Topological and geometrical quantum computation in cohesive Khovanov homotopy type theory
Abstract
Abstract
The recently proposed Cohesive Homotopy Type Theory is exploited as a formal foundation for central concepts in Topological and Geometrical Quantum Computation. Specifically the Cohesive Homotopy Type Theory provides a formal, logical approach to concepts like smoothness, cohomology and Khovanov homology; and such approach permits to clarify the quantum algorithms in the context of Topological and Geometrical Quantum Computation. In particular we consider the socalled a "openclosed stringy topological quantum computera" which is a theoretical topological quantum computer that employs a system of openclosed strings whose worldsheets are openclosed cobordisms. The openclosed stringy topological computer is able to compute the Khovanov homology for tangles and for hence it is a universal quantum computer given than any quantum computation is reduced to an instance of computation of the Khovanov homology for tangles. The universal algebra in this case is the Frobenius Algebra and the possible openclosed stringy topological quantum computers are forming a symmetric monoidal category which is equivalent to the category of knowledgeable Frobenius algebras. Then the mathematical design of an openclosed stringy topological quantum computer is involved with computations and theorem proving for generalized Frobenius algebras. Such computations and theorem proving can be performed automatically using the Automated Theorem Provers with the TPTP language and the SMTsolver Z3 with the SMTLIB language. Some examples of application of ATPs and SMTsolvers in the mathematical setup of an openclosed stringy topological quantum computer will be provided. © 2015 SPIE.
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