Controllability of Quantum Bits from the von Neumann Architecture to Quantum Computing

In this paper we consider the (state) reachability and controllability problems of special two-level quantum systems, the so-called quantum bits via externally applied electro-magnetic field. The system is described by special a bilinear right-invariant model whose state varies on the Lie group of 2 x 2 special unitary matrices. We show that if two or more independent controls are used, then every state can be achieved in arbitrary time using finite energy. The mathematical construction is motivated by the demand of manipulating (or logically operating on) the state of quantum bits, and the results provide some insight into the feasibility of realizing given operations in quantum computers. The paper has a special actuality: both computer science and quantum mechanics are closely related to the famous Hungarian mathematician John von Neumann, and this year there is the 50 th Anniversary of his early and sorrowful death.