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

International Conference on Cryptography and Information Security (VCRIS)

A Model of Quantum Tunneling with Knots and Braid Operators

Phuong-Nam Nguyen

Published Oct 30, 2025• 0 citations
Publisher

Abstract

Topological quantum computing is an emerging field that leverages the mathematical structures of knots and braids to encode and manipulate quantum information. However, the topological nature of quantum tunneling remains largely unexplored. In this work, we introduce a novel paradigm for quantum tunneling based on the knot representation of quantum wavefunctions, where Reidemeister moves—interpreted as braid group operators—govern the wavefunction’s evolution. Specifically, we propose four distinct models for topological quantum tunneling: (1) tunneling via Reidemeister transitions, (2) tunneling on a ring-domain, (3) tunneling under space-time curvature, and (4) tunneling influenced by gravitational potential. By establishing a rigorous connection between knot theory, braid group structures, and quantum tunneling, our approach provides a new framework for understanding quantum transport in complex environments. The broader impact of this work lies in bridging topological quantum mechanics with quantum biology and life sciences, opening new avenues for interdisciplinary research in quantum information processing and biophysical modeling.

DOI
10.1109/vcris68011.2025.11250559

Related papers

Unitary braid matrices: bridge between topological and quantum entanglements

B Abdesellam, A Chakrabati

Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions.Two distinct classes are considered.Their (nonlocal) unitary actions on separable pure product states of three identical subsystems (i.e., the spin projections of three particles) are explicitly evaluated for all dimensions.This, for our classes, is shown to generate entangled superposition of four terms in the base space.The 3-body and 2-body entanglements (in three 2-body subsystems), the 3 tangles, and 2 tangles are explicitly evaluated for each class.For our matrices, these are parametrized.Varying parameters they can be made to sweep over the domain (0, 1).Thus, braiding operators corresponding to over-and undercrossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entanglements.For higher dimensions, starting with different initial triplets one can entangle by turns, each state with all the rest.A specific coupling of three angular momenta is briefly discussed to throw more light on three body entanglements.

Journal of Physical Mathematics•Jan 1, 2010•0 citations

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

Unitary braid matrices: bridge between topological and quantum entanglements

B. Abdesselam, Amitabha Chakrabarti

Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions. Two distinct classes are considered. Their (non-local) unitary actions on separable pure product states of three identical subsystems (say, the spin projections of three particles) are explicitly evaluated for all dimensions. This, for our classes, is shown to generate entangled superposition of four terms in the base space. The 3-body and 2-body entanglements (in three 2-body subsystems), the 3-tangles and 2-tangles are explicitly evaluated for each class. For our matrices, these are parametrized. Varying parameters they can be made to sweep over the domain (0,1).Thus braiding operators corresponding to over- and under-crossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entanglements. For higher dimensions, starting with different initial triplets one can entangle by turns, each state with all the rest. A specific coupling of three angular momenta is briefly discussed to throw more light on three body entanglements.

arXiv:0911.0906•Journal of Physical Mathematics•Jan 1, 2010•2 citations

Topological Quantum Computing and 3-Manifolds

Torsten Aßelmeyer-Maluga

In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston’s geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al.

arXiv:2102.04452•Quantum Reports•Feb 5, 2021•4 citations

Quantum Entanglement Geometry Information Theory and Topological Field Theory: A Connection

Jincheng Zhang

This paper investigates the interplay between quantum entanglement and topological field theory, proposing a framework for quantum entanglement that leverages the principles of topological geometry to reveal the fundamental nature of entanglement and explore its intricate relationship with topological structures. We aim to provide a novel approach to understanding entanglement, moving beyond traditional interpretations and exploring potential connections within the realm of topological physics. The core of this work focuses on a rigorous mathematical formulation of entanglement through the lens of topological invariants, ultimately seeking to uncover deeper insights into the essence of quantum entanglement. The proposed mechanism involves a deep integration of quantum mechanics and topological geometry, offering a potential paradigm shift in our understanding of this fascinating phenomenon. The paper concludes with a discussion of the implications of this approach for both quantum information theory and the broader field of topological physics.

•Sep 1, 2026•0 citations

Quantum Entanglement Topological Topology

Jincheng Zhang

Quantum Entanglement Topological Topology (QET) introduces a novel framework for studying the interplay between topological structures and quantum entanglement. This work proposes a computational paradigm predicated on the unique properties of quantum entanglement, aiming to provide a more profound understanding of how these two concepts interact. The core mechanism leverages entanglement to construct a system that dynamically modifies and explores topological properties, offering a potential pathway for investigating quantum entanglement's influence on topological phenomena. This research explores the implications of this framework for understanding the emergence and manipulation of complex topological structures, and ultimately, assessing its potential for applications in quantum information processing and materials science.

•Sep 2, 2026•0 citations
Topological quantum computing is an emerging field that leverages the mathematical structures of knots and braids to encode and manipulate quantum information. However, the topological nature of quantum tunneling remains largely unexplored. In this work, we introduce a novel paradigm for quantum tunneling based on the knot representation of quantum wavefunctions, where Reidemeister moves—interpreted as braid group operators—govern the wavefunction’s evolution. Specifically, we propose four distinct models for topological quantum tunneling: (1) tunneling via Reidemeister transitions, (2) tunneling on a ring-domain, (3) tunneling under space-time curvature, and (4) tunneling influenced by gravitational potential. By establishing a rigorous connection between knot theory, braid group structures, and quantum tunneling, our approach provides a new framework for understanding quantum transport in complex environments. The broader impact of this work lies in bridging topological quantum mechanics with quantum biology and life sciences, opening new avenues for interdisciplinary research in quantum information processing and biophysical modeling.
Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions.Two distinct classes are considered.Their (nonlocal) unitary actions on separable pure product states of three identical subsystems (i.e., the spin projections of three particles) are explicitly evaluated for all dimensions.This, for our classes, is shown to generate entangled superposition of four terms in the base space.The 3-body and 2-body entanglements (in three 2-body subsystems), the 3 tangles, and 2 tangles are explicitly evaluated for each class.For our matrices, these are parametrized.Varying parameters they can be made to sweep over the domain (0, 1).Thus, braiding operators corresponding to over-and undercrossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entanglements.For higher dimensions, starting with different initial triplets one can entangle by turns, each state with all the rest.A specific coupling of three angular momenta is briefly discussed to throw more light on three body entanglements.
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.
Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions. Two distinct classes are considered. Their (non-local) unitary actions on separable pure product states of three identical subsystems (say, the spin projections of three particles) are explicitly evaluated for all dimensions. This, for our classes, is shown to generate entangled superposition of four terms in the base space. The 3-body and 2-body entanglements (in three 2-body subsystems), the 3-tangles and 2-tangles are explicitly evaluated for each class. For our matrices, these are parametrized. Varying parameters they can be made to sweep over the domain (0,1).Thus braiding operators corresponding to over- and under-crossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entanglements. For higher dimensions, starting with different initial triplets one can entangle by turns, each state with all the rest. A specific coupling of three angular momenta is briefly discussed to throw more light on three body entanglements.
In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston’s geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al.
This paper investigates the interplay between quantum entanglement and topological field theory, proposing a framework for quantum entanglement that leverages the principles of topological geometry to reveal the fundamental nature of entanglement and explore its intricate relationship with topological structures. We aim to provide a novel approach to understanding entanglement, moving beyond traditional interpretations and exploring potential connections within the realm of topological physics. The core of this work focuses on a rigorous mathematical formulation of entanglement through the lens of topological invariants, ultimately seeking to uncover deeper insights into the essence of quantum entanglement. The proposed mechanism involves a deep integration of quantum mechanics and topological geometry, offering a potential paradigm shift in our understanding of this fascinating phenomenon. The paper concludes with a discussion of the implications of this approach for both quantum information theory and the broader field of topological physics.
Quantum Entanglement Topological Topology (QET) introduces a novel framework for studying the interplay between topological structures and quantum entanglement. This work proposes a computational paradigm predicated on the unique properties of quantum entanglement, aiming to provide a more profound understanding of how these two concepts interact. The core mechanism leverages entanglement to construct a system that dynamically modifies and explores topological properties, offering a potential pathway for investigating quantum entanglement's influence on topological phenomena. This research explores the implications of this framework for understanding the emergence and manipulation of complex topological structures, and ultimately, assessing its potential for applications in quantum information processing and materials science.