Spatial search on a two-dimensional lattice with long-range interactions

Quantum-walk-based algorithms that search a marked location among $N$ locations on a $d$-dimensional lattice succeeds in time $O(\sqrt{N})$ for $dg2$, while this is not found to be possible when $d=2$. In this paper, we consider a spatial search algorithm using continuous-time quantum walk on a two-dimensional square lattice with the existence of additional long-range edges. We examined such a search on a probabilistic graph model where an edge connecting non-nearest-neighbor lattice points $i$ and $j$ apart by a distance $|i\ensuremath{-}j|$ is added by probability ${p}_{ij}={|i\ensuremath{-}j|}^{\ensuremath{-}\ensuremath{\alpha}}\phantom{\rule{1em}{0ex}}(\ensuremath{\alpha}\ensuremath{\ge}0)$. Through numerical analysis, we found that the search succeeds in time $O(\sqrt{N})$ when $\ensuremath{\alpha}\ensuremath{\le}{\ensuremath{\alpha}}_{c}=2.4\ifmmode\pm\else\textpm\fi{}0.1$. For $\ensuremath{\alpha}g2$, the expectation value of the additional long-range edges on each node scales as a constant when $N\ensuremath{\rightarrow}\ensuremath{\infty}$, which means that search time of $O(\sqrt{N})$ is achieved on a graph with average degree scaling as a constant.