A Complete Equational Presentation of Qudit Circuits via Polycontrolled PROPs

Abstract

High-dimensional quantum computation needs a native circuit-level equational theory for qudits. We give the first finite schematic equational theory that is sound and complete for exact unitary qudit circuits in every finite dimension at least two. Circuits are built from local gates, sequential and parallel composition, and value-controls; equality is derivable exactly when the standard unitary denotations agree. For each dimension, a finite list of local bounded-arity axiom schemata presents the theory, and the diagrammatic shapes do not depend on d. Primitive value-control makes control on a chosen basis value part of the language, so local rules generate the internal algebra of controlled operations within the circuit PROP. This gives a finite, dimension-uniform basis for exact equational reasoning about qudit circuits.