## Abstract

We present a comprehensive review of photonic implementations of discrete-time quantum walks (DTQW) in the spatial and temporal domains. Moreover, we introduce a novel scheme for DTQWs using transverse spatial modes of single photons and programmable spatial light modulators (SLM) to manipulate them. We discuss current applications of such photonic DTQW architectures in quantum simulation of topological effects in photonic systems.

### Keywords

- quantum walks
- spatial-multiplexing
- time-multiplexing
- spatial light modulators
- geometric phase
- Zak phase
- topology

## 1. Introduction

Quantum computation is an interdisciplinary field that encompasses several interconnected branches such as quantum algorithms, quantum information, and quantum communication. There are several advantages associated with quantum information processing that have positioned quantum computation as a key resource in advanced modern science and technologies. Among the promising conjectures predicted by quantum information and communication, we find the development of more powerful algorithms that may allow to significantly increase the processing capacity and may enable the quantum simulation of complex physical systems and mathematical problems for which we know no classical digital computer algorithm that could efficiently simulate them at present.

Quantum algorithms are the main building blocks of quantum information and quantum communication strategies. Nevertheless, building superior quantum algorithms is a challenging task due to the complexities of quantum mechanics itself, and because quantum algorithms are required to demonstrate that they can outperform their classical counterparts, in order to be considered an evolutionary advantage. Therefore quantum algorithms must be more efficient than any existing classical protocol. In this context, quantum walks, i.e., the quantum mechanical counterpart of classical random walks, can be regarded as a sophisticated tool for building quantum algorithms for quantum information and quantum communication that has been shown to constitute a universal model for quantum computation [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13].

The quantum walk is one of the most striking manifestations of how quantum interference leads to a strong departure between quantum and classical phenomena [2, 3, 14]. In the discrete version of the quantum walk, namely the discrete-time quantum walk (DTQW) [15], the time evolution is described in terms of a series o discrete time-steps. DTQWs provide for a flexible architecture for the investigation of a large number of complex topological and holonomical effects, in the experimental [16, 17, 18] and theoretical domains [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Moreover, DTQWs are robust algorithm for modeling a large number of time-varying processes, ranging from energy transfer in chains of spins [32, 33] to energy transport in biological systems [34]. Furthermore, DTQWs allow to study multi-dimensional quantum interference effects [35, 36, 37, 38] and can outline a route for authentication of quantum complexity [39, 40] and universal quantum computation [41]. In addition, quantum walks involving multiple particles guarantee a relentless tool for encoding quantum information in an exponentially large Hilbert space [42], as well as for simulations in quantum chemical, biological and physical systems [43], in 1D and 2D geometries [44, 45, 46].

In this Chapter, we present a comprehensive review of photonic realizations of DTQW in both, the spatial [47] and the temporal [48] realms, based on spatial-multiplexing and time-multiplexing techniques, respectively. Moreover, we present a novel scheme for photonic DTQW exploiting transverse spatial modes of photons and programmable spatial light modulators (SLM) to manipulate the modes [3]. In contrast to all previous multiplexed implementations, this novel approach warrants quantum simulation of arbitrary discrete time-steps, only limited by the spatial resolution of the SLM itself. We also deliberate about possible applications of such photonic DTQW platforms in quantum simulation of topological phenomena in photonic systems, and the implementation of non-local quantum coin operations, based on two-photon hybrid entanglement. Part of this review is based on the work by the Author, selected as the cover story of a Special Issue on Quantum Topology, for the journal Crystals (MDPI) in 2017 [2].

## 2. Theoretical framework

As a starter, we describe the theoretical framework for the mathematical description of DTQWs, and applications in the generation and detection of non-trivial geometric-phase structures, in 1D DTQW platforms. The basic discrete step in the DTQW is mathematically described by a unitary quantum evolution operator

written in the well-known

This unitary operation is followed by a spin- or polarization-dependent translation

with

The quantum evolution operator for a discrete time-step is generated by a Hamiltonian

and

## 3. Photonic DTQWs

### 3.1 Multiplexed DTQWs in the spatial domain

The original strategy for implementation of photonic DTQW via spatial-mode multiplexing was first introduced by Broome * et al.* [47]. The dimension of the Hilbert space for the spatial DTQW is determined by

### 3.2 Multiplexed DTQW in the temporal domain

The strategy for implementation of photonic DTQW via temporal-mode multiplexing was first introduced in Ref. [48]. The dimension of the Hilbert space for the DTQW is determined by a unique spatial mode

### 3.3 DTQW using spatial light modulators (SLM) and transverse spatial modes

We will identify the lattice points of a DTQW in a 1D geometry by the transverse spatial modes of a single photon. More specific, for photonic propagation in * et al.* in an intricate setup [55], encoding th subspace of the quantum coin in the upper and lower regions of the

For an unbiased coin operator, we have:

Considering the following initial state for the quantum walker

with

Thus, the corresponding probability distribution characterizing the quantum walker, after

In order to analyze DTQW in 1D, within the framework described above, we present a realistic optical setup which can be divided into two modules. The first module is destined to prepare the initial state of Eq. (3) for an arbitrary value of

#### 3.3.1 DTQW preparation optical module

In Figure 2(b) a sketch of the optical module proposed in order to prepare the input state of the quantum walker, corresponding to the

Let us consider * et al.* [61]. Within this approach, it is possible to prepare arbitrary states of the form

where

In order to prepare the state given by Eq. (3) starting by the input state given by Eq. (6), it is required to implement polarization rotations conditioned on the transverse-mode positions, as described by the unitary operator

where

transforms

Spatially-dependent polarization rotations can be implemented by means of an SLM programmed for such task [62]. There are several different techniques for the various types of existing SLMs, which enable each pixel of the SLM device to work effectively as programmable polarization rotator [63, 64]. The details of these techniques are beyond the scope of the present work. With such programmable SLM, the transformation (7) onto the state (6) can be implemented by manipulating the transverse spatial modes of

This concludes the description of the proposed preparation optical module for arbitrary walker-coin state in the

#### 3.3.2 DTQW one-step propagation module

The quantum coin operator is the quantum analogue of a walker throwing a coin, and deciding whether to proceed to the left or to right, depending on whether the coin falls heads or tail. By encoding the left and right information in the 2 dimensional photon polarization basis

## 4. DTQW: applications in topology and geometry

Geometric phases acquired during quantum evolution of a particle can have different origins. The Berry phase [65] is a type of geometric phase that can be assigned to quantum particles which return their initial state adiabatically, while recording the path information on a geometric phase (

A number of physical consequences can be attached to geometric phases, such as the modification of material properties in solids, for example the conductivity in Graphene [67, 68], the emergence of surface edge-states in topological insulators, whose surface electrons experience a geometric phase [69], the modification of molecular chemical reactions [70], and more recently geometric phases have been predicted to have implications for quantum technology, via the elusive Majorana particle [71].

In this review, we report on the progress in the characterization of geometry and topology of DTQW architectures consisting of a unitary step

## 5. Topology and the geometric Zak phase

The physical concept of geometric phase, such as Berry or Zak phase, is intimately linked to the concept of holonomy of a manifold. Holonomy from a geometrical standpoint: within the framework of differential geometry, an holonomy group

here

stating that the vector field norm

where

The geometric concept of holonomy can be depicted for a manifold

This sphere represents a surface

By taking a vector

For the applications in DTQW that we intend to consider, we can express the condition above in terms of a complex unit vector

In order to find the solid angle

On the other hand, the phase

Note that

the previous step makes use of the Stokes. It can be noted that the integrand is invariant under the Gauge transformations, meaning:

This integral can be written explicitly in terms of the coordinate system, obtaining:

which is the solid angle subtended by

Within the framework of quantum mechanics, one can replace the mathematical complex vector

A complex basis

This phase is of course is base dependent, but the holonomy is independent from that choice of basis. Holonomy can be defined by an adiabatic travel around a curve

The phase

By simple generalization of the arguments given above in the differential geometrical context, it follows that this phase is simply

Note that the phase is dependent on the choice of path

The natural language for an holonomy in this context is in terms of principal bundles. There exists a natural metric

This tensor is Gauge invariant

One may define a “distance” between two states by

The interpretation of distance is as follows. For two states

this follows from the fact that the product of a symmetric tensor by an antisymmetric one is zero. Note that, for a 2-dimensional spin system:

gives the canonical metric on

A subtle remark is in order, in relation to the coloquial use of the words geometry, holonomy, and topology. The holonomy of a manifold

We can define such a geometric phase as the holonomy for an abstract connection in a principal bundle

In the following section, we present applications of these mathematical concepts within the context of quantum mechanical problems, in particular of DTQWs.

### 5.1 Applications via spatial multiplexing: split-step DTQW

In this Section we analyze in detail two cases of topologically non-trivial Zak phase landscape, where thee Zak phase is the equivalent to the Berry phase across the Brillouin zone. The first, the so-called split-step DTQW is implemented by applying two consecutive conditional translations

The dispersion relation, which expresses the quasi-energy

In order to decompose the DTQW Hamiltonian of the system in terms of Pauli matrices

We now turn to our second example of topologically non-trivial DTQW.

### 5.2 Applications via temporal multiplexed: DTQW with non-commuting rotations

As a second non-trivial example, we introduce a DTQW consisting of two sequential non-commuting rotations

The 3D-norm required for expressing the Hamiltonian in the Pauli basis, results in:

The dispersion relation for the DTQW with non-commuting rotations results in:

it cam be easily verified that we recover a Dirac-like dispersion relation for

As readily mentioned, the described system exhibits a non-trivial phase diagram consisting of a large number of discrete gapless points for different quasi-momenta. Such singular points can be regarded as topological defects in parameter space. Each gapless points represent topological boundaries of dimension zero, where topological invariant, such as the winding number

## 6. Geometric phase calculation

We now provide expressions for the geometric phase, the so-called Zak phase acquired due to quantum evolution across the Brillouin Zone, in the two aforementioned scenarios. These two scenarios are characterized by a generic Hamiltonian of the form:

The specific Hamiltonians for each scenario differ by a constant factor, and by the specific expressions of the normal vector

In general the Hamiltonian in the Pauli basis is given by the following matrix:

and is characterized by the eigenvalues, which represent the eigenenergies of the system:

By diagonalizing this generic Hamiltonian, we find thee normalized eigenvectors for the generic Hamiltonian are given by:

It is to be noted that the scaling factor

The geometric Zak phase (

We will now apply these concepts to the specific examples reviewed in the previous sections.

### 6.1 Split-step DTQW

We will calculate the Zak phase for two types of DTQW, the first one is the so-called split-step DTQW [19, 73]. It consists of a DTQW with unitary step

In particular, we consider the case in which the normal vector

There are two possible angle choices that lead to

from where it follows that

A numerical simulation of the Zak phase for the split-step DTQW is depicted in Figure 4a.

### 6.2 DTQW with non-commuting rotations

The particular DTQW with non-commuting rotations presented in previous Sections can be readily implemented via temporal multiplexing approaches. To this end, we recall that the unitary step results in

where

angular functions as defined above.

In this scenario, calculation of Zak phase in terms of the Hamiltonian eigenvectors (

Making use of expression (41), the geometric Zak phase results in:

Note that, for the case of DTQW with non-commuting rotations, the consequences of setting the norm to be fully transverse (i.e.,

A brief discussion is in order, it is well known that the Zak phase is Gauge dependent —that is, it depends on the particular choice of origin of the unit cell [74]. Therefore, in general it is not uniquely defined and cannot be considered a topological invariant. Nevertheless, a related topological invariant quantity can be defined in terms of the Zak phase * difference* between two states (

A time-multiplexed experimental scheme, which can be readily implemented to obtain the Zak phase difference between two states at a given time-step

## 7. Conclusions

In this Book Chapter, we reported a review of novel approaches to photonic discrete-time quantum walk (DTQW) platforms. Namely, we discussed implementations via spatial-multiplexing or temporal-multiplexing schemes, and we introduced a novel scheme for implementations based on transverse spatial modes of photons, which are in turn controlled by spatial light modulators (SLMs). While the number of discrete time-steps (* split-step* DTQW and for the case of DTQW with non-commuting rotations, which are implemented via spatial and temporal mode-multiplexing, respectively.

## Acknowledgments

The author gratefully acknowledges Leonardo Neves, Osvaldo Santillan, and Mohammad Hafezi. G.P. gratefully acknowledges financial support from PICT2015-0710 grant, and UBACYT PDE 2017 Raices programme.

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