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-neighbor 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 high fidelity, is possible in one-dimensional spin chains with long-range interactions that decay as $1/r^{\alpha }$ with distance $r$. In particular, near unit fidelity is achieved for $\alpha \approx 1$ and, in the limit $n\to \infty$, we find a continuous transition from a region where optimal spatial search does exist ($\alpha <1.5$) to where it does not ($\alpha >1.5$). Numerically, we show that spatial search is robust to dephasing noise and that, for reasonable chain lengths, $\alpha \lesssim 1.2$ should be sufficient to demonstrate optimal spatial search experimentally with near unit fidelity.

• Received 29 October 2020
• Revised 9 April 2021
• Accepted 7 May 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.240502