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 $r$.

中文翻译：

---

具有快速反转的量子路由

我们提出了在交互约束下实现量子位任意排列的方法。我们的协议利用以前的方法来快速反转沿路径的量子位的顺序。给定长度为 $n$ 的路径上的最近邻交互，我们证明存在一个常数 $\epsilon \approx 0.034$，使得量子路由时间至多为 $(1-\epsilon)n$，而任何交换基于协议至少需要时间$n-1$。这代表了第一个相对于基于交换的路由方法的已知量子优势，并且还为网格等现实架构提供了改进的量子路由时间。此外，我们表明，我们的算法在均匀随机排列的预期中接近 $2n/3$ 的量子路由时间，而基于交换的协议渐近地需要 $n$ 时间。此外，我们考虑路由 $k \le n$ 量子位的稀疏排列，并给出在路径上量子路由时间最多 $n/3 + O(k^2)$ 和最多 $2r/3 + O(k^ 2)$ 在半径为 $r$ 的一般图上。