Equational reasoning is central to quantum circuit optimisation and verification: one replaces subcircuits by provably equivalent ones using a fixed set of rewrite rules viewed as equations. A finite rule set is most informative when it separates the genuine algebra of a circuit fragment from the structural treatment of wires. This paper gives six near-Clifford fragments a common PROP treatment, where wire permutations are structural: qubit Clifford, real Clifford, Clifford+T (up to two qubits), Clifford+CS (up to three qubits), CNOT-dihedral, and qutrit Clifford. Starting from prior completeness theorems, we transfer completeness into this setting and remove redundant non-structural rules, then check minimality by separating interpretations tailored to individual axioms; the resulting presentations are minimal in all arities for qubit Clifford, real Clifford, and CNOT-dihedral, minimal in bounded ranges for the remaining fragments, and comparable by one transfer-and-separation pattern.

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.