Defining Quantum Control Flow

A remarkable difference between quantum and classical programs is that the control flow of the former can be either classical or quantum. One of the key issues in the theory of quantum programming languages is defining and understanding quantum control flow. A functional language with quantum control flow was defined by Altenkirch and Grattage [\textit{Proc. LICS'05}, pp. 249-258]. This paper extends their work, and we introduce a general quantum control structure by defining three new quantum program constructs, namely quantum guarded command, quantum choice and quantum recursion. We clarify the relation between quantum choices and probabilistic choices. An interesting difference between quantum recursions with classical control flows and with quantum control flows is revealed.