论文标题
具有一维远程相互作用的最佳量子空间搜索
Optimal quantum spatial search with one-dimensional long-range interactions
论文作者
论文摘要
连续时间量子步行可用于解决空间搜索问题，这是许多量子算法的重要组成部分，其在$ \ mathcal o（\ sqrt n）$ $ n $条目中以$ \ mathcal o（\ sqrt n）$ time运行的速度比其经典对应物更快。然而，自然界中发现的模型的能力在很大程度上没有探索 - 例如，在一个维度上，仅考虑到迄今为止最近的邻居汉密尔顿人，尚不存在二次加速。在这里，我们证明，最佳的空间搜索，即使用$ \ Mathcal O（\ sqrt N）$运行时间和较大的保真度，在具有远距离交互的一维旋转链中可以使用，其远距离交互作用为$ 1/r^α$，带有距离$ r $。特别是，以$α\约1 $的价格实现了近乎单位的保真度，并且在限制$ n \ to \ infty $的情况下，我们发现了从一个最佳空间搜索确实存在（$α<1.5 $）的地区连续过渡到它不存在的区域（$ $α> 1.5 $）。从数值上讲，我们表明空间搜索对于降低噪声是可靠的，并且对于现实的条件，$α\ Lessim 1.2 $应该足以在实验上以接近单位的保真度在实验上展示最佳的空间搜索。
Continuous-time quantum walks can be used to solve the spatial search problem, which is an essential component for many quantum algorithms that run quadratically faster than their classical counterpart, in $\mathcal O(\sqrt n)$ time for $n$ entries. However the capability of models found in nature is largely unexplored - e.g., in one dimension only nearest-neighbour Hamiltonians have been considered so far, for which the quadratic speedup does not exist. Here, we prove that optimal spatial search, namely with $\mathcal O(\sqrt n)$ run time and large fidelity, is possible in one-dimensional spin chains with long-range interactions that decay as $1/r^α$ with distance $r$. In particular, near unit fidelity is achieved for $α\approx 1$ and, in the limit $n\to\infty$, we find a continuous transition from a region where optimal spatial search does exist ($α<1.5$) to where it does not ($α>1.5$). Numerically, we show that spatial search is robust to dephasing noise and that, for realistic conditions, $α\lesssim 1.2$ should be sufficient to demonstrate optimal spatial search experimentally with near unit fidelity.
