Algebras and universal quantum computations with higher dimensional systems

Here is discussed application of the Weyl pair to construction of universal set of quantum gates for high-dimensional quantum system. An application of Lie algebras (Hamiltonians) for construction of universal gates is revisited first. It is shown next, how for quantum computation with qubits can be used two-dimensional analog of this Cayley-Weyl matrix algebras, i.e. Clifford algebras, and discussed well known applications to product operator formalism in NMR, Jordan-Wigner construction in fermionic quantum computations. It is introduced universal set of quantum gates for higher dimensional system (“qudit”), as some generalization of these models. Finally it is briefly mentioned possible application of such algebraic methods to design of quantum processors (programmable gates arrays) and discussed generalization to quantum computation with continuous variables.