The Geometry of Parallelism. Classical, Probabilistic, and Quantum Effects

We introduce a Geometry of Interaction model for higher-order quantum computation, and prove its adequacy for a full quantum programming language in which entanglement, duplication, and recursion are all available. Our model comes with a multi-token machine, a proof net system, and a PCF-style language. The approach we develop is not specific to quantum computation, and our model is an instance of a new framework whose main feature is the ability to model commutative effects in a parallel setting. Being based on a multi-token machine equipped with a memory, it has a concrete nature which makes it well suited for building low-level operational descriptions of higher-order languages.