Categorical Models of Quantum Circuits

Abstract

It is shown that equations that hold in appropriate monoidal categories have an explicit representation in terms of quantum circuits. Hence, we adjust and map the graphical calculus of Abramsky and Coecke's categorical axiomatization of quantum theory onto quantum circuits, making a suitable extension applicable to problems stated in the language of quantum information science. Viewed in this new way, circuit diagrams themselves now become arrows in a Category, making quantum circuits a special case of a much more general mathematical framework. By building a precise connection between the quantum circuit language and the categorical model, we were able to use this new framework to produce results new to both areas. This should lead to more cross communication between the field of Categorical Quantum Theory, and Quantum Information Science.