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 $epsilon approx 0.034$ 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 $k le n$ 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 $r$.
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