AI-Assisted Discovery of CNOT Decompositions for Quantum Permutations on Constrained Topologies

Abstract

Efficient quantum circuit compilation is challenging for quantum computers, where limited qubit connectivity can introduce substantial overhead in the implementation of logical operations. Recent AI-assisted synthesis methods have shown considerable potential for reducing circuit complexity, but their role in uncovering principles that can guide the design of new deterministic compilation algorithms remains largely unexplored. In this work, we investigate this perspective for the synthesis of quantum permutations into CNOT circuits under constrained qubit connectivity. Using AlphaCNOT as an exploratory tool, we identify a recurring decomposition for remote transpositions that improves on the construction based on adjacent SWAP routing. For two qubits at graph distance $d>1$, the discovered pattern yields a topology-legal implementation using at most $6d-4$ CNOT gates, compared to the $6d-3$ gates required by the corresponding SWAP-based construction. We formalize this observation through a constructive proof to define Global Transpositions, a deterministic synthesis method that searches for a minimum-cost decomposition of an entire permutation into topology-aware logical transpositions. The resulting method subsumes optimal SWAP routing within its synthesis model and provides certified upper bounds on the topology-constrained CNOT cost. Our experiments also reveal circuits that cannot be expressed as concatenations of independently synthesized transpositions. Instead, these circuits exploit intermediate linear transformations that temporarily leave the permutation subgroup and combine partial effects of multiple logical operations, a mechanism we call intermediate CNOT fusion. Our results show that AI-assisted synthesis can support not only circuit optimization, but also the discovery of reusable identities, provable constructions, and hidden mechanisms in quantum circuit compilation.