Operadic foundations for higher-order (supermap) composition

Construct an operad whose objects are boxes-with-holes representing higher-order quantum processes (supermaps), define operations that encode all reasonable ways these boxes-with-holes can be composed (including parallel and sequential composition while avoiding causal loops), and demonstrate that algebras of this operad yield a satisfactory and faithful theory of supermaps beyond mild adaptations of existing categorical structures such as polycategories.

Background

The paper discusses higher-order processes (supermaps), which are informally depicted as boxes with holes into which other processes can be plugged. Existing approaches often model these using polycategories or related categorical structures with restricted composition to avoid causal loops.

The authors question whether these adaptations fully capture the intended compositional behavior of supermaps and suggest the possibility of constructing an operad tailored to these objects and their compositions, with algebras providing a more natural and comprehensive framework.