# Dynamical Triangulation Induced by Quantum Walk

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## Abstract

**:**

## 1. Introduction

## 2. Recap: Quantum Walk on the Triangular Grid and Pachner Moves

#### 2.1. Quantum Walk on the Triangular Grid

- first, rotate each triangle according to its label: if triangle v’s label is s, we set$${\tilde{\psi}}^{s}\left(\right)open="("\; close=")">t+\frac{\u03f5}{2},v,k$$
- then, apply a unitary to each edge:$$\tilde{\psi}(t+\u03f5,v,k)=W\tilde{\psi}\left(\right)open="("\; close=")">t+\frac{\u03f5}{2},v,k$$

#### 2.2. Pachner Moves

## 3. Coupling the Quantum Walker with Pachner Moves

#### 3.1. The Grid

#### 3.2. Evolution of the Grid under Pachner Moves

**Lemma**

**1.**

**Proof.**

**Corollary**

**1.**

#### 3.3. The Quantum Walker

- first, each triangle rotates its internal components: if triangle v’s label is $({s}_{1},{s}_{2},{s}_{3})$,$${\tilde{\psi}}^{{s}_{k}}\left(\right)open="("\; close=")">t+\frac{\u03f5}{2},v,k$$
- second, we apply W to each edge:$$\tilde{\psi}(t+\u03f5,v,k)=W\tilde{\psi}\left(\right)open="("\; close=")">t+\frac{\u03f5}{2},v,k$$

- third, we apply the Pachner moves for this timestep.

#### 3.4. Evolution of the Walker during $1-\mathrm{to}-3$ and $2-\mathrm{to}-2$ Pachner Moves

#### 3.5. Evolution of the Walker during $3-\mathrm{to}-1$ Pachner Moves

**Lemma**

**2.**

**Proof.**

#### 3.6. When to Make Pachner Moves

## 4. Discrete Equation of the Walker

**Notation**

**1.**

**Notation**

**2.**

- if triangle v’s label is $({s}_{1},{s}_{2},{s}_{3})$, we write:$$\psi (t,v:k)={\psi}^{{s}_{k}}(t,v,k)$$$$\psi (t,v:k:v.k)=\psi (t,v,k)$$
- if triangle v carries the ↑ component of its $k-\mathrm{th}$ side and its neighbor $v.k$ on that side carries the ↓ component, we write:$$\psi \left(\right)open="("\; close=")">t,\begin{array}{c}v\\ k\\ v.k\end{array}$$

## 5. Numerical Simulations

## 6. Summary and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Time goes down with one time step between each figure vertically. x and y axis are spatial dimensions. On the left is the evolution of the usual quantum walk, on the right is the evolution of the walk with Pachner moves.

**Figure 8.**Translation of the component internal to w on its first side during a $3-\mathrm{to}-1$ Pachner move.

**Figure 10.**From the left to the right: (

**a**) Number of wells (

**top**) and local curvature (

**bottom**) in a ball of a radius 1, with $\beta =3\alpha $, fit as ${t}^{a}{e}^{-b{t}^{2}}c$; (

**b**) The parameter b as function of $\alpha $; (

**c**) The time $\mathtt{tmax}$ of the exponential cut as function of $\alpha $. The initial condition $\psi (t=0)=\frac{1}{\sqrt{3}}$ on each of the three internal components of the origin triangle and 0 elsewhere.

**Figure 11.**Gradient of the logarithm of the variance with $\beta =3\alpha $: it always converges to 2. The initial condition $\psi (t=0)=\frac{1}{\sqrt{3}}$ on each of the three internal components of the origin triangle and 0 elsewhere.

**Figure 12.**Heatmap after 100 steps of evolution. On the left $\alpha ={10}^{-4}$, on the right $\alpha ={10}^{-1}$. In both case $\beta =3\alpha $.

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**MDPI and ACS Style**

Aristote, Q.; Eon, N.; Di Molfetta, G.
Dynamical Triangulation Induced by Quantum Walk. *Symmetry* **2020**, *12*, 128.
https://doi.org/10.3390/sym12010128

**AMA Style**

Aristote Q, Eon N, Di Molfetta G.
Dynamical Triangulation Induced by Quantum Walk. *Symmetry*. 2020; 12(1):128.
https://doi.org/10.3390/sym12010128

**Chicago/Turabian Style**

Aristote, Quentin, Nathanaël Eon, and Giuseppe Di Molfetta.
2020. "Dynamical Triangulation Induced by Quantum Walk" *Symmetry* 12, no. 1: 128.
https://doi.org/10.3390/sym12010128