This upload contains the manuscript, LaTeX source, and figures for the second paper in the Klein Bottle QEC series. Research Overview We introduce a one-parameter family of Klein bottle stabilizer codes C(δ) defined by shifting the orientation-reversing boundary identification of an Lx × Ly lattice. Although all members share identical topology (Klein bottle, GSD=4), each produces a distinct and theoretically predictable syndrome fingerprint P(δ) when a topological excitation is prepared. Key Results We verify all four fingerprints of the Lx=4, Ly=2 family on the 156-qubit IBM Fez processor (Heron r2), with statistical significance Z = 316–606σ (job d71582469uic73cl1q5g, 8192 shots each). Topological Protection The encoding map δ → P(δ) is injective and robust under noise. Because δ is embedded in the circuit topology rather than the quantum state, local errors cannot corrupt the parameter directly. Misidentifying δ requires ≥ d simultaneous errors, providing a misidentification probability scaling as pᵈ (where d=4) versus p¹ for standard gate-encoded parameters. Contents paper.pdf – Full manuscript paper.tex – LaTeX source (self-contained) fig_delta_lattice.pdf – Lattice diagrams and fingerprint encoding maps fig_delta_fingerprints.pdf – Hardware results: pattern frequencies and Z-scores fig_delta_depth.pdf – Signal strength vs. circuit depth analysis README.md – Detailed job IDs, API instructions, and reproduction notes Related Resources Paper 1 (Discovery): doi:10.5281/zenodo.19202945 Live API: https://kleincode.pythonanywhere.com GitHub: https://github.com/theoricline/klein-bottle-qec License: Creative Commons Attribution 4.0 International (CC BY 4.0)
This upload contains the manuscript, LaTeX source, and figures for the second paper in the Klein Bottle QEC series. Research Overview We introduce a one-parameter family of Klein bottle stabilizer codes C(δ) defined by shifting the orientation-reversing boundary identification of an Lx × Ly lattice. Although all members share identical topology (Klein bottle, GSD=4), each produces a distinct and theoretically predictable syndrome fingerprint P(δ) when a topological excitation is prepared. Key Results We verify all four fingerprints of the Lx=4, Ly=2 family on the 156-qubit IBM Fez processor (Heron r2), with statistical significance Z = 316–606σ (job d71582469uic73cl1q5g, 8192 shots each). Topological Protection The encoding map δ → P(δ) is injective and robust under noise. Because δ is embedded in the circuit topology rather than the quantum state, local errors cannot corrupt the parameter directly. Misidentifying δ requires ≥ d simultaneous errors, providing a misidentification probability scaling as pᵈ (where d=4) versus p¹ for standard gate-encoded parameters. Contents paper.pdf – Full manuscript paper.tex – LaTeX source (self-contained) fig_delta_lattice.pdf – Lattice diagrams and fingerprint encoding maps fig_delta_fingerprints.pdf – Hardware results: pattern frequencies and Z-scores fig_delta_depth.pdf – Signal strength vs. circuit depth analysis README.md – Detailed job IDs, API instructions, and reproduction notes Related Resources Paper 1 (Discovery): doi:10.5281/zenodo.19202945 Live API: https://kleincode.pythonanywhere.com GitHub: https://github.com/theoricline/klein-bottle-qec License: Creative Commons Attribution 4.0 International (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.
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.
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
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 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.
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