Topological quantum computation of the Dold-Thom functor

Abstract

A possible topological quantum computation of the Dold-Thom functor is presented. The method that will be used is the following: a) Certain 1+1-topological quantum field theories valued in symmetric bimonoidal categories are converted into stable homotopical data, using a machinery recently introduced by Elmendorf and Mandell; b) we exploit, in this framework, two recent results (independent of each other) on refinements of Khovanov homology: our refinement into a module over the connective k-theory spectrum and a stronger result by Lipshitz and Sarkar refining Khovanov homology into a stable homotopy type; c) starting from the Khovanov homotopy the Dold-Thom functor is constructed; d) the full construction is formulated as a topological quantum algorithm. It is conjectured that the Jones polynomial can be described as the analytical index of certain Dirac operator defined in the context of the Khovanov homotopy using the Dold-Thom functor. As a line for future research is interesting to study the corresponding supersymmetric model for which the Khovanov-Dirac operator plays the role of a supercharge.

Topological and geometrical quantum computation in cohesive Khovanov homotopy type theory

“…The evolution in this case is given by (10) Then (10) is interpreted as a topological quantum computer which is able to compute indexes for operators which are super-charges of given super-symmetric σ-models specified by the a lagrangian with the form (11) where the temporal covariant derivative is defined by (12) and (13) Proc. of SPIE Vol.…”

“…Recent advances in topological quantum computation [1] have been focused towards the possible quantum algorithms for Khovanov homology and its generalizations [2,3,4]; including Khovanov Homotopy [5,6,7,8,9,10] , the corresponding the Steenrod square operation Sq 2 [7,10,11] and the Dold-Thom functor [8,13] . Also recently the so called cohesive homotopy type theory was proposed as a logical framework for the formulation of the basic notions of differential geometry (fiber bundles, connections, curvatures, characteristics classes) in the spirit of the synthetic differential geometry [14].…”