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
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
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
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
Realizing the full potential of quantum computation requires Quantum Error Correction (QEC). QEC reduces error rates by encoding logical information across redundant physical qubits, enabling errors to be detected and corrected. A common decoder used for this task is Minimum Weight Perfect Matching (MWPM) a graph-based algorithm that relies on edge weights to identify the most likely error chains. In this work, we propose a data-driven decoder named Neural Minimum Weight Perfect Matching (NMWPM). Our decoder utilizes a hybrid architecture that integrates Graph Neural Networks (GNNs) to extract local syndrome features and Transformers to capture long-range global dependencies, which are then used to predict dynamic edge weights for the MWPM decoder. To facilitate training through the non-differentiable MWPM algorithm, we formulate a novel proxy loss function that enables end-to-end optimization. Our findings on the toric code under depolarizing noise demonstrate thresholds of 17.9% and 10.95%, nearing the 18.9% and 11.0% maximum likelihood bounds, highlighting the advantage of hybrid decoders that combine the predictive capabilities of neural networks with the algorithmic structure of classical matching.
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.
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.