Noncommutative tori and universal sets of non-binary quantum gates

Abstract

A problem of universality in simulation of evolution of quantum system and in theory of quantum computations is 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 (quant-ph/0010071) application of Clifford algebras to constructions of universal sets of binary quantum gates $U_k \in U(2^n)$ was shown. For application of a similar approach to non-binary quantum gates $U_k \in U(l^n)$ in present work is used rational noncommutative torus ${\Bbb T}^{2n}_{1/l}$. A set of universal non-binary two-gates is presented here as one example.