Minimizing the number of CNOT gates required to synthesize a Clifford operator is a central problem in quantum circuit optimization. We extend the link–middle–cut (LMC) framework for linear reversible circuits to Clifford operators represented by bi
Minimizing the number of CNOT gates required to synthesize a Clifford operator is a central problem in quantum circuit optimization. We extend the link–middle–cut (LMC) framework for linear reversible circuits to Clifford operators represented by binary symplectic tableaux. By introducing block-support connectivity graphs and determinantal invariants, we obtain efficiently computable lower bounds on ancilla-free CNOT complexity. We prove that these bounds are tight for qubit permutations: a permutation of n qubits with k cycles requires exactly 3(n-k) CNOT gates, even when arbitrary one-qubit Clifford gates are available. We also derive efficiently computable bounds for synthesis up to a permutation of the output qubits, corresponding to free qubit relabeling. Finally, we further demonstrate the utility of the bound by using it as an admissible heuristic for A^*-based Clifford synthesis and show that, on benchmark instances, the resulting search can prove optimality beyond what was possible with earlier SAT-based methods, and that it may be used in combination with earlier A^* heuristics to improve the CNOT counts obtained through heuristic search.