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Journal of Functional Analysis

On the Structure of Classical and Quantum Flows

Luigi Accardi, Anilesh Mohari, Centro V. Volterra

Published Feb 1, 1996• 32 citations
Publisher
DOI
10.1006/jfan.1996.0015

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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.
We have previously shown that the flow in open quantum dots may be described in terms of meandering open orbits which separate regions of trapped flows around vortices. Furthermore the topology of the flow follows from a simple geometrical construction in which the classical ballistic trajectories are first computed and then all crossings are replaced by hyperbolic flow separation and for which all trapped flows form quantised closed orbits. In the present paper we extend this picture to a computational methodology for developing quantum hydrodynamic descriptions of flows in the presence of quantised vortices by extension from classical hydrodynamics. For coherent transport we again encounter the Bohm quantum potential but the singularity structure is predetermined by applying a fictitious repulsion which prevents trajectories and hence streamlines crossing. This leads to vortex producing terms in the hydrodynamic equations. The discussion is completed by including the effects of non-hermitian potentials into the Hamiltonian in order to simulate dissipation and phase breaking flows.