Quantum Gates and Clifford Algebras

Clifford algebras are used for definition of spinors. Because of using spin-1/2 systems as an adequate model of quantum bit, a relation of the algebras with quantum information science has physical reasons. But there are simple mathematical properties of the algebras those also justifies such applications. First, any complex Clifford algebra with 2n generators, Cll(2n, C), has representation as algebra of all 2 n × 2 n complex matrices and so includes unitary matrix of any quantum n-gate. An arbitrary element of whole algebra corresponds to general form of linear complex transformation. The last property is also useful because linear operators are not necessary should be unitary if they used for description of restriction of some unitary operator to subspace. The second advantage is simple algebraic structure of Cll(2n) that can be expressed via tenzor product of standard " building units " and similar with behavior of composite quantum systems. The compact notation with 2n generators also can be used in software for modeling of simple quantum circuits by modern conventional computers. The standard blocks may be based on three classical groups: 2 × 2 complex and real unimodular matrices and group of Weyl spinors, SU (2). The last group may have more close relation with nonrela-tivistic quantum systems as spinor representation of group of 3D rotations , SO(3). The second one, SL(2, IR), is widely used for theory of quantum error correction codes [1] together with a complexification, the first group SL(2, C) of Pauli spinors. More exactly, they are based on group algebras: C(2×2), IR(2×2) and H, i.e. all 2 × 2 complex and real matrices and quaternions respectively.