Quantum differentials on boolean algebras and other applications

Quantum or noncommutative differential geometry should have a significant role in quantum computing, e.g. in the transfer of ideas from quantum gravity to quantum computing. In this talk I will give a short introduction to quantum geometry on unital algebras using string diagrams, then focus on two examples. Firstly, a noncommutative differential structure on a set is the same thing as a directed graph on the set as vertices. As an application, we  extend the boolean algebra of subsets of a set to include subsets of arrows, with $\cup$ and $\cap$ between vertices and arrows necessarily noncommutative. We show that de Morgan duality still holds, as a diffeomorphism. As another application we naturally ``quantise" the 2-vertex graph to a ``discrete Schroedinger process" on $\psi\in \Bbb C^2$, constructed quantum geometrically, and find that $|\psi|^2$ undergoes a generalised Markov process. If time, we will also see that the algebra of 2 x 2 matrices itself has a natural curved quantum Riemannian geometry with quantum geodesic flows on it.