Quantum gates and quantum algorithms with Clifford algebra technique

Abstract

We use our Clifford algebra technique, that is nilpotents and projectors which are binomials of the Clifford algebra objects ${\gamma }^a$ with the property $\{{\gamma }^a,{\gamma }^b{\}}_{+}=2{\eta }^{ab}$, for representing quantum gates and quantum algorithms needed in quantum computers in an elegant way. We identify $n$-qubits with 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. 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.