# Topology and Holonomy in Discrete-time Quantum Walks

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## Abstract

**:**

## 1. Introduction

## 2. Holonomy, Topology and the Berry phase

## 3. Discrete-Time Quantum Walks

#### 3.1. Split-Step Quantum Walk

#### 3.2. Quantum Walk with Non-Commuting Rotations

## 4. Zak Phase Calculation

#### 4.1. Split-Step Quantum Walk

#### 4.2. Quantum Walk with Non-Commuting Rotations

#### 4.3. Discussion

## 5. Experimental Realization

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Non-trivial phase diagram for the quantum walk with consecutive non-commuting rotations, indicating gapless Dirac points where quasi-energy gap closses for different values of quasi-momentum: squares ($k=0$), pentagons ($\left|k\right|=\pi $), romboids ($k=+\pi /2$), and circles ($k=-\pi /2$). These discrete Dirac points represent topological boundaries of dimension zero. They endow the system with a non-trivial topology.

**Figure 2.**(

**a**) non-trivial geometric Zak phase landscape for “split-step” quantum walk, obtained analytically; (

**b**) non-trivial geometric Zak phase landscape for the quantum walk with non-commuting rotations, obtained by numeric integration.

**Figure 3.**(

**a**) schematic of experimental setup to study the proposed system; (

**b**) implementation of non-commuting rotations: ${R}_{y}\left(\theta \right)$ is implemented via an Half-Wave Plate (HWP) at angle $\alpha =\theta /2$. ${R}_{x}\left(\varphi \right)$ is implemented by a sequence of Quarter-Wave Plate (QWPs) with fast axes oriented vertically and horizontally, respectively. In between the QWPs, an HPW oriented at $\beta =\varphi /2$ determines the angle for the second rotation; (

**c**) histogram of arrival times, after a trigger event at $t=0$.

**Figure 4.**(

**a**) measured proability distributions for $N=7$ steps in Hadamard QW with $\theta =\pi /4$, $\varphi =0$, and input state $|{\psi}_{0}^{+}\rangle $; (

**b**) difference between experiment and theory is within $20\%$, and is mainly ascribed to different soures of polarization dependent losses, spurious reflections, and shot-noise.

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**MDPI and ACS Style**

Puentes, G.
Topology and Holonomy in Discrete-time Quantum Walks. *Crystals* **2017**, *7*, 122.
https://doi.org/10.3390/cryst7050122

**AMA Style**

Puentes G.
Topology and Holonomy in Discrete-time Quantum Walks. *Crystals*. 2017; 7(5):122.
https://doi.org/10.3390/cryst7050122

**Chicago/Turabian Style**

Puentes, Graciana.
2017. "Topology and Holonomy in Discrete-time Quantum Walks" *Crystals* 7, no. 5: 122.
https://doi.org/10.3390/cryst7050122