Simpler Presentations for Many Fragments of Quantum Circuits

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. We study such reasoning through finite equational theories, presenting restricted quantum gate fragments as symmetric monoidal categories (PROPs), where wire permutations are treated as structural and separated cleanly from fragment-specific gate axioms. For six widely used near-Clifford fragments: qubit Clifford, real Clifford, Clifford+T (up to two qubits), Clifford+CS (up to three qubits) and CNOT-dihedral, we transfer the completeness results of prior work into our PROP framework. Beyond completeness, we address minimality (axiom independence). Using uniform separating interpretations into simple semantic targets, we prove minimality for several fragments (including all arities for qubit Clifford, real Clifford, and CNOT-dihedral), and bounded minimality for the remaining cases. Overall, our presentations significantly reduce rule counts compared to prior work and provide a reusable categorical framework for constructing complete and often minimal rewrite systems for quantum circuit fragments.