Continuous-Time Quantum Search on Balanced Trees

TL;DR

This paper investigates how the position of a target node in a balanced tree affects the efficiency of continuous-time quantum search algorithms, revealing a transition in complexity from square-root to linear time.

Contribution

It provides analytical and numerical evidence that the quantum search complexity on balanced trees varies with node position, linking it to path-based centrality measures.

Findings

Quantum search time scales from N^{0.5} to N as target moves from root to leaves.

Search complexity correlates with path-based centrality of the target node.

Results highlight the impact of network heterogeneity on quantum search performance.

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 $\sim N^{\beta}$ changes uniformly from $\beta=0.5$ to $\beta=1$ 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.