Quantum lambda calculus

Question

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?

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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.

Answer 1

here is a half-baked answer: I know that Ugo Dal Lago at University of Bologna has been studying quantum lambda calculus. You may want to check his publications and perhaps this one in particular:

Quantum implicit computational complexity by U. Dal Lago, A. Masini, M. Zorzi.

I am saying it's a half-baked answer, because I haven't had chance to read any of his works.

Answer 2

Apologies in advance for the shameless plug, but there is a paper of mine on a quantum lambda calculus that you may find interesting. It is called The Dagger Lambda Calculus and provides a higher-order representation for the diagrammatic circuits that the categorical school of quantum computation have introduced:

http://arxiv.org/abs/1406.1633

You can also check my talk on YouTube for more info:

https://www.youtube.com/watch?v=2pDPVd1BukI

Other works in the area include the Selinger-Valiron quantum lambda calculi, and the lambda calculus by Andre van Tonder: [Sel04a], [Sel04b], [vTD03], [vT04], [SV04], [SV08], [SV10].