Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.30406

Planar Contact Structures with Calabi-Yau Fillings and Topological Quantum Computation

Atsuhide Mori

math.GTmath-phmath.SGquant-phSubmitted Sep 24, 2026• Announced Sep 28, 2026
v1 · Sep 24, 2026v2 · Oct 1, 2026
arXivHTMLPDF

Abstract

We study planar open books obtained by lifting braids through branched covers of the disk D^2, together with the quantum operations in the Ising representation. We note a criterion for the associated Stein fillings to be Calabi-Yau (CY). Among positive factorizations of a fixed monodromy, every CY factorization has minimal length, and its filling minimizes χand b_2. An application to Baykur's recent examples gives one planar contact 3-manifold with infinitely many non-homeomorphic CY fillings. The cover there is of degree \ge 6. At degree 4 a single CY filling forces every filling to be CY, and the filling is unique in a certain case; at degree \le 3 the filling is unique and is CY under a mild condition. Admissible cuts of the covering disk decompose the openbook into subopenbooks. Every positive factorization then localizes to the pieces, and the CY condition holds exactly when it holds locally. This makes the state space a direct sum of tensor products indexed by the compatible parity choices with at most two qubits in each factor when the pieces have degree \le 4, which is also the range in which the CY condition depends only on the monodromy. For a particular degree 4 cover, the points, lines and flags of the two-qubit doily are realized by the system of its subopenbooks. Fifteen liftable braids there share the same quantum operation and the same Stein filling, and are separated only by which subopenbook systems they admit. A choice of tensor-product structure is thus carried by the lift and not by the braid group representation, which suggests a link between contact topology and quantum entanglement.

Comments
18 pages. Generative AI (ChatGPT, Claude) used for literature search and verification, checking elementary computations, and mathematical discussion; see the declaration before the references

Related papers

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

Atsuhide Mori

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.

math.GTCross-listedarXiv:2610.25878•Oct 20, 2026

Quantum entanglement, Calabi-Yau manifolds, and noncommutative algebraic geometry

Shinnosuke Okawa, Kazushi Ueda

We relate SLOCC equivalence classes of qudit states to moduli spaces of Calabi-Yau manifolds equipped with a collection of line bundles. The cases of 3 qutrits and 4 qubits are also related to noncommutative algebraic geometry.

arXiv:1402.3768•Feb 16, 2014•3 citations

Topological qubit design and leakage

Robert Ainsworth, Johannes K. Slingerland

We examine how best to design qubits for use in topological quantum computation. These qubits are topological Hilbert spaces associated with small groups of anyons. Operations are performed on these by exchanging the anyons. One might argue that in order to have as many simple single-qubit operations as possible, the number of anyons per group should be maximized. However, we show that there is a maximal number of particles per qubit, namely 4, and more generally a maximal number of particles for qudits of dimension d . We also look at the possibility of having topological qubits for which one can perform two-qubit gates without leakage into non-computational states. It turns out that the requirement that all two-qubit gates are leakage free is very restrictive and this property can only be realized for two-qubit systems related to Ising-like anyon models, which do not allow for universal quantum computation by braiding. Our results follow directly from the representation theory of braid groups, which implies that they are valid for all anyon models. We also make some remarks about generalizations to other exchange groups.

arXiv:1102.5029•New Journal of Physics•Jun 30, 2011•69 citations

A₂ Root System-Driven Cross-Disciplinary Closed Loop: Qiskit Experimental Verification of Geometry, Quantum, and Topology Issues

Yang Ou, W. Y. Sun

Taking the A₂root system as the sole core symmetric carrier, this paper constructs a cross-disciplinary self-consistent loop of "algebraic coding-geometric realization-quantum stabilization-topological verification". Through structure-preserving mappings and Qiskit quantum simulations, it achieves in-depth integration and empirical verification of 3-dimensional Calabi-Yau (CY) manifolds, quantum stable systems, and the Poincaré conjecture. The core results include: 1) A constructive proof of the exclusive correspondence between the A₂ root system and 3-dimensional CY manifolds—the rank 2 of the root system strictly determines the Hodge number , and the Cartan subalgebra forms an inner product-preserving mapping with the Kähler moduli space; 2) Establishment of the A₂ root system-driven quantum system theory: algebraic preparation of 4 Bell states via the structure-preserving mapping , realization of noise-resistant stabilization with fidelity ≈1.0 in noisy environments based on the dual reflection mechanism of polarized root systems, and further construction of a 2-qubit time crystal (TIME crystal), verifying period doubling (response period 4.0 = 2× driving period 2.0) and spontaneous time-translation symmetry breaking; 3) Proof of the equivalence between supersymmetric score locking () and the simply connectedness of CY manifolds (). Combined with the convergence of Ricci flow curvature (error < 10⁻⁴), it provides a specific instance verification of the Poincaré conjecture under "CY manifold + quantum constraints".

•Nov 11, 2025•1 citation

Topological aspects of quantum entanglement

Louis Hirsch Kauffman, Eshan Mehrotra

Kauffman and Lomonaco (New J Phys 4:73.1–73.18, 2002. arXiv:quant-ph/0401090 , New J Phys 6:134.1–134.40, 2004) explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In Alagic et al. (Yang–Baxter operators need quantum entanglement to distinguish knots, 2015. arXiv:1507.05979v1 ), it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang–Baxter equation. We show that the arguments used by Alagic et al. (2015) generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang–Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of Alagic et al. (2015). We also give a class of R matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give an example of an SU(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three-strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing. The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement and the $$ER = EPR$$ E R = E P R hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.

arXiv:1611.08047•Quantum Information Processing•Feb 2, 2019•20 citations

Quantum entanglement, unitary braid representation and Temperley-Lieb algebra

C.-L. Ho, A. I. Solomon, C.-H. Oh

Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly , from separable basis states, important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.

arXiv:1011.6229•Europhysics Letters (EPL)•Nov 1, 2010•11 citations
We study planar open books obtained by lifting braids through branched covers of the disk D^2, together with the quantum operations in the Ising representation. We note a criterion for the associated Stein fillings to be Calabi-Yau (CY). Among positive factorizations of a fixed monodromy, every CY factorization has minimal length, and its filling minimizes χand b_2. An application to Baykur's recent examples gives one planar contact 3-manifold with infinitely many non-homeomorphic CY fillings. The cover there is of degree \ge 6. At degree 4 a single CY filling forces every filling to be CY, and the filling is unique in a certain case; at degree \le 3 the filling is unique and is CY under a mild condition. Admissible cuts of the covering disk decompose the openbook into subopenbooks. Every positive factorization then localizes to the pieces, and the CY condition holds exactly when it holds locally. This makes the state space a direct sum of tensor products indexed by the compatible parity choices with at most two qubits in each factor when the pieces have degree \le 4, which is also the range in which the CY condition depends only on the monodromy. For a particular degree 4 cover, the points, lines and flags of the two-qubit doily are realized by the system of its subopenbooks. Fifteen liftable braids there share the same quantum operation and the same Stein filling, and are separated only by which subopenbook systems they admit. A choice of tensor-product structure is thus carried by the lift and not by the braid group representation, which suggests a link between contact topology and quantum entanglement.
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.
We examine how best to design qubits for use in topological quantum computation. These qubits are topological Hilbert spaces associated with small groups of anyons. Operations are performed on these by exchanging the anyons. One might argue that in order to have as many simple single-qubit operations as possible, the number of anyons per group should be maximized. However, we show that there is a maximal number of particles per qubit, namely 4, and more generally a maximal number of particles for qudits of dimension d . We also look at the possibility of having topological qubits for which one can perform two-qubit gates without leakage into non-computational states. It turns out that the requirement that all two-qubit gates are leakage free is very restrictive and this property can only be realized for two-qubit systems related to Ising-like anyon models, which do not allow for universal quantum computation by braiding. Our results follow directly from the representation theory of braid groups, which implies that they are valid for all anyon models. We also make some remarks about generalizations to other exchange groups.
Taking the A₂root system as the sole core symmetric carrier, this paper constructs a cross-disciplinary self-consistent loop of "algebraic coding-geometric realization-quantum stabilization-topological verification". Through structure-preserving mappings and Qiskit quantum simulations, it achieves in-depth integration and empirical verification of 3-dimensional Calabi-Yau (CY) manifolds, quantum stable systems, and the Poincaré conjecture. The core results include: 1) A constructive proof of the exclusive correspondence between the A₂ root system and 3-dimensional CY manifolds—the rank 2 of the root system strictly determines the Hodge number , and the Cartan subalgebra forms an inner product-preserving mapping with the Kähler moduli space; 2) Establishment of the A₂ root system-driven quantum system theory: algebraic preparation of 4 Bell states via the structure-preserving mapping , realization of noise-resistant stabilization with fidelity ≈1.0 in noisy environments based on the dual reflection mechanism of polarized root systems, and further construction of a 2-qubit time crystal (TIME crystal), verifying period doubling (response period 4.0 = 2× driving period 2.0) and spontaneous time-translation symmetry breaking; 3) Proof of the equivalence between supersymmetric score locking () and the simply connectedness of CY manifolds (). Combined with the convergence of Ricci flow curvature (error < 10⁻⁴), it provides a specific instance verification of the Poincaré conjecture under "CY manifold + quantum constraints".
Kauffman and Lomonaco (New J Phys 4:73.1–73.18, 2002. arXiv:quant-ph/0401090 , New J Phys 6:134.1–134.40, 2004) explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In Alagic et al. (Yang–Baxter operators need quantum entanglement to distinguish knots, 2015. arXiv:1507.05979v1 ), it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang–Baxter equation. We show that the arguments used by Alagic et al. (2015) generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang–Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of Alagic et al. (2015). We also give a class of R matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give an example of an SU(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three-strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing. The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement and the ER=EPRER = EPRER=EPR E R = E P R hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.
Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly , from separable basis states, important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.