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arXiv:2610.25878

Non-planar contact structures with Calabi-Yau Fillings and topological quantum computation

Atsuhide Mori

math.GTmath-phmath.SGquant-phSubmitted Oct 19, 2026• Announced Oct 20, 2026• Revised Oct 20, 2026 (v2)
v1 · Oct 19, 2026v2 · Oct 20, 2026
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Abstract

This paper is a non-planar counterpart of the author's earlier work on planar contact structures and topological quantum computation. On a planar page every branch point is paired by a liftable arc, and the pairing matches the Ising representation exactly, in that fixing the channels of the pairs already fixes the state. Here the page has genus g and connected binding, presented as a simple degree-three cover of the disk. Every maximal system of disjoint liftable arcs then has exactly g members and leaves two branch points unused, and the half-twist interchanging the two points is not liftable, although its cube is and lifts to the identity. Ising fusion does not see that leftover pair, determining its channel from the other g and acting there only by scalars, whereas Fibonacci fusion can leave the channel undetermined. The powers of the cube then act by ten distinct relative phases on a single joint eigenspace, while the Calabi-Yau Stein filling stays fixed. An integral residue criterion supplies the Calabi-Yau test, a factorization localized on a contact connected sum is automatically Calabi-Yau, and, unlike in the planar case, a Calabi-Yau factorization need not be the shortest.

Comments
13 pages. v2: shortened for readability. Companion to arXiv:2609.30406

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This paper is a non-planar counterpart of the author's earlier work on planar contact structures and topological quantum computation. On a planar page every branch point is paired by a liftable arc, and the pairing matches the Ising representation exactly, in that fixing the channels of the pairs already fixes the state. Here the page has genus g and connected binding, presented as a simple degree-three cover of the disk. Every maximal system of disjoint liftable arcs then has exactly g members and leaves two branch points unused, and the half-twist interchanging the two points is not liftable, although its cube is and lifts to the identity. Ising fusion does not see that leftover pair, determining its channel from the other g and acting there only by scalars, whereas Fibonacci fusion can leave the channel undetermined. The powers of the cube then act by ten distinct relative phases on a single joint eigenspace, while the Calabi-Yau Stein filling stays fixed. An integral residue criterion supplies the Calabi-Yau test, a factorization localized on a contact connected sum is automatically Calabi-Yau, and, unlike in the planar case, a Calabi-Yau factorization need not be the shortest.
Non-Clifford gates, used to generate quantum magic, are essential for universal quantum computation. We show that non-Clifford gates arise naturally from path integrals in topological quantum field theories, where their magic-generating properties are determined by the algebraic data of the theory. In Chern-Simons theory, we construct the Ising interaction gate, whose generator is prepared by path integration over simple three-boundary manifolds, and show that it produces non-local magic away from discrete Clifford points. We show that the Toffoli gate is obstructed in SU(2)1SU(2)_1SU(2)1​ by the Z2\mathbb{Z}_2Z2​ fusion structure, while SU(2)3SU(2)_3SU(2)3​ is the minimal theory supporting the required conditional logic, given the density of the mapping class group in the projective unitary group on the manifold boundary. Finally, we demonstrate that the T gate arises as a path integral in Dijkgraaf-Witten theory, with gauge group Z4\mathbb{Z}_4Z4​, where a single Dehn twist on the boundary torus produces the gate without approximation. These results show that topological path integrals construct gates in multiple levels of the Clifford hierarchy, and across distinct classes of field theories, with implications for topological quantum computing.
We investigate a promising conformal field theory realization scheme for topological quantum computation based on the Fibonacci anyons, which are believed to be realized as quasiparticle excitations in the Z3\mathbb{Z}_3Z3​ parafermion fractional quantum Hall state in the second Landau level with filling factor ν=12/5ν=12/5ν=12/5. These anyons are non-Abelian and are known to be capable of universal topological quantum computation. The quantum information is encoded in the fusion channels of pairs of such non-Abelian anyons and is protected from noise and decoherence by the topological properties of these systems.The quantum gates are realized by braiding of these anyons. We propose here an implementation of the nnn-qubit topological quantum register in terms of 2n+22n+22n+2 Fibonacci anyons. The matrices emerging from the anyon exchanges, i.e. the generators of the braid group for one qubit are derived from the coordinate wave functions of a large number of electron holes and 4 Fibonacci anyons which can furthermore be represented as correlation functions in Z3\mathbb{Z}_3Z3​ parafermionic two-dimensional conformal field theory. The representations of the braid groups for more than 4 anyons are obtained by fusing pairs of anyons before braiding, thus reducing eventually the system to 4 anyons.
We study planar openbooks obtained by lifting braids through branched covers of D^2, together with the quantum operations in the Ising representation. We give a criterion for the Stein fillings to be Calabi-Yau (CY). Among positive factorizations of a fixed monodromy, a CY one has minimal length and its filling minimizes χand b_2. Applied to Baykur's recent examples, this gives a planar contact 3-manifold with infinitely many non-homeomorphic CY fillings, the cover there having degree \ge 6. At degree 4 a single CY filling forces every filling to be CY, and in one case the filling is unique; at degree \le 3 it is unique, and CY under a mild condition. Admissible cuts of D^2 decompose the openbook into subopenbooks. Every positive factorization then localizes, and the CY condition holds exactly when it holds locally. The state space is then a direct sum of tensor products indexed by the compatible parity choices, with at most two qubits per factor when the pieces have degree ≤4\le 4≤4, the same range in which the CY condition depends only on the monodromy. For one degree 4 cover, the points, lines and flags of the two-qubit doily are realized by its subopenbooks. Fifteen liftable braids there share the same quantum operation and Stein filling, and are separated only by the subopenbook systems they admit. A choice of tensor-product structure is thus carried by the lift, not by the braid group representation, suggesting a link between contact topology and quantum entanglement.
ABSTRACT We propose a framework in which physical reality is fundamentally a two-dimensional topological surface M², and quantum correlations—observed in Bell-type experiments as CHSH values above the classical bound—arise from the non-factorizability of anyonic fusion spaces built on M² rather than from nonlocal causal correlations. The structural label is the topological triple λ = (n, g, φ_T): winding number, genus, and topological phase. We show (i) that the d1b Fibonacci fusion space does not admit a tensor-product decomposition into spatially separated Alice/Bob Hilbert spaces; (ii) that on the n=8 Fibonacci fusion space of dimension D=F_7=13, using a sequential measurement protocol whose second setting contains the cross-bipartition generator σ_4, CHSH values up to |S|=3.406 are attained; and (iii) that λ carries independent operational signatures at the level of reduced channel dynamics and winding-number quantization. (A third claimed signature via surface topology is retracted in v1.6 — see Changelog below.) The framework yields a specific correction to the quantum prediction, C_TCH(a,b) = −cos θ + ε·f(n,g,θ). The dimensional hierarchy 0D → 1D → 2D → 3D_projected is derived from three premises: existence, finitude, distinguishability. TCH is categorically distinct from superdeterminism under the timeless TQFT ontology adopted throughout. VERSION 1.6 CHANGELOG Round 1 — Retraction of Predictions 1 and 6 following empirical tests (22 April 2026): Prediction 1 (detection-order autocorrelation ACF(1) ∈ [0.24, 0.59] in randomized-setting Bell experiments) was tested by reanalysis of Hensen et al. 2015 (N=245, MDE ≈ 0.18 below the predicted lower bound). Measured ACF(1) = −0.017 ± 0.062, p=0.87. Null hypothesis not rejected. Prediction retracted. Prediction 6 (genus-dependent Bell-fraction plateau as topology-specific signature) was tested on Fibonacci braiding simulations across S², T², T²\p, Σ_2, Σ_3. The Σ_2/Σ_3 plateau at 93.75% ± 0.52pp is exactly reproduced by a non-topological random-diagonal-phase control, and a Haar-random U(5) baseline yields 100% Bell-fraction. Topology-specific interpretation retracted. Coupled passages retracted: black-hole-handle enhancement bridge, ACF-as-Bohm-discriminator, "Autocorrelation as a theorem" paragraph, Abstract ACF-range sentence and Claim (iii) third signature (BH-α v2). Round 2 — Integration of Track-A (derivation check) and Track-B (structural audit) findings (25 April 2026): §III.C has been substantially sharpened without altering the numerical result |S|=3.406: Observable-edge labels corrected to (x_3, x_{n-2}=x_6) to match the calculation. For n=8 with total charge VAC, Bob's observable M_B = parity(x_6) = −I identically on all 13 basis states. The CHSH functional reduces to a one-party quantity with algebraic maximum 4; |S|=3.406 is approximately 85% of this maximum. Not a genuine Tsirelson violation in the EPR-bipartite sense. Haar-random U(13) probes without any braid structure yield |S| ∈ [3.16, 3.75] over 200 samples; the phenomenon is generic to non-factorized D=13 observables, not Fibonacci-MTC-exclusive. What remains specific is the analytical handle: only σ_4 fails to commute with M_A. Mechanism paragraph corrected: σ_4 appears in three of the four settings (A, A', B'), not only B'. §VIII.A Limitations table extended with three entries (Haar-U(13) universality; Fourier R²=0.38 at L=12; Z_2-MI universality baseline open). New scope-boundary paragraph identifies framing elements (TCH label, dimensional-emergence narrative, neutral-monistic positioning, Van-Raamsdonk references) as not entering the §III.C derivation; their status is deferred to Paper 3 (in preparation). §IV.A qualifier on Z_2-MI (no Haar-baseline yet). §IV.C context note on 91.8% Bell-fraction. §VI.C genus component of harmonic overtones removed. §I.C positive autocorrelation qualified as fixed-setting. §VIII.C Prediction 3 flagged as structurally coupled to Prediction 1. §VIII.D Protocol-dependence retitled "interpretive, not evidential". §VIII.F Conclusion list-sentence: "quantitative support" replaced by "first numerical construction". NOT AFFECTED: mathematical core §III (d1b non-factorisation, D=13 CHSH calculation, bipartition control ‖B‖=2.000 in 8000/8000), dimensional hierarchy §II, measurement barrier §VI (apart from §VI.C genus term), response to Conway-Kochen §VIII.B, Predictions 2, 2b, 2c, 4, 5, 7, 8. Retracted passages marked "[Retracted in v1.6]" in place; original v1.5 text preserved inline for transparency.