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

Mario Vélez; Juan Ospina

doi:10.1117/12.918886

8 May 2012 Possible universal quantum algorithms for generalized Khovanov homology and the Rasmussen's invariant

Mario Vélez, Juan Ospina

Proceedings Volume 8400, Quantum Information and Computation X; 84000W (2012) https://doi.org/10.1117/12.918886
Event: SPIE Defense, Security, and Sensing, 2012, Baltimore, Maryland, United States

Abstract

Possible quantum algorithms for generalized Khovanov homology and the Rasmussen's invariant are proposed. Such algorithms are resulting from adaptations of the recently proposed Kauffman's 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 Rasmussen's 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.

Citation Download Citation

Mario Vélez and Juan Ospina "Possible universal quantum algorithms for generalized Khovanov homology and the Rasmussen's invariant", Proc. SPIE 8400, Quantum Information and Computation X, 84000W (8 May 2012); https://doi.org/10.1117/12.918886

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