-Based Readout of a Classical Parameter in a Family of Non-Orientable Stabilizer Codes

We introduce a one-parameter family of Klein bottle stabilizer codes C(δ), δ ∈ {0,1,2,3}, defined by shifting the orientation-reversing boundary identification of a 4×2 lattice. Although all four codes share identical topology (Klein bottle, GSD=4), each produces a distinct and theoretically predictable syndrome fingerprint P(δ) when a topological excitation is prepared. We define the encoding map E: δ → P(δ) and show it is injective and robust under noise: δ can be recovered from syndrome measurements alone, without access to circuit metadata, because δ is embedded in the circuit topology rather than the quantum state. The misidentification probability is expected to scale as p^d versus p^1 for a gate-encoded parameter. Key results (job d71582469uic73cl1q5g, IBM Fez, 8192 shots each): δ Pattern P(δ) f_K Z 0 10000001 42.1% 606σ 1 00010001 27.5% 394σ 2 00100001 22.2% 316σ 3 01000001 33.2% 476σ All four patterns match theory exactly. Syndrome qubit 0 fires in all cases; the second firing qubit steps 7→4→5→6 as δ advances, providing a direct readout of the boundary condition. Topology kill test (job d73rfc4vllmc73ansvk0): Klein circuit produced 10000001 at 47.99% while an identical Fake Klein (one edge changed) produced 0.39% — 122.8× enhancement at Z=691σ — confirming the signal is topology-determined, not hardware resonance. Author: Leonardo Roma, Independent Researcher, Ceccano, Italy Date: March 2026 Live demo: https://kleincode.pythonanywhere.com Companion paper (Paper 1): https://doi.org/10.5281/zenodo.19284050 All hardware experiments performed on IBM Fez (Heron r2, 156 qubits) via IBM Quantum Runtime. Code: MIT license. Paper: CC BY 4.0.