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 n, we show that there exists a constant epsilon approximate to 0.034 such that the quantum routing time is at most (1 - epsilon)n, whereas any SWAP-based protocol needs at least time n - 1. 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 2n/3 in expectation for uniformly random permutations, whereas SWAP-based protocols require time n asymptotically. Additionally, we consider sparse permutations that route k <= n qubits and give algorithms with quantum routing time at most n/3 + O(k(2)) on paths and at most 2r/3 + O(k(2)) on general graphs with radius r.