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

TL;DR

This paper critiques and corrects the proof of quantum spatial search optimality on Erdős-Rényi graphs, refining the conditions under which the search is proven to be optimal.

Contribution

It identifies issues in previous proofs and provides corrected, improved conditions for the optimality of quantum spatial search on Erdős-Rényi graphs.

Findings

Corrects the proof of optimality conditions for quantum search

Improves the threshold to p=ω(log(n)/n) for transition rate γ=1/λ₁

Highlights potential issues with perturbation theory application

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.