Multimarked Spatial Search by Continuous-Time Quantum Walk
Pedro H. G. Lugão, Renato Portugal, Mohamed Sabri, Hajime Tanaka
TL;DR
This work advances multimarked spatial search by continuous-time quantum walk on graphs, introducing a practical framework that reduces the spectral problem to a real symmetric matrix $M^{\lambda}$ whose determinant encodes the eigenvalues of the search Hamiltonian $H=-\gamma A-\sum_{w\in W}|w\rangle\langle w|$. By analyzing the asymptotics of the dominant eigenvalues $\lambda^{\pm}$, the authors derive how to choose the coupling $\gamma$ and extract the optimal running time and success probability, even with multiple marked vertices. Applying the framework to Johnson graphs $J(n,k)$ with two marked vertices, they show that, for fixed $k$ and known separation $\delta$, the search runs in time $t_{\mathrm{run}}=\tfrac{\pi\sqrt{N}}{2\sqrt{2}}$ with asymptotic success probability $p_{\mathrm{succ}}=1+o(1)$ as $n\to\infty$, and discuss how to handle unknown $\delta$ by repeating the procedure. The results provide a concrete method to assess and implement quantum-walk-based spatial searches on arbitrary graphs and illuminate how marked-vertex placement influences quantum speedups.
Abstract
The quantum-walk-based spatial search problem aims to find a marked vertex using a quantum walk on a graph with marked vertices. We describe a framework for determining the computational complexity of spatial search by continuous-time quantum walk on arbitrary graphs by providing a recipe for finding the optimal running time and the success probability of the algorithm. The quantum walk is driven by a Hamiltonian derived from the adjacency matrix of the graph modified by the presence of the marked vertices. The success of our framework depends on the knowledge of the eigenvalues and eigenvectors of the adjacency matrix. The spectrum of the Hamiltonian is subsequently obtained from the roots of the determinant of a real symmetric matrix $M$, the dimensions of which depend on the number of marked vertices. The eigenvectors are determined from a basis of the kernel of $M$. We show each step of the framework by solving the spatial searching problem on the Johnson graphs with a fixed diameter and with two marked vertices. Our calculations show that the optimal running time is $O(\sqrt{N})$ with an asymptotic probability of $1+o(1)$, where $N$ is the number of vertices.