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 pᵈ versus p¹ for a gate-encoded parameter. Key results (job d71582469uic73cl1q5g, IBM Fez, 8192 shots each): δ Pattern P (δ) fK 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.