Optimal Quantum Spatial Search with One-Dimensional Long-Range Interactions

Continuous-time quantum walks can be used to solve the spatial search problem, which is an essential component for many algorithms that run quadratically faster than their classical counterpart, in $\mathcal{O}(\sqrt{n})$ time $n$ entries. However, capability of models found nature largely unexplored---e.g., one dimension only nearest-neighbor Hamiltonians have been considered so far, quadratic speedup does not exist. Here, we prove optimal search, namely with and high fidelity, possible one-dimensional spin chains long-range interactions decay as $1/{r}^{\ensuremath{\alpha}}$ distance $r$. In particular, near unit fidelity achieved $\ensuremath{\alpha}\ensuremath{\approx}1$ and, limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, find a continuous transition from region where exist ($\ensuremath{\alpha}<1.5$) it ($\ensuremath{\alpha}>1.5$). Numerically, show robust dephasing noise that, reasonable chain lengths, $\ensuremath{\alpha}\ensuremath{\lesssim}1.2$ should sufficient demonstrate experimentally fidelity.