Quantum search on the two-dimensional lattice using the staggered model with Hamiltonians

TL;DR

This paper proves that the staggered quantum walk model with Hamiltonians enables efficient quantum search on a 2D lattice, outperforming non-Hamiltonian versions and classical algorithms, with implications for quantum algorithm design and implementation.

Contribution

It provides a rigorous proof of the efficiency of the staggered quantum walk model with Hamiltonians for 2D lattice search, validating its theoretical potential.

Findings

Hamiltonian-based staggered models are efficient for 2D quantum search

Non-Hamiltonian staggered models are as slow as classical algorithms

Numerical results support the theoretical efficiency of the Hamiltonian approach

Abstract

Quantum search on the two-dimensional lattice with one marked vertex and cyclic boundary conditions is an important problem in the context of quantum algorithms with an interesting unfolding. It avails to test the ability of quantum walk models to provide efficient algorithms from the theoretical side and means to implement quantum walks in laboratories from the practical side. In this paper, we rigorously prove that the recent-proposed staggered quantum walk model provides an efficient quantum search on the two-dimensional lattice, if the reflection operators associated with the graph tessellations are used as Hamiltonians, which is an important theoretical result for validating the staggered model with Hamiltonians. Numerical results show that on the two-dimensional lattice staggered models without Hamiltonians are not as efficient as the one described in this paper and are, in fact, as slow as classical random-walk-based algorithms.