ar X iv : q ua nt - p h / 06 05 21 8 v 1 2 5 M ay 2 00 6 Quantum Loop Programs ∗

Loop is a powerful program construct in classical computation, but its power is still not exploited fully in quantum computation. The exploitation of such power definitely requires a deep understanding of the mechanism of quantum loop programs. In this paper, we introduce a general scheme of quantum loops. The computational process of a quantum loop is then described. Moreover, the notions of termination and almost termination are proposed for quantum loops. The function computed by a quantum loop is also defined. To illustrate these notions, we carefully examine two simplest classes of quantum loop programs: one qubit quantum loops, and two qubit quantum loops defined by controlled gates. In particular, we find a necessary and sufficient condition for termination of a general quantum loop on any mixed input state. A necessary and sufficient condition for almost termination on a pure input state is given too.