Two qutrits universal quantum gates from the nine-dimensional unitary solutions of the Yang-Baxter equation

Abstract

Using the Kauffman-Lomonaco method, some two-qutrits universal quantum gates are derived from the nine-dimensional unitary solutions of the Yang-Baxter equations associated with algebraic structures like the partial transpose operator and the dihedral group, which admit three dimensional representations. The Yang-Baxterization method given by Zhang-Kauffman-Ge is continuously used to obtain two-qutrits quantum gates and certain Hamiltonians for the evolution of the quantum gates are obtained, being such Hamiltonians interpreted as physical Hamiltonians of chain of particles of spin 1. Finally, the generalization for systems of two qudits is presented in the case of Yang-Baxterization of representations of braided monoidal algebra like the BH algebra and the bicolored Birman-Wenzl-Muraki algebra For these algebras the corresponding two-qudits quantum gates are constructed jointly with the associated Hamiltonians interpreted like physical chains of particles with spin d . It is conjectured that the derived two-qdits quantum gates and the Hamiltonians may be implemented over bi-dimensional lattice systems like anyons systems or more generally over any physical systems ruled by the Yang-Baxter equations.