Quantum Circuits: From a Network to a One-Way Model

We present a translation from the standard network model of quantum computation to the one-way model of quantum computation. The translation is compositional, i.e., it preserves the structure of computations, which allows us to abstract the concrete realizations in a monadic abstraction layer.We briefly review the By-Product and Measurement Calculus approaches to combining circuits in the one-way model, and show how the procedures can be expressed in the exact same notation, bearing a direct relationship with the representation of circuits in the standard network model, and using monads. Discussions on improving the abstraction yield us to introducing an alternative approach to combining circuits in the one-way model: a graphical one.