We introduce a Geometry of Interaction model for higher-order quantum computation, and prove 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 model is obtained as an instance of a new framework:

• we equip Geometry of Interaction with the ability to capture \emph{choice effects} in a parametric flexible way, via a memory structure;
• we develop a general notion of parallel abstract rewrite system, which allows us to deal with the combination of parallelism and probabilistic choice in an infinitary setting;
• our construction produces a multi-token machine, a proof-net system and a PCF-style language, where the multi-token machine is an adequate model of nets reduction, and the nets an adequate model of PCF-style terms rewriting.

Being based on a multi-token machine associated to a memory, our model has a concrete nature which makes it well suited to build low-level operational descriptions of higher-order languages.