Quantum Gates and Quantum Algorithms with Clifford Algebra Technique

We use the Clifford algebra technique (J. Math. Phys. 43:5782, 2002 ; J. Math. Phys. 44:4817, 2003 ), that is nilpotents and projectors which are binomials of the Clifford algebra objects γ a with the property { γ a , γ b } + =2 η ab , for representing quantum gates and quantum algorithms needed in quantum computers in a simple and an elegant way. We identify n -qubits with the spinor representations of the group SO (1,3) for a system of n spinors. Representations are expressed in terms of products of projectors and nilpotents; we pay attention also on the nonrelativistic limit. An algorithm for extracting a particular information out of a general superposition of 2 n qubit states is presented. It reproduces for a particular choice of the initial state the Grover’s algorithm (Proc. 28th Annual ACM Symp. Theory Comput. 212, 1996 ).