This talk will explain the structure of Khovanov homology for knots and links and will present a quantum algorithm for its computation that is based on the use of combinatorial Hodge theory. By using combinatorial Hodge theory we can formulate the problem of computing the homology in terms of the determination of the dimension of the eigenvalue one subspace of a unitary operator that is associated with the boundary and coboundary operators of the Khovanov complex. We then use the quantum phase estimation algorithm to find the dimension of this subspace. This is a first foray into the problems of producing a quantum algorithm for the significant knot and link invariant Khovanov homology. There are other deeper problems related to the structure of the chain complex and we shall discuss these difficulties. This paper represents joint work with Sam Lomonaco and Nadya Shirakova.
This talk will explain the structure of Khovanov homology for knots and links and will present a quantum algorithm for its computation that is based on the use of combinatorial Hodge theory. By using combinatorial Hodge theory we can formulate the problem of computing the homology in terms of the determination of the dimension of the eigenvalue one subspace of a unitary operator that is associated with the boundary and coboundary operators of the Khovanov complex. We then use the quantum phase estimation algorithm to find the dimension of this subspace. This is a first foray into the problems of producing a quantum algorithm for the significant knot and link invariant Khovanov homology. There are other deeper problems related to the structure of the chain complex and we shall discuss these difficulties. This paper represents joint work with Sam Lomonaco and Nadya Shirakova.
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.
It has been conjectured that quantum entanglement operators can be lifted to braiding operators by the way of the topological quantum field theory axioms set forth by Witten and Atiyah. Moreover, it can be readily shown that quantum link invariants need entanglement to construct topological invariants. Given these results and the already dense mathematical framework underlying topology and quantum field theory, we propose that, through the usage of quantum algebra and bracket models, we can identify a significant area of overlap where entangling R-matrix solutions to the Yang-Baxter equation can be used to construct invariants of knots and links. Such a connection illuminates the relationship between quantum and topological entanglement, and allows us to draw novel insights about properties of entangling gates, for example the question of what it is that allows some entangling R-matrices to generate topological invariants when others do not. In conclusion, by studying the boundary between topological and quantum entanglement we construct novel topological invariants that can have significant impact on the study of quantum computing and can help make progress towards the problem of representing the Artin braid group as unitary matrices.
. This paper is an introduction to relationships between quantum topology and quantum computing. We show how knots are related not just to braiding and quantum operators, but to quantum set theoretical foundations, algebras of fermions, and we show how the operation of negation in logic, seen as both a value and an operator, can generate the fusion algebra for a Majorana fermion. We call negation in this mode the mark, as it operates on itself to change from marked to unmarked states. The mark viewed recursively as a simplest discrete dynamical system naturally generates the fermion algebra, the quaternions and the braid group representations related to Majorana fermions. The paper begins with these fundamentals. It then discusses unitary solutions to the Yang-Baxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and gives an exposition of knot-theoretic recoupling theory, its relationship with topological quantum field theory and applies these methods to produce unitary representations of the braid groups that are dense in the unitary groups. These methods are rooted in the bracket state sum model for the Jones polynomial. A self-contained study of the quantum universal Fibonacci model is given. Results are applied to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and theWitten-Reshetikhin-Turaev invariant of three manifolds. Two constructions are given for the Fibonacci model, one based in Temperley-Lieb recoupling theory, the other quite elementary and also based on the Temperley-Lieb algebra. This paper begins an exploration of quantum epistemology in relation to the structure of discrimination as the underpinning of basic logic, perception and measurement. Introduction. This paper is an introduction to relationships between quantum topology and quantum computing. We take a foundational approach, showing how knots are related not just to braiding and quantum operators, but to quantum set theoretical foundations and algebras of fermions. We show how the operation of negation in logic, seen as both a value and an operator, can generate the fusion algebra for a Majorana fermion, a particle that is its own anti-particle and interacts with itself either to annihilate itself or to produce itself. We call negation in this mode the mark , as it operates on itself to change from marked to unmarked states.
Quantum entanglement, a peculiar phenomenon in quantum mechanics, offers a fundamentally different approach to computation than classical systems. This research explores the potential of leveraging entanglement as a core building block for computational complexity, proposing a novel method for quantifying the difficulty of algorithms through the development of an entanglement metric. The core claim is that certain computational tasks exhibit inherent complexity that cannot be adequately explained by classical complexity theory, and that this approach could lead to a deeper understanding of the limits of computation. The paper details the creation of an entanglement metric, a mathematical framework designed to directly relate entanglement properties to computational resource requirements, and demonstrates its ability to accurately estimate the complexity of specific algorithms. This work represents a significant step towards a more rigorous and predictive analysis of computational challenges, potentially uncovering previously unrecognized limitations within the realm of algorithmic complexity.
Khovanov homology is a topological knot invariant that categorifies the Jones polynomial, recognizes the unknot, and is conjectured to appear as an observable in 4D supersymmetric Yang--Mills theory. Despite its rich mathematical and physical significance, the computational complexity of Khovanov homology remains largely unknown. To address this challenge, this work initiates the study of efficient quantum algorithms for Khovanov homology. We provide simple proofs that increasingly accurate additive approximations to the ranks of Khovanov homology are DQC1-hard, BQP-hard, and #P-hard, respectively. For the first two approximation regimes, we propose a novel quantum algorithm. Our algorithm is efficient provided the corresponding Hodge Laplacian thermalizes in polynomial time and has a sufficiently large spectral gap, for which we give numerical and analytical evidence. Our approach introduces a pre-thermalization procedure that allows our quantum algorithm to succeed even if the Betti numbers of Khovanov homology are much smaller than the dimensions of the corresponding chain spaces, overcoming a limitation of prior quantum homology algorithms. We introduce novel connections between Khovanov homology and graph theory to derive analytic lower bounds on the spectral gap.
We present a comprehensive quantum algorithm for computing Khovanov cohomology, a categorification of the Jones polynomial, together with a clear narrative that connects its motivation, construction, and implementation. By simulating Laplacian operators derived from the Khovanov cochain complex, we estimate cohomological dimensions using quantum phase estimation, thermal sampling, and Quantum Singular Value Transformation (QSVT). The framework explicitly exploits the Frobenius algebra structure and cobordism maps, showing how these topological tools translate naturally into unitary quantum operations. Worked examples for the trefoil knot and Hopf link illustrate each step, from encoding the cube of resolutions to constructing differentials and Laplacians. This narrative approach not only refines and operationalizes the Laplacian method introduced by previous works but also provides a practical roadmap for extending categorified invariants into the realm of quantum algorithms suitable for near-term devices.