# On the Amplitude Amplification of Quantum States Corresponding to the Solutions of the Partition Problem

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction and Preliminaries

## 2. The Effect of Applying Hadamard-S-Hadamard Gates to ${|\mathbf{0}\rangle}^{{\otimes}^{\mathit{n}}}$

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 3. Doubling the Amplitudes of the Solution States of the PP

- $w(y)$ is even$$\begin{array}{cc}\hfill {i}^{w(y)}{(-1)}^{w(y)}& +{i}^{n-w(y)}={i}^{w(y)}{(-1)}^{w(y)}+{i}^{n+w(y)}=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={i}^{w(y)}(1+{i}^{n})\hfill \end{array}$$
- $w(y)$ is odd$$\begin{array}{cc}\hfill {i}^{w(y)}{(-1)}^{w(y)}& +{i}^{n-w(y)}={i}^{w(y)}{(-1)}^{w(y)}-{i}^{n+w(y)}=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =-{i}^{w(y)}(1+{i}^{n})\hfill \end{array}$$

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**Left**) The amplitudes of ${a}_{z}$ for $n=3$. (

**Right**) The amplitudes of ${a}_{z}$ for $n=4$. In order to get the final amplitudes one should multiply them by a suitable normalization factor.

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**MDPI and ACS Style**

Mezzini, M.; Paulet, J.J.; Cuartero, F.; Cruz, H.I.; Pelayo, F.L.
On the Amplitude Amplification of Quantum States Corresponding to the Solutions of the Partition Problem. *Mathematics* **2021**, *9*, 2027.
https://doi.org/10.3390/math9172027

**AMA Style**

Mezzini M, Paulet JJ, Cuartero F, Cruz HI, Pelayo FL.
On the Amplitude Amplification of Quantum States Corresponding to the Solutions of the Partition Problem. *Mathematics*. 2021; 9(17):2027.
https://doi.org/10.3390/math9172027

**Chicago/Turabian Style**

Mezzini, Mauro, Jose J. Paulet, Fernando Cuartero, Hernan I. Cruz, and Fernando L. Pelayo.
2021. "On the Amplitude Amplification of Quantum States Corresponding to the Solutions of the Partition Problem" *Mathematics* 9, no. 17: 2027.
https://doi.org/10.3390/math9172027