Comprehensive experimental account of non-orientable stabilizer codes on superconducting quantum hardware. This work consolidates and extends a five-paper series on the Klein bottle stabilizer code, and reports the first experimental realisation of the real projective plane (RP²) stabilizer code. Key results: Klein bottle code confirmed at distances d=2,3,4 with a working MWPM decoder (antipodal shortcut, 6/6 hardware validation at Z=688–730σ); transversal Clifford group with O(1) gate overhead; 12 logical qubits on 156 physical qubits at CV=0.01; δ-family confirmed at d=2,3,4. RP² 4×4 code: both boundary twists confirmed simultaneously (266–571σ, zero noise bits in dominant), logical qubit verified with Z_L contrast 0.885, GSD=2 confirmed. Transversal T^⊗32 ruled out analytically; code deformation identified as the open problem. Raw syndrome count data for all experiments available at https://github.com/theoricline/klein-bottle-qec/data Paper 1: Code existence, GSD=4, kill test Z=691σ https://doi.org/10.5281/zenodo.19284050 Paper 2: δ-family, topological parameter encoding https://doi.org/10.5281/zenodo.19286677 Paper 3: Non-orientable logical algebra, holonomy https://doi.org/10.5281/zenodo.19287977 Paper 4: Six Simultaneous Non-Orientable Stabilizer Codes https://doi.org/10.5281/zenodo.19333513 Paper 5: MWPM Decoder for a Non-Orientable Stabilizer Code https://doi.org/10.5281/zenodo.19451825
Comprehensive experimental account of non-orientable stabilizer codes on superconducting quantum hardware. This work consolidates and extends a five-paper series on the Klein bottle stabilizer code, and reports the first experimental realisation of the real projective plane (RP²) stabilizer code. Key results: Klein bottle code confirmed at distances d=2,3,4 with a working MWPM decoder (antipodal shortcut, 6/6 hardware validation at Z=688–730σ); transversal Clifford group with O(1) gate overhead; 12 logical qubits on 156 physical qubits at CV=0.01; δ-family confirmed at d=2,3,4. RP² 4×4 code: both boundary twists confirmed simultaneously (266–571σ, zero noise bits in dominant), logical qubit verified with Z_L contrast 0.885, GSD=2 confirmed. Transversal T^⊗32 ruled out analytically; code deformation identified as the open problem. Raw syndrome count data for all experiments available at https://github.com/theoricline/klein-bottle-qec/data Paper 1: Code existence, GSD=4, kill test Z=691σ https://doi.org/10.5281/zenodo.19284050 Paper 2: δ-family, topological parameter encoding https://doi.org/10.5281/zenodo.19286677 Paper 3: Non-orientable logical algebra, holonomy https://doi.org/10.5281/zenodo.19287977 Paper 4: Six Simultaneous Non-Orientable Stabilizer Codes https://doi.org/10.5281/zenodo.19333513 Paper 5: MWPM Decoder for a Non-Orientable Stabilizer Code https://doi.org/10.5281/zenodo.19451825
We present the first minimum weight perfect matching (MWPM) decoder designed for a non-orientable stabilizer code: the Klein bottle quantum error-correcting code. The decoder incorporates an antipodal shortcut: in the Klein bottle code, the non-orientable boundary identification connects two syndrome qubits via a single edge, giving them defect-graph distance 1 rather than the toric Manhattan distance of 4. A standard toric MWPM decoder fails on 25% of single-qubit errors (the four vertical edges on the non-orientable boundary); the Klein decoder corrects all 64 single-qubit errors across all four logical sectors analytically. Hardware validation on IBM Fez (Heron r2, 156 qubits) confirms 6/6 correction scenarios at Z = 688–730σ, including both antipodal-edge scenarios. Multi-session reproducibility across six calibration sessions and double-error logical fidelity (98.57% single-error, one identified degenerate pair at f = 0.51%) are reported in the appendix. This is the fifth paper in a series on the experimental characterisation of the Klein bottle stabilizer code on superconducting quantum hardware. Companion papers: Paper 1: Code existence, GSD=4, kill test Z=691σ https://doi.org/10.5281/zenodo.19284050 Paper 2: δ-family, topological parameter encoding https://doi.org/10.5281/zenodo.19286677 Paper 3: Non-orientable logical algebra, holonomy https://doi.org/10.5281/zenodo.19287977 Paper 4: Six Simultaneous Non-Orientable Stabilizer Codes https://doi.org/10.5281/zenodo.19333513
Six simultaneous Klein bottle stabilizer codes on IBM Fez (Heron r2, 156 qubits). 12 logical qubits. 144 physical qubits. CV=0.01. Reproduced across 4 independent calibration sessions on 3 IBM Heron r2 processors (IBM Fez, IBM Marrakesh, IBM Kingston) on 2026-03-29 and 2026-03-30. CV=0.01 in all sessions. Jobs ID: d741hmp8qmgc73flhrtg, d7552n5bjrds73ebv320, d755bc7i6gpc73etvf7g, d755csvi6gpc73etvh60. Author: Leonardo Roma, Independent Researcher, Ceccano, ItalyEmail: leonardo.roma@protonmail.comDate: March 2026Live demo: https://kleincode.pythonanywhere.comGitHub: https://github.com/theoricline/klein-bottle-qec Companion papers:Paper 1: Code existence, GSD=4, kill test Z=691σ https://doi.org/10.5281/zenodo.19284050Paper 2: δ-family, topological parameter encoding https://doi.org/10.5281/zenodo.19286677Paper 3: Non-orientable logical algebra, holonomy https://doi.org/10.5281/zenodo.19287977
This upload contains the manuscript, LaTeX source, and Python code for the paper: "Non-Orientable Topology in a Stabilizer Circuit: Experimental Demonstration of a Klein Bottle Quantum Error-Correcting Code" We report the first experimental implementation of a Klein bottle stabilizer code on superconducting quantum hardware. The code achieves a 138× enhancement of the predicted antipodal syndrome signature (44.2% frequency, Z=499σ) on IBM Fez, confirmed on three independent processors (IBM Fez, Marrakesh, Torino) and in four simultaneous parallel code instances. The code uses 25% fewer physical qubits than the square toric code at equivalent code distance. The Python code implements the Klein bottle stabilizer code on a 4×2 lattice, reproduces the antipodal signature from the initial 26.3% to the optimised 44.2%, and includes hardware verification across three processors and MWPM logical error rate simulation. All hardware job IDs are provided in the README for independent reproducibility. Contents: paper.pdf — manuscript latex/ — LaTeX source and figures code/ — Python scripts (circuit builder, verification, simulation) README.md — instructions and complete hardware job ID table All hardware experiments were performed on IBM Fez (Heron r2, 156 qubits) via IBM Quantum Runtime. The code is released under MIT license; the manuscript under CC BY 4.0.
We demonstrate that the logical operator algebra of the Klein bottle stabilizer code is non-orientable, confirming that the code encodes non-orientable topology at the level of the logical Hilbert space, not merely at the level of the stabilizer construction. Using dynamic circuits with mid-circuit reset on IBM Fez (Heron r2, 156 qubits), we apply the b-cycle logical gate X_L2 to each of the four logical sectors and measure the resulting syndrome in a second round. The Klein code produces different sector transitions than the toric code under the identical gate, confirming that X_L2 anticommutes with Z_L1 in Klein but commutes in the toric code. No orientable stabilizer code can produce this result. Key results: Sector Job ID Klein R2 Z Toric R2 Z |10>_L (holonomy) d73rqh5koquc73e22ni0 10000010 at 11.79% 90σ 10001011 at 22.96% 272σ |00>_L d73u5ttkoquc73e251eg 10000001 at 13.9% 99σ 10001000 at 21.9% 312σ |10>_L d73u5ttkoquc73e251eg 10000010 at 12.7% 105σ 10001011 at 18.0% 230σ |01>_L d73u5ttkoquc73e251eg 10001000 at 11.9% 45σ 10000001 at 16.0% 70σ |11>_L d73u854vllmc73anvom0 dominant confirmed — dominant confirmed — All dominant patterns match theoretical predictions exactly. Cross-contamination below 1.2% in all cases. The transition table: Klein: |00> ↔ |11>, |10> ↔ |01> (X_L2 flips BOTH Z_L1 and Z_L2) Toric: |00> ↔ |01>, |10> ↔ |11> (X_L2 flips ONLY Z_L2) The two holonomy maps act on different sector pairs, the non-orientable twist permutes the logical sector labels relative to the orientable case. This is the third in a series of three papers establishing complete experimental characterisation of the Klein bottle stabilizer code: Paper 1 (doi:10.5281/zenodo.19284050): Code exists on hardware, GSD=4, kill test Z=691σ Paper 2 (doi:10.5281/zenodo.19286677): Boundary parameter δ is tunable, Z=316–606σ Paper 3 (this paper): Logical algebra is non-orientable, Z=45–312σ Live demo: https://kleincode.pythonanywhere.com Paper 1: https://doi.org/10.5281/zenodo.19284050 Paper 2: https://doi.org/10.5281/zenodo.19286677 All hardware experiments performed on IBM Fez (Heron r2, 156 qubits) via IBM Quantum Runtime. Code: MIT license. Paper: CC BY 4.0.
The first experimental implementation of a non-orientable stabilizer code on quantum hardware. We encode the boundary conditions of a Klein bottle onto the 156-qubit IBM Fez processor and measure a 138× enhancement of the predicted antipodal syndrome signature (44.2% frequency, Z=499σ). The code uses 25% fewer physical qubits than the square toric code at equivalent code distance. Key results confirmed on hardware: Result Value Best antipodal frequency 44.2% (seed=77, depth=112) Best enhancement over toric 138× Best Z-score 499σ Orientation-even sectors 13% fewer syndrome violations Orientation-odd sectors 26% more syndrome violations Logical error rate (MWPM) K/T ≈ 1.0 (equivalent to toric) Qubit efficiency 24 vs 32 qubits (25% reduction vs square toric d=4) Processors confirmed IBM Fez, Marrakesh, Torino Parallel codes on 156q chip 4 simultaneous (Z=404–721σ each) Author: Leonardo Roma, Independent Researcher, Ceccano, ItalyDate: March 2026Live demo: kleincode.pythonanywhere.com Additional verification Topology kill test (job d73rfc4vllmc73ansvk0) Klein circuit (prep edge v(3,1)) produced 10000001 at 47.99% while an identical Fake Klein (one edge changed) produced 0.39% — 122.8× enhancement at Z=691σ. The Fake Klein produced its own predicted pattern 10001000 at 33.39%, proving topology determines where the signal lands, not hardware resonance. Holonomy — non-orientable algebra (job d73rqh5koquc73e22ni0) Dynamic circuit test (mid-circuit reset). B-cycle gate applied to a-anyon sector, syndrome measured in a second round. Klein: 10000010 at 11.79%, Z=90σ. Toric: 10001011 at 22.96%, Z=272σ. Cross-contamination < 1.2%. Confirms b-cycle has non-trivial action on a-cycle — the algebraic signature of non-orientable topology. All hardware experiments performed on IBM Fez (Heron r2, 156 qubits) via IBM Quantum Runtime. Code: MIT license. Paper: CC BY 4.0.
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.