Abstract.One perspective on quantum algorithms is that they are classical algorithms having access to a special kind of memory with exotic properties. This perspective suggests that, even in the case of quantum algorithms, the control flow notions of sequencing, conditionals, loops, and recursion are entirely classical. There is however, another notion of control flow, that is itself quantum. The notion of quantum conditional expression is reasonably well-understood: the execution of the two expressions becomes itself a superposition of executions. The quantum counterpart of loops and recursion is however not believed to be meaningful in its most general form.In this paper, we argue that, under the right circumstances, a reasonable notion of quantum loops and recursion is possible. To this aim, we first propose a classical, typed, reversible language with lists and fixpoints. We then extend this language to the closed quantum domain (without measurements) by allowing linear combinations of terms and restricting fixpoints to structurally recursive fixpoints whose termination proofs match the proofs of convergence of sequences in infinitedimensional Hilbert spaces. We additionally give an operational semantics for the quantum language in the spirit of algebraic lambda-calculi and illustrate its expressiveness by modeling several common unitary operations.
We develop a sound and complete equational theory for the functional quantum
programming language QML. The soundness and completeness of the theory are with
respect to the previously-developed denotational semantics of QML. The
completeness proof also gives rise to a normalisation algorithm following the
normalisation by evaluation approach. The current work focuses on the pure
fragment of QML omitting measurements.Comment: To appear in ENTCS, 3rd International Workshop on Quantum Programming
Languages, 2005. 21 Page
We show that the model of quantum computation based on density matrices and superoperators can be decomposed into a pure classical (functional) part and an effectful part modelling probabilities and measurement. The effectful part can be modelled using a generalisation of monads called arrows. We express the resulting executable model of quantum computing in the Haskell programming language using its special syntax for arrow computations. However, the embedding in Haskell is not perfect: a faithful model of quantum computing requires type capabilities that are not directly expressible in Haskell.
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