Continuous-time quantum search on balanced trees

Abstract

We examine the effect of network heterogeneity on the performance of quantum search algorithms. To this end, we study quantum search on a tree for the oracle Hamiltonian formulation employed by continuous-time quantum walks. We use analytical and numerical arguments to show that the exponent of the asymptotic running time $\ensuremath{\sim}{N}^{\ensuremath{\beta}}$ changes uniformly from $\ensuremath{\beta}=0.5$ to $\ensuremath{\beta}=1$ as the searched-for site is moved from the root of the tree towards the leaves. These results imply that the time complexity of the quantum search algorithm on a balanced tree is closely correlated with certain path-based centrality measures of the searched-for site.