Searching via nonlinear quantum walk on the 2D-grid

TL;DR

This paper extends a nonlinear quantum walk search algorithm to a 2D grid, demonstrating it retains quantum speedup with numerical evidence and parameter optimization for practical implementation.

Contribution

It generalizes a nonlinear quantum walk search algorithm to 2D grids and analyzes its complexity and optimal parameters through numerical simulations.

Findings

Search time scales as O(N^{1/4} log^{3/4} N)

Probability of finding the marked vertex is O(1 / log N)

Optimal parameters exist to mitigate measurement precision effects

Abstract

We provide numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer \cite{meyer2013nonlinear}, rephrased in terms of quantum walks with effective nonlinear phase, can be extended to the finite 2-dimensional grid, keeping the same computational advantage \BHg{with} respect to the classical algorithms. For this purpose, we have considered the free lattice Hamiltonian, with linear dispersion relation introduced by Childs and Ge \cite{Childs_2014}. The numerical simulations showed that the walker finds the marked vertex in $O(N^{1/4} \log^{3/4} N) $ steps, with probability $O(1/\log N)$, for an overall complexity of $O(N^{1/4}\log^{7/4}N)$. We also proved that there exists an optimal choice of the walker parameters to avoid that the time measurement precision affects the complexity searching time of the algorithm.