🤖 AI Summary
This work addresses the problem of syntactic modeling of circuits—represented as string diagrams—in terms of their structural connectivity. We propose a novel circuit grammar framework grounded in order theory: circuits are modeled as partially ordered sets (posets) equipped with designated input and output interfaces, where connectivity is fully encoded by the partial order; circuit rewriting is formalized via structure-preserving morphisms, and a factorization theorem for such morphisms is established. Crucially, we introduce formal concept analysis into circuit theory for the first time: we prove that all circuits sharing the same connectivity correspond bijectively to a unique concept lattice, which precisely characterizes their minimal (canonical) representative. Furthermore, we construct the dual lattice structure. This establishes a rigorous, computable correspondence between circuit connectivity and lattice-theoretic structures, providing a principled, algebraic foundation for modeling quantum causality and enabling causal decomposition of unitary transformations.
📝 Abstract
We take an order-theoretic approach to circuit (string diagram) syntax, treating a circuit as a partial order with additional input-output structure. We define morphisms between circuits and prove a factorisation theorem showing that these can, in the finite case, be regarded as formalising a notion of syntactical circuit rewrites, with quotient maps in particular corresponding to gate composition. We then consider the connectivity of a circuit, expressed as a binary relation between its inputs and outputs, and characterise the concept lattice from formal concept analysis as the unique smallest circuit that admits morphisms from all other circuits with the same connectivity. This has significance for quantum causality, particularly to the study of causal decompositions of unitary transformations. We close by constructing the circuit characterised by the dual statement.