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arXiv · 2009.07800

Searching via nonlinear quantum walk on the 2D-grid

Basile Herzog·Giuseppe Di Molfetta
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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.

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Basile Herzog, Giuseppe Di Molfetta. 2020-09-16. Searching via nonlinear quantum walk on the 2D-grid. https://arxiv.org/abs/2009.07800

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