# Searching via Nonlinear Quantum Walk on the 2D-Grid

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## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. The Linear Algorithm

Algorithm 1: Searching algorithm for the $2d$-grid |

#### 2.2. Adding a Nonlinearity

Algorithm 2: Searching with a nonlinear algorithm for the $2d$-grid |

## 3. Numerical Results

## 4. Scale Analysis

- A strictly linear regime, when $\delta \left(t\right)=0$, which appears periodically with a period$${T}_{0}\sim 1/\mathsf{\Delta}\lambda \sim {\displaystyle \frac{1}{E}}=O\left(\sqrt{NlogN}\right)$$
- A nonlinear regime, when $\delta \left(t\right)$ is maximum. In this case, $(1+cg\delta \left(t\right))\sim cg\delta \left(t\right)=O\left(\sqrt{NlogN}\right)$ and consequently the period is constant:$${T}_{1}\sim 1/\mathsf{\Delta}\lambda =O\left(1\right).$$

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The $2d$—grid and its factorisation into cells (grey squares). The marked vertex (circled) is disconnected from its neighbors.

**Figure 3.**The probability $p\left(t\right)=|\langle \mathsf{\Gamma}|{e}^{-iHt}{|s\rangle |}^{2}$ for different values of c−$N=900$.

**Figure 6.**$N=900$—Probability over $|\mathsf{\Gamma}\rangle $ for different values of c and numerical resolution of Equation (43).

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**MDPI and ACS Style**

Di Molfetta, G.; Herzog, B.
Searching via Nonlinear Quantum Walk on the 2D-Grid. *Algorithms* **2020**, *13*, 305.
https://doi.org/10.3390/a13110305

**AMA Style**

Di Molfetta G, Herzog B.
Searching via Nonlinear Quantum Walk on the 2D-Grid. *Algorithms*. 2020; 13(11):305.
https://doi.org/10.3390/a13110305

**Chicago/Turabian Style**

Di Molfetta, Giuseppe, and Basile Herzog.
2020. "Searching via Nonlinear Quantum Walk on the 2D-Grid" *Algorithms* 13, no. 11: 305.
https://doi.org/10.3390/a13110305