Hybrid simplification rules for boundaries of quantum circuits

We describe rules to simplify quantum circuits at their boundaries, i.e. at state preparation and measurement. There, any strictly incoherent operation may be pushed into a pre- or post-processing of classical data. The rules can greatly simplify the implementation of quantum circuits and are particularly useful for hybrid algorithms on noisy intermediate-scale quantum hardware, e.g. in the context of quantum simulation. Finally we illustrate how circuit cutting can enable the rules to be applied to more locations.