Deterministic Search on Complete Bipartite Graphs by Continuous Time Quantum Walk

TL;DR

This paper introduces a deterministic quantum search algorithm on complete bipartite graphs using continuous-time quantum walks, including a quantum counting method, with constant-time Hamiltonian simulation.

Contribution

It generalizes Grover's algorithm to bipartite graphs with multiple marked states and achieves constant-time Hamiltonian simulation.

Findings

Deterministic search algorithm for bipartite graphs

Quantum counting method for multiple marked states

Constant-time Hamiltonian simulation

Abstract

This paper presents a deterministic search algorithm on complete bipartite graphs. Our algorithm adopts the simple form of alternating iterations of an oracle and a continuous-time quantum walk operator, which is a generalization of Grover's search algorithm. We address the most general case of multiple marked states, so there is a problem of estimating the number of marked states. To this end, we construct a quantum counting algorithm based on the spectrum structure of the search operator. To implement the continuous-time quantum walk operator, we perform Hamiltonian simulation in the quantum circuit model. We achieve simulation in constant time, that is, the complexity of the quantum circuit does not scale with the evolution time.