Higher-order circuits
Matt Wilson
TL;DR
The paper develops an axiomatic framework for higher-order circuits by treating theories as $\mathcal{P}$-enriched strict monoidal categories over a symmetric polycategory $\mathcal{P}$, augmented with cotensors to model gaps and higher-order wiring. It carefully defines polycategorical composition of holes, enrichment laws, and coherence conditions (Frobenius, copy, and braid), arguing that this setup naturally captures salient aspects of higher-order quantum processes without resorting to closed monoidal structures. A key contribution is showing that any such higher-order circuit theory embeds into the theory of strong profunctors $\mathbf{StrProf}[\mathcal{C}]$, via a faithful multifunctor from the operational closure $\mathcal{P}^{\#}$, thereby establishing a principled upper bound on what higher-order circuits can express. The work also provides a convenient cotensor notation, a diagrammatic function-box calculus, and several illustrative examples (including compact closed, precausal, and locally-applicable transformations), highlighting potential applications to resource theories and future generalisations beyond strictness. Overall, the framework unifies disparate approaches to higher-order processes under enrichment in symmetric polycategories and clarifies the algebraic structure that governs their composition and interaction.
Abstract
We write down a series of basic laws for (strict) higher-order circuit diagrams. More precisely, we define higher-order circuit theories in terms of: (a) nesting, (b) temporal and spatial composition, and (c) equivalence between lower-order bipartite processes and higher-order bipartite states. In category-theoretic terms, these laws are expressed using enrichment and cotensors in symmetric polycategories, along with a frobenius-like coherence between them. We describe how these laws capture the salient features of higher-order quantum theory, and discover an upper bound for higher-order circuits: any higher-order circuit theory embeds into the theory of strong profunctors.