Mario Vélez

Numerical Simulations of a Possible Hypercomputational Quantum Algorithm

The hypercomputers compute functions or numbers, or more generally solve problems or carry out tasks, that cannot be computed or solved by a Turing machine. Several numerical simulations of a possible hypercomputational algorithm based on quantum computations previously constructed by the authors are presented. The hypercomputability of our algorithm is based on the fact that this algorithm could solve a classically non-computable decision problem, the Hilbert’s tenth problem. The numerical simulations were realized for three types of Diophantine equations: with and without solutions in non-negative integers, and without solutions by way of various traditional mathematical packages.

Tutte Polynomials and Topological Quantum Algorithms in Social Network Analysis for Epidemiology, Bio-surveillance and Bio-security

The Tutte polynomial and the Aharonov-Arab-Ebal-Landau algorithm are applied to Social Network Analysis (SNA) for Epidemiology, Biosurveillance and Biosecurity. We use the methods of Algebraic Computational SNA and of Topological Quantum Computation. The Tutte polynomial is used to describe both the evolution of a social network as the reduced network when some nodes are deleted in an original network and the basic reproductive number for a spatial model with bi-networks, borders and memories. We obtain explicit equations that relate evaluations of the Tutte polynomial with epidemiological parameters such as infectiousness, diffusivity and percolation. We claim, finally, that future topological quantum computers will be very important tools in Epidemiology and that the representation of social networks as ribbon graphs will permit the full application of the Bollobas-Riordan-Tutte polynomial with all its combinatorial universality to be epidemiologically relevant.

Possible topological quantum computation via Khovanov homology: D-brane topological quantum computer

A model of a D-Brane Topological Quantum Computer (DBTQC) is presented and sustained. The model is based on four-dimensional TQFTs of the Donaldson-Witten and Seiber-Witten kinds. It is argued that the DBTQC is able to compute Khovanov homology for knots, links and graphs. The DBTQC physically incorporates the mathematical process of categorification according to which the invariant polynomials for knots, links and graphs such as Jones, HOMFLY, Tutte and Bollobás-Riordan polynomials can be computed as the Euler characteristics corresponding to special homology complexes associated with knots, links and graphs. The DBTQC is conjectured as a powerful universal quantum computer in the sense that the DBTQC computes Khovanov homology which is considered like powerful that the Jones polynomial.

Possible quantum algorithms for the Bollobas-Riordan-Tutte polynomial of a ribbon graph

Three possible quantum algorithms, for the computation of the Bollobás-Riordan-Tutte polynomial of a given ribbon graph, are presented and discussed. The first possible algorithm is based on the spanning quasi-trees expansion for generalized Tutte polynomials of generalized graphs and on a quantum version of the Binary Decision Diagram (BDD) for quasi-trees . The second possible algorithm is based on the relation between the Kauffman bracket and the Tutte polynomial; and with an application of the recently introduced Aharonov-Arad-Eban-Landau quantum algorithm. The third possible algorithm is based on the relation between the HOMFLY polynomial and the Tutte polynomial; and with an application of the Wocjan-Yard quantum algorithm. It is claimed that these possible algorithms may be more efficient that the best known classical algorithms. These three algorithms may have interesting applications in computer science at general or in computational biology and bio-informatics in particular. A line for future research based on the categorification project is mentioned.

Biomedical Computer Vision Using Computer Algebra: Analysis of a Case of Rhinocerebral Mucormycosis in a Diabetic Boy

Computer algebra is applied to biomedical computer vision. Specifically certain biomedical images resulting from a case of rhinocerebral mucormysocis in a diabetic boy are analyzed using techniques in computational geometry and in algebraic-geometric topology. We apply convolution and deblurring via diffusion equation from the side of computational geometry and knot theory, graph theory and singular homology form the side of algebraic-geometric topology. Our strategy consists in to represent the biomedical images using algebraic structures in such way that the peculiarities of the images are represented using algebraic complexities. With our strategy we obtain an automatic procedure for the analysis and the diagnostic based on biomedical images.

Using computer algebra for Yang-Baxterization applied to quantum computing

Using Computer Algebra Software (Mathematica and Maple), the recently introduced topic of Yang- Baxterization applied to quantum computing, is explored from the mathematical and computational views. Some algorithms of computer algebra were elaborated with the aim to make the calculations to obtain some of results that were originally presented in the paper by Shang-Kauffman-Ge. Also certain new results about computational Yang-baxterization are presented. We obtain some Hamiltonians for hypothetical physical systems which can be realized within the domain of spin chains and certain diffusion process. We conclude that it is possible to have real physical systems on which implement, via Yang-baxterization, the standard quantum gates with topological protection. Finally some lines for future research are deligned.

Possible universal quantum algorithms for generalized Khovanov homology and the Rasmussen's invariant

Possible quantum algorithms for generalized Khovanov homology and the Rasmussens invariant are proposed. Such algorithms are resulting from adaptations of the recently proposed Kauffmans algorithm for the standard Khovanov homology. The method that was applied consists in to write the relevant quantum invariant as the trace of a certain unitary operator and then to compute the trace using the Hadamard test. We apply such method to the quantum computation of the Jones polynomial, HOMFLY polynomial, Chromatic polynomial, Tutte polynomial and Bollobàs- Riordan polynomial. These polynomials are quantum topological invariants for knots, links, graphs and ribbon graphs respectively. The Jones polynomial is categorified by the standard Khovanov homology and the others polynomials are categorified by generalized Khovanov homologies, such as the Khovanov-Rozansky homology and the graph homologies. The algorithm for the Rasmussens invariant is obtained using the gauge theory; and the recently introduced program of homotopyfication is linked with the super-symmetric quantum mechanics. It is claimed that a new program of analytification could be development from the homotopyfication using the celebrated Atiyah-Singer theorem and its super-symmetric interpretations. It is hoped that the super-symmetric quantum mechanics provides the hardware for the implementation of the proposed quantum algorithms.

Possible Universal Quantum Algorithms for Generalized Turaev-Viro invariants

An emergent trend in quantum computation is the topological quantum computation (TQC). Briefly, TQC results from the application of quantum computation with the aim to solve the problems of quantum topology such as topological invariants for knots and links (Jones polynomials, HOMFLY polynomials, Khovanov polynomials); topological invariants for graphs (Tutte polynomial and Bollobás-Riordan polynomial); topological invariants for 3-manifolds (Reshetiskin-Turaev, Turaev-Viro and Turaer-Viro-Ocneanu invariants) and topological invariants for 4-manifolds( Crane-Yetter invariants). In a few words, TQC is concerned with the formulation of quantum algorithms for the computation of these topological invariants in quantum topology. Given that one of the fundamental achievements of quantum topology was the discovery of strong connections between monoidal categories and 3-dimensional manifolds, in TQC is possible and necessary to exploit such connections with the purpose to formulate universal quantum algorithms for topological invariants of 3-manifolds. In the present work we make an exploration of such possibilities. Specifically we search for universal quantum algorithms for generalized Turaev-Viro invariants of 3-manifolds such as the Turaev-Viro-Ocneanu invariants, the Kashaev-Baseilhac-Benedetti invariants of 3-manifolds with links and the Geer-Kashaev-Turaev invariants of 3-manifolds with a link and a principal bundle. We also look for physical systems (three dimensional topological insulators and three-dimensional gravity) over which implement the resulting universal topological quantum algorithms.

Quantum algorithms for virtual Jones polynomials via Thistlethwaite theorems

Recently a quantum algorithm for the Jones polynomial of virtual links was proposed by Kauffman and Dye via the implementation of the virtual braid group in anyonic topological quantum computation when the virtual crossings are considered as generalized swap gates. Also recently, a mathematical method for the computation of the Jones polynomial of a given virtual link in terms of the relative Tuttle polynomial of its face (Tait) graph with some suitable variable substitutions was proposed by Diao and Hetyei. The method of Diao and Hetyei is offered as an alternative to the ribbon graph approach according to which the Tutte polynomial of a given virtual link is computed in terms of the Bollobás- Riordan polynomial of the corresponding ribbon graph. The method of Diao and Hetyei can be considered as an extension of the celebrated Thistlethwaite theorem according to which invariant polynomials for knots and links are derived from invariant polynomials for graphs. Starting from these ideas we propose a quantum algorithm for the Jones polynomial of a given virtual link in terms of the generalized Tutte polynomials by exploiting the Thistlethwaite theorem and the Kauffman algorithm . Our method is claimed as the quantum version of the Diao-Hetyei method. Possible supersymmetric implementations of our algortihm are discussed jointly with its formulations using topological quantum lambda calculus.

Gravitational topological quantum computation

A new model in topological quantum computing, named Gravitational Topological Quantum Computing (GTQC), is introduced as an alternative respect to the Anyonic Topological Quantum Computing and DNA Computing. In the new model the quantum computer is the quantum space-time itself and the corresponding quantum algorithms refer to the computation of topological invariants for knots, links and tangles. Some applications of GTQC in quantum complexity theory and computability theory are discussed, particularly it is conjectured that the Khovanov polynomial for knots and links is more hard than #P-hard; and that the homeomorphism problem, which is noncomputable, maybe can be computed after all via a hyper-computer based on GTQC.