Higher-order circuits

TL;DR

This paper develops a formal framework for higher-order circuit diagrams using category theory, capturing key features of higher-order quantum theory and establishing an upper bound for such circuit theories.

Contribution

It introduces a set of laws for higher-order circuits based on category theory, unifying nesting, composition, and equivalence in a rigorous mathematical structure.

Findings

Higher-order circuit theories embed into strong profunctors.

The laws capture essential features of higher-order quantum theory.

A formal categorical framework for higher-order circuits is established.

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.