# Improving Quantum Search on Simple Graphs by Pretty Good Structured Oracles

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## Abstract

**:**

## 1. Introduction

## 2. Quantum Spatial Search on Graphs

## 3. Quantum Search by Structured Oracles

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Localization probability for ring graphs. The two panels show ${p}_{wr}(\tau ,\lambda )$ as a function of $\tau $ for two rings with $N=11$ (

**left**) and $N=17$ (

**right**), and for different values of $\lambda $ (chosen among those maximizing the localization probability in the temporal range considered). From bottom to top (referring to the short time region, where the curves do not intersect) we have ${p}_{wr}(\tau ,\lambda )$ for $\lambda =0.9,1.0,1.1,1.2,1.3,1.4$ in the left plot and for $\lambda =0.4,0.5,0.6,0.7,0.8,0.9$ in the right one.

**Figure 2.**Localization probability for ring graphs with a three-site symmetric oracle. The two panels show ${p}_{wr}(\tau ,\lambda )$ as a function of $\tau $ for two rings with $N=11$ and $\lambda =1.3$ (

**left**) and $N=17$ and $\lambda =0.8$ (

**right**), and for different values of $\alpha $ (chosen among those leading to a localization probability larger than that for $\alpha =0$). The red curves in both panels denote the results obtained for negative values of $\alpha $ whereas the green ones are for positive values of $\alpha $. Solid lines in the left panel correspond to $\left|\alpha \right|=0.9$ and dashed lines to $\left|\alpha \right|=0.8$. In the right panel we have $\left|\alpha \right|=0.2$ (solid lines) and $\left|\alpha \right|=0.3$ (dashed lines).

**Figure 3.**(

**Left**): The solutions of Equations (8) and (9), denoted by ${\lambda}_{\mathrm{PG}}$ and ${c}_{\mathrm{PG}}$, as a function of N. The two horizontal dashed lines denote the values ${\lambda}_{\mathrm{PG}}=2$ and ${c}_{\mathrm{PG}}=1$, respectively. (

**Right**): The red squares denote the pretty good oracle (PGO) localization probability ${p}_{\mathrm{PG}}$, defined in Equation (10), as a function of N. The blue circles denote the corresponding localization probability for an unstructured oracle. The upper and lower dashed curves correspond to the functions $1.6/{N}^{0.31}$ and $4/N$, respectively.

**Figure 4.**The equivalent search time ${\tau}_{e}$ of Equation (11) for a structured PGO (red squares) and for an unstructured oracle (blue circles) as a function of the evolution time for two ring graphs with sizes $N=11$ (

**left**) and $N=21$ (

**right**). The vertical dashed line indicates the time at which the maximum PGO localization probability is obtained.

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**MDPI and ACS Style**

Paris, M.G.A.; Benedetti, C.; Olivares, S.
Improving Quantum Search on Simple Graphs by *Pretty Good* Structured Oracles. *Symmetry* **2021**, *13*, 96.
https://doi.org/10.3390/sym13010096

**AMA Style**

Paris MGA, Benedetti C, Olivares S.
Improving Quantum Search on Simple Graphs by *Pretty Good* Structured Oracles. *Symmetry*. 2021; 13(1):96.
https://doi.org/10.3390/sym13010096

**Chicago/Turabian Style**

Paris, Matteo G. A., Claudia Benedetti, and Stefano Olivares.
2021. "Improving Quantum Search on Simple Graphs by *Pretty Good* Structured Oracles" *Symmetry* 13, no. 1: 96.
https://doi.org/10.3390/sym13010096