Spatial Search by Quantum Walk is Optimal for Almost all Graphs.

The problem of finding a marked node in a graph can be solved by the spatial search algorithm based on continuous-time quantum walks (CTQW). However, this algorithm is known to run in optimal time only for a handful of graphs. In this work, we prove that for Erd\"os-Renyi random graphs, i.e.\ graphs of $n$ vertices where each edge exists with probability $p$, search by CTQW is \textit{almost surely} optimal as long as $p\geq \log^{3/2}(n)/n$. Consequently, we show that quantum spatial search is in fact optimal for \emph{almost all} graphs, meaning that the fraction of graphs of $n$ vertices for which this optimality holds tends to one in the asymptotic limit. We obtain this result by provin…