We present methods for implementing arbitrary permutations of qubits under interaction constraints. Our protocols make use of previous methods for rapidly reversing the order of qubits along a path. Given nearest-neighbor interactions on a path of length , we show that there exists a constant such that the quantum routing time is at most , whereas any SWAP-based protocol needs at least time . This represents the first known quantum advantage over swAP-based routing methods and also gives improved quantum routing times for realistic architectures such as grids. Furthermore, we show that our algorithm approaches a quantum routing time of in expectation for uniformly random permutations, whereas SWAP-based protocols require time asymptotically. Additionally, we consider sparse permutations that route qubits and give algorithms with quantum routing time at most on paths and at most on general graphs with radius .