Quantum routing with fast reversals

We present methods for implementing arbitrary permutations of qubits under interaction constraints. Our protocols make use previous rapidly reversing the order along a path. Given nearest-neighbor interactions on path length $n$, we show that there exists constant $\epsilon \approx 0.034$ such quantum routing time is at most $(1-\epsilon)n$, whereas any swap-based protocol needs least $n-1$. This represents first known advantage over and also gives improved times realistic architectures as grids. Furthermore, our algorithm approaches $2n/3$ in expectation uniformly random permutations, require $n$ asymptotically. Additionally, consider sparse route $k \le n$ give algorithms with $n/3 + O(k^2)$ paths $2r/3 general graphs radius $r$.