Vertices cannot be hidden from quantum spatial search for almost all random graphs

TL;DR

This paper demonstrates that quantum spatial search can locate all nodes in almost all Erdős-Rényi random graphs under certain edge probability conditions, highlighting the method's effectiveness and limitations.

Contribution

It establishes conditions under which quantum spatial search can find all nodes in Erdős-Rényi graphs, providing tight bounds for adjacency and Laplacian matrices.

Findings

Quantum search finds all nodes for p above thresholds

Property fails for p below thresholds due to connectivity issues

Results are tight for Laplacian matrix conditions

Abstract

In this paper we show that all nodes can be found optimally for almost all random Erd\H{o}s-R\'enyi ${\mathcal G}(n,p)$ graphs using continuous-time quantum spatial search procedure. This works for both adjacency and Laplacian matrices, though under different conditions. The first one requires $p=\omega(\log^8(n)/n)$, while the seconds requires $p\geq(1+\varepsilon)\log (n)/n$, where $\varepsilon>0$. The proof was made by analyzing the convergence of eigenvectors corresponding to outlying eigenvalues in the $\|\cdot\|_\infty $ norm. At the same time for $p<(1-\varepsilon)\log(n)/n$, the property does not hold for any matrix, due to the connectivity issues. Hence, our derivation concerning Laplacian matrix is tight.