Topological properties of reachable sets and the control of quantum bits

We consider the problem of controlling the state of a two-level quantum system (quantum bit) via an externally applied electro-magnetic field. The describing model is a bilinear right-invariant system whose state varies on the Lie group of 2 x 2 special unitary matrices. We study the topological structure of the reachable sets. If two or more independent controls are used, then every state can be achieved in arbitrary time, However, this is no longer true if only one control is available and, in this case, we give an exact characterization of states reachable in arbitrary time. We prove small time local controllability for any state and the existence of a critical time which is the smallest time after which every transfer of state is possible. We provide upper and lower bounds for such a time. The mathematical development is motivated by the problem of manipulating the state of a quantum bit. Every transfer of state may be interpreted as a quantum logic operation and not every logic operation can be obtained in arbitrary time. The analysis we present provides information about the feasibility of a given operation as well as estimates for the speed of a quantum computer. (C) 2000 Elsevier Science B.V. All rights reserved.