Abstract
Based on the connection between the categorical derivation of classical programs from specifications and a category-theoretic approach to quantum information, this paper contributes to extending the laws of classical program algebra to quantum programming. This aims at building correct-by-construction quantum circuits to be deployed on quantum devices such as those available through the IBM Q Experience. Reversibility is ensured by minimal complements. Such complementation is extended inductively to encompass catamorphisms on lists (vulgo folds), giving rise to the corresponding recursion scheme in reversible computation. The same idea is then applied to the setting of quantum programming, where computation is expressed by unitary transformations. This yields the notion of 'quantamorphism', a structural form of quantum recursion implementing cycles and folds on lists with quantum control flow. By Kleisli correspondence, quantamorphisms can be written as monadic functional programs with quantum parameters. This enables the use of Haskell, a monadic functional programming language, to perform the experimental work. Such calculated quantum programs prepared in Haskell are pushed through Quipper and the Qiskit interface to IBM Q quantum devices. The generated quantum circuits - often quite large - exhibit the predicted behaviour. However, running them on real quantum devices naturally incurs a significant amount of errors. As quantum technology is rapidly evolving, an increase in reliability is likely in the future, allowing for our programs to run more accurately.