Polycontrolled PROPs for Qudit Circuits: A Uniform Complete Equational Theory For Arbitrary Finite Dimension

We present a finite schematic axiomatisation of quantum circuits over d-level systems (qudits), uniform in every finite dimension d >= 2. For each d we define a PROP equipped with a family of control functors, treating control as a primitive categorical constructor. Using a translation between qudit circuits and the LOPP calculus for linear optics based on d-ary Gray codes, we obtain for each d a finite set of local axiom schemata that is sound and complete for unitary d-level circuits: two circuits denote the same unitary if and only if they are inter-derivable using axioms involving at most three wires. The generators are compatible with standard universal qudit gate families, yielding a sound equational basis for circuit rewriting and optimisation-by-rewriting. Conceptually, this extends the qubit circuit completeness results of Clément et al.\ to arbitrary finite dimension, and instantiates the control-as-constructor approach of Delorme and Perdrix in this setting, while keeping the axiom shapes uniform in d.