Systematic Optimization of Quantum Circuits via Toffoli Permutation and Local Search

The synthesis of efficient reversible logic circuits is critical for fault-tolerant quantum computing. While standard libraries often treat the Toffoli gate as an atomic unit, its realization in the NCV library requires decomposition into 5 elementary gates. In this work, we propose a novel, systematic four-phase optimization methodology. First, we generate an optimal circuit in the strict NCT library (NOT, CNOT, Toffoli) using automated synthesis. Second, we replace each Toffoli gate with a specific NCV decomposition, permuting the control lines such that the final internal CNOT gate matches the subsequent linear gate in the circuit, allowing us to apply a cancellation rule (X ·X = I). Finally, post-synthesis local searches are applied using Mixed- Integer Non-Linear Programming (MINLP) techniques to further reduce the total gate count. Applying this method to the MIG, TSG, and MKG gates, we achieve Quantum Costs of 7, 14, and 12 respectively, matching or beating the best-known literature benchmarks.