I stumbled upon "the geometry of quantum computation" --- to quote the abstract:
Determining the quantum circuit complexity of a unitary operation is closely related to the problem of finding minimal length paths in a particular curved geometry
Which is extremely interesting, since one can hope that we can develop techniques to find optimal unitaries that represent an operation. Indeed, the abstract continues:
... We develop many analytic solutions to the geodesic equation, and a set of invariants that completely determine the geodesics. We investigate the problem of finding minimal geodesics through a desired unitary, U, and develop a procedure which allows us to deform the (known) geodesics of a simple and well understood metric to the geodesics of the metric of interest in quantum computation
My question is whether there exists a similar geometric structure for lambda calculus? Trivially, I can think of "embedding" classical computation inside a quantum circuit by using the usual Toffoli gate trick of producing the inputs and the output to make it reversible.
However, perhaps the geometry of classical computation is more "rigid" than the quantum world, which would be very interesting. I'd like pointers to theories that try to study lambda calculus using differential geometry, in the spirit of the above paper.
