Controlled quantum search on structured databases

TL;DR

This paper introduces quantum algorithms for searching structured databases like Cayley trees, achieving near-perfect success rates and optimal runtimes by controlling the number of stages and edge weights, with robustness to perturbations.

Contribution

It develops a multi-stage quantum search method for structured graphs, optimizing runtime and success probability, and demonstrates how to merge stages for a faster, single-stage search.

Findings

Success probability close to 100% on Cayley trees with large branching factors.

Runtime proportional to N^{(2r-1)/2r} for Cayley trees of height r.

Merging multi-stage search into a single stage achieves a runtime proportional to √N.

Abstract

We present quantum algorithms to search for marked vertices in structured databases with low connectivity. Adopting a multi-stage search process, we achieve a success probability close to $100\%$ on Cayley trees with large branching factors. We find that the number of stages required is given by the height of the Cayley tree. At each stage, the jumping rate should be chosen as different values. The dominant term of the runtime in the search process is proportional to $N^{(2r-1)/2r}$ for the Cayley tree of height $r$ with $N$ vertices. We further find that one can control the number of stages by adjusting the weight of the edges in the graphs. The multi-stage search process can be merged into a single stage, and then an optimal runtime proportional to $\sqrt{N}$ is achieved, yielding a substantial speedup. The search process is quite robust under various kinds of small perturbations.