Noncommutative tori and universal sets of nonbinary quantum gates

Abstract

We address the problem of universality in simulation of evolution of quantum system and in theory of quantum computations related with the possibility of expression or approximation of arbitrary unitary transformation by composition of specific unitary transformations (quantum gates) from given set. In an earlier paper application of Clifford algebras to constructions of universal sets of binary quantum gates $U_k{\text{∈U(2}}^n)$ was shown. For application of a similar approach to nonbinary quantum gates $U_k{\mathrm{\in U(l}}^n\mathrm{),}$ in present work we used rational noncommutative torus ${\mathbb{T}}_{\text{1/l}}^{\text{2n}}.$ A set of universal nonbinary two-gates is presented here as one example.