Classically, there are 3 popular ways to think about computation: Turing machine, circuits, and lambda-calculus (I use this as a catch all for most functional views). All 3 have been fruitful ways to think about different types of problems, and different fields use different formulation for this reason.

When I work with quantum computing, however, I only ever think about the circuit model. Originally, QC was defined in terms of quantum Turing machines but as far as I understand, this definition (although equivalent to quantum circuits if both are formulated carefully) has not been nearly as fruitful. The 3rd formulation (in terms of lambda-calculus or similar functional settings) I am completely unfamiliar with. Hence my questions:

• What are useful definitions of quantum lambda-calculus (or other functional paradigms)?

• What subfields of QIP gain deeper insight from using this formulation instead of the circuit model?

###Notes

I am aware that I am ignoring many other popular formalisms like cellular automata, RAM-models, etc. I exclude these mostly because I don't have experience with thinking in terms of these models classically, let alone quantumly.

I am also aware that there are popular alternatives in the quantum setting, such as measurement-based, topological, and adiabatic. I do not discuss them because I am not familiar with the classical counterparts.