Comment to Spatial Search by Quantum Walk is Optimal for Almost all Graphs

Abstract

This comment is to correct the proof of optimality of quantum spatial search for Erd\H{o}s-R\'enyi graphs presented in `Spatial Search by Quantum Walk is Optimal for Almost all Graphs' (https://doi.org/10.1103/PhysRevLett.116.100501). The authors claim that if $p\geq \frac{\log^{3/2}(n)}{n}$, then the CTQW-based search is optimal for almost all graphs. Below we point the issues found in the main paper, and propose corrections, which in fact improve the result to $p=\omega(\log(n)/n)$ in case of transition rate $\gamma = 1/\lambda_1$. In the case of the proof for simplified transition rate $1/(np)$ we pointed a possible issue with applying perturbation theory.