Curved space-time Dirac equation in the honeycomb and triangular lattice 

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Discrete time quantum walk

To illustrate what a quantum random walk is, we introduce the simplest, and most studied model of quantum walk, i.e the discrete-time quantum walk (DTQW) on a infinite line. Of course, DTQWs can be defined in more sophisticated struc-tures like cycles or higher dimensional lattices. However, as the QW on the line is a simple model, it helps us to understand the most relevant properties. Let us consider the dynamical evolution of the DTQW described by the Hilbert space H = Hp ⊗ Hc. Hp is the Hilbert space spanned by the positions of the particle. In the case of the discrete line, it is spanned by the basis {|pi}p∈Z. Hc is the internal degree of freedom, which is referred as the ’coin’-space. In the simplest case, the coin space is spanned by the two basis states {|↑i , |↓i}, which play the role of the spin-12 of the particle. The evolution of the system is determined by the application of a unitary op-erator, instead of a stochastic matrix as in the CRW. The unitary operator which acts on the total Hilbert space is given by: U =T(IP ⊗C). (1.3).
Therefore, the evolution of the DTQW after j time steps is written as: |ψ(j)i = Uj |ψ(0)i , (1.4).
where IP is the identity operator in HP . C is the coin operator, which only acts on HC , whereas T is the conditional shift operator, which involves the whole Hilbert space H. The conditional shift operator T allows the particle to move one step for-ward/backward, depending on the internal degree of freedom, e.g the walker goes forward if it has spin up |↑i) , and goes backwards in case of spin down |↓i. This is represented by: X∈ (1.5) T = |p + 1i hp| ⊗ |↑i h↑| + |p − 1i hp| ⊗ |↓i h↓| . p Z On the other hand, IP ⊗C acts only on the coin space, playing the role of ’tossing’ the quantum coin. C rotates the internal degree of freedom of the walker. Since a unitary transformation is quite arbitrary, we can define a family of walk with different behaviors. The most general unitary rotation can be chosen as: C(α, θ, γ, φ) = eiα −e−iφ sin θ e−iγ cos θ! (1.6) eiγ cos θ eiφ sin θ.
where the four parameters (α, θ, γ, φ) are real. In the case where θ = π4 , α = −γ = π2 and φ = −π2 , the quantum coin becomes the so-called Hadamard coin: CH=√2 1 −1! . (1.7) 1 1 1.
The Hadamard coin CH is said to be balanced, which means that applying the evolution operator U:
1 |↑i ⊗ |0i → √ (|↑i + |↓i) ⊗ |0i.

Quantum walk in momentum space

The analytical study of the discrete time quantum walk using the Discrete Time Fourier Transform (DTFT) was introduced first by Nayak and Vishwanath [101]. For simplicity, we will consider the Hadamard coin, which due to its translational invariance, permits a simple description in the Fourier domain. Let us describe the position of the walker as a two component spinor, being at position p at time-step j, described by: ψ(p, j) = ψL(p, j)! . (1.12) ψR(p, j).
Upon identifying this notation with the notation in operator terms, we can write the spinor as: |ψ(p, j)i = ψR(p, j) |↑i + ψL(p, j) |↓i . (1.13).
Applying the unitary evolution operator, Eq.(1.3), to a state at time j, it is possi-ble to relate it to the state at time j + 1. Thus, the dynamics in matrix notation is written as: ψ(p, j + 1) = M+ψ(p + 1, j) + M−ψ(p − 1, j). (1.14) using the notation: M+=√2 0 0 ! M−=√2 1 −1 ! . (1.15) 1 1 1 1 0 0.
Therefore, the finite difference equations after one iteration of the unitary evolution operator U, for each amplitude ψR(p, j) and ψL(j, p) are given by: 1 ψR(p, j + 1) = √ (ψR(p + 1, j) + ψL(p + 1, j)) 2 1 ψL(p, j + 1) = √ (ψR(p − 1, j) − ψL(p − 1, j)) . (1.16).
In this way, the analysis of the QW on the line, for the Hadamard coin, reduces to solving a two dimensional linear recurrence system. The discrete Fourier transform of the wave function ψ(p, j), over Z is defined by: ˜ X ikp (1.17) ψ(k, t) = ψ(p, j)e p where k ∈ [−π, π] is the quasi-momentum. The inverse Fourier transform is given by: 1 π ψ(p, j) = 2π Z
−π ψ˜(k, j)e−ikpdk (1.18).

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QW continuous limit

It is already well-studied the connections between the DTQW and their contin-uous limit [45, 46], however it is necessary give an introduction to this topic because it constitutes the basic mathematical technique to prove that DTQW can be used for the purpose of quantum simulation. Homogeneous QW In this basic example we consider a quantum walk defined on a discrete one-dimensional space and discrete time. The evolution of this QW is driven by a U(2) coin, which acts on a walker represented by a two-component field ψ. The discrete space points are labeled by p ∈ Z, and the time steps are labeled by j ∈ N. The finite difference equation reads ψj↑+1,p! = Q(α, θ, ξ, ζ)T ψj↓+1,p where the operators T and Q are given by: ψ↑ ! Tψj,p = j,p+1.

Table of contents :

Acknowledgment
Resumen
Résum
Abstract
List of Figures
Introduction
1 Quantum walks: an introduction 
1.1 Classical random walk
1.2 Discrete time quantum walk
1.2.1 Quantum walk in momentum space
1.2.2 The asymptotic probability distribution in the long time limit
2 Quantum walks as a quantum simulators 
2.1 Quantum simulation
2.2 QW continuous limit
2.3 Localization
2.4 Domain wall model
2.5 Publication: « Fermion confinement via quantum walks in (2 + 1)- dimensional and (3 + 1)-dimensional) space-time
2.6 Bound states in the Dirac equation
3 Gauge invariance in DTQW 
3.1 Gauge invariance in electromagnetism
3.1.1 Gauge invariance in quantum mechanics
3.1.2 Electromagnetic gauge invariance in relativistic quantum mechanics
3.1.3 Discrete local invariance in LGT
3.2 Publication: « Electromagnetic lattice gauge invariance in two-dimensional discrete-time quantum walks »
4 Quantum walks over the honeycomb and triangular lattice 
4.1 Motivation
4.1.1 Spatial search on hexagonal and triangular lattices
4.1.2 Localization on the honeycomb and triangular lattices
4.1.3 Topological phases in the triangular lattice
4.1.4 Quantum walks over graphene structures
4.2 Publication: « Dirac equation as a quantum walk over the honeycomb and triangular lattices »
5 Curved space-time Dirac equation in the honeycomb and triangular lattice 
5.1 Motivation
5.2 The Dirac equation in a curved space time
5.2.1 General covariance
5.2.2 Affine connection
5.2.3 Spin connection
5.3 Continuum Deformation Mechanics
5.4 Crystallographic defects
5.5 Publication: « From curved spacetime to spacetime-dependent local unitaries over the honeycomb and triangular Quantum Walks »
6 Perspectives and Conclusions 
6.1 Tetrahedral QW
6.2 Conclusions
6.3 Perspectives

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