Optimizing quantum circuit synthesis for permutations using recursion

We describe a family of recursive methods for the synthesis of qubit permutations on quantum computers with limited qubit connectivity. Two objectives are of importance: circuit size and depth. In each case we combine a scalable heuristic with a nonscalable, yet exact, synthesis. Our algorithms are applicable to generic connectivity constraints, scale favorably, and achieve closeto-optimal performance in many cases. We demonstrate the utility of these algorithms by optimizing the compilation of Quantum Volume circuits, and to disprove an old conjecture on reversals being the hardest permutation on a path.