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Control flow of quantum programs is often divided into two different classes:classical and quantum. Quantum programs with classical control flow have theirconditional branching determined by the classical outcome of measurements, andthese collapse quantum data. Conversely, quantum control flow is coherent, i.e.it does not perturb quantum data; quantum walk-based algorithms are practicalexamples where coherent quantum feedback plays a major role. This dissertationhas two main contributions: (i) a categorical study of coherent quantumiteration and (ii) the introduction of weak while loops.  (i) The objective is to endow categories of quantum processes with a tracedmonoidal structure capable of modelling iterative quantum loops. To this end,the trace of a morphism is calculated via the execution formula, which adds upthe contribution of all possible paths of the control flow. Haghverdi's uniquedecomposition categories are generalised to admit additive inverses andequipped with convergence criteria using basic topology. In this setting, it ispossible to prove the validity of the execution formula as a categorical traceon certain categories of quantum processes.  (ii) A weak while loop is a classical control flow primitive that offers atrade-off between the collapse caused on each iteration and the amount ofinformation gained. The trade-off may be adjusted by tuning a parameter and, incertain situations, it is possible to set its value so that we may control thealgorithm without sacrificing its quantum speed-up. As an example, it is shownthat Grover's search problem can be implemented using a weak while loop,maintaining the same time complexity as the standard Grover's algorithm (aspreviously shown by Mizel).

Unbounded loops in quantum programs: categories and weak while loops