  This dissertation has two main parts. The first part deals with questions
relating to Haghverdi and Scott's notion of partially traced categories. The
main result is a representation theorem for such categories: we prove that
every partially traced category can be faithfully embedded in a totally traced
category. Also conversely, every monoidal subcategory of a totally traced
category is partially traced, so this characterizes the partially traced
categories completely. The main technique we use is based on Freyd's
paracategories, along with a partial version of Joyal, Street, and Verity's Int
construction. Along the way, we discuss some new examples of partially traced
categories, mostly arising in the context of quantum computation.
  The second part deals with the construction of categorical models of
higher-order quantum computation. We construct a concrete semantic model of
Selinger and Valiron's quantum lambda calculus, which has been an open problem
until now. We do this by considering presheaf categories over appropriate base
categories arising from first-order quantum computation. The main technical
ingredients are Day's convolution theory and Kelly and Freyd's notion of
continuity of functors. We first give an abstract description of the properties
required of the base categories for the model construction to work; then
exhibit a specific example of base categories satisfying these properties.
