## 1. Introduction

In their origins [1–5], quantum walks (QWs) were thought as the quantum-mechanical generalization of the standard random walk in one dimension: the mathematical model describing the motion of a particle which follows a path that consists of a succession of jumps with fixed length whose direction depends on the random outcome of flipping a coin. In the quantum version, the coin toss is replaced by the action of a unitary operator upon some intrinsic degree of freedom of the system, a quantum observable with only two possible eigenvalues: for example, the spin of an electron, the polarization of a photon, or the chirality of a molecule.

After this preliminary analysis, it became clear that the similitude between these two processes was mainly formal and that random and QWs displayed divergent properties [6]. The most remarkable of these discrepancies is perhaps the ability of unbiased QWs to spread over the line, not as the square root of the elapsed time, the fingerprint of any diffusion process, but with constant speed [7]. This higher rate of percolation enables the formulation of quantum algorithms [8, 9] that can tackle some problems in a more efficient way than their classical analogs: For instance, QWs are very promising resources for optimal searching [10–12]. Today, QWs have exceeded the boundaries of quantum computation and attracted the attention of researchers from other fields as, for example, information theory or game theory [13–16].

As a consequence of this wide interest, diverse extensions of the discrete-time QW on the line have been considered in the past. Most of these variations are related with the properties of the unitary coin operator [17], backbone of the novel features of the process. Thus, one can find in the literature QWs whose evolution depends on more than one coin [18–20], QWs that suffer from decoherence [21, 22], or QWs driven by inhomogeneous, site-dependent coins [23–28]. There are also precedents where the temporal variability of the QW is explicit: in the form of a recursive rule for the coin selection, as in the so-called Fibonacci QWs [29, 30], through a given function that determines the value of the coin parameters [31–33], or by means of an auxiliary random process that modifies properties of the coin [34].

The main goal in most of these seminal papers is to find out new and exciting features that the considered modifications introduce in the behavior of the system, like the emergence of quasiperiodic patterns or the induction of dynamic localization. Recent works [35–37], however, have also regarded the issue from the opposite point of view, by exploring the conditions under which the evolution of the system results unchanged. In particular, Montero [37] considers the case of a discrete-time QW on the line with a time-dependent coin, a unitary operator with changing phase factors.

These phase factors are three parameters that appear in the definition of the coin operator whose relevance has been sometimes ignored in the past: When these phases are static magnitudes, they are superfluous [38], but if they are dynamic quantities, they can substantially modify the evolution of the system. This fact does not close the door to the possibility that a set of well-tuned variable phase factors can keep the process unchanged from a probabilistic perspective. This defines a control mechanism that can compensate externally induced decoherence and introduces a nontrivial invariance to be added to other well-known symmetries of QWs [39–41].

In this Chapter, we will review the approach taken in [37] and consider a generalization of it. Now, the evolution of the discrete-time quantum walker on the line will be subjected to the introduction of a fully inhomogeneous coin operator: The properties of the unitary operator will depend both on the location and on the present time through the action of the aforementioned phase factors. This extra variability leads to additional constraints to be satisfied by these magnitudes if one wants to guarantee that the properties of the motion of the walker remain unaltered. Finally, we will connect our results with those appearing in the study of Di Molfetta et al. [36], where the authors considered how the inclusion of time- and site-dependent phase factors in the coin operator of a quantum walk on the line may induce some dynamics which, in the continuous limit, can be linked with the propagation of a Dirac spinor coupled to some external electromagnetic field. We will also explore the implications of this mapping here.

## 2. Fundamentals of QWs

We begin this Chapter with a survey of the fundamental concepts required in the designing of discrete QWs on the line. In its simplest version, the particle represented by the walker can occupy detached and numerable locations on a one-dimensional space. This space of positions may be just a topological space (a graph or a chain, for instance) or can be endowed with a metric. In such a case, it is usual to consider that the sites are separated by a fixed distance

Up to this point, there is no significant difference between random and quantum walks. The major distinction is found in the nature of the random event that determines the progress of the particle. While in a world governed by the laws of classical mechanics, randomness is the way in which we describe the uncertain effect of multiple (and usually uncontrollable) external agents acting upon a system, in the realms of quantum mechanics randomness is not an exogenous ingredient. This means that we can use some internal degree of freedom in the quantum system with two possible eigenvalues (the spin, the polarization, or the chirality) as a proxy for the coin and understand that any change in this inner property is the result of the act of tossing. Therefore, to represent the state of the walker, we need two different Hilbert spaces:

where we have introduced the wave-function components

Now, we have to consider the mechanism that connects these two properties, position and quirality, which eventually leads to a model for the dynamics of

In a second step, the shift operator *moves* the walker depending on the result obtained after the last toss: [2] -

Therefore, the state of the system at a later time

and the complete evolution of the system is determined once

Needless to say that the linearity and the translational invariance of the problem ensure that the solution for a general initial state can be recovered by direct superposition of the evolution of Eq. (7), Eqs. (14) to (17) later.

The similarities and dissimilarities between classical and QWs must be grounded on the analysis of the probability mass function (PMF) of the process,

where

On the basis of the values of the moduli of

or the value of

another interesting magnitude that can be connected with the local magnetization of the system if the internal degree of freedom has its origin in the spin of the particle [45].

### 2.1. General solution

The evolution operator induces the following set of recursive equations in the wave-function components,

and

whose general solution [38] can be written in a compact way by using

and the nonzero components of the wave function at time

since

and

where

and

It is noted that in this picture the evolution of each component depends only on their own initial values. In fact, it can be shown [38] that

Even though the expression for

Equation (16) is recovered from the above relationship once one considers the initial condition

Observe how *p* in random walks and that the rest of parameters represent mathematical degrees of freedom without correspondence in the physical world. This impression can be strengthened by computing the value of the PMF in simple examples as, for instance, when *n* coincides with *t*: in this case,

This conclusion is illusory, however. It is well known [19] that _{,} and **Figure 1** illustrates this fact. In the upper panel, we observe how the probability is distributed unevenly for positive and negative values of *n*, although

### 2.2. Stationary PMF

**Figure 1** also shows us that the disparity in the bias is not the most striking aspect that distinguishes QWs from their classical analogues. These differences can be appreciated more easily when one considers the stationary limit [34]. It can be shown [38] that for

in the range **Figure 1**, the agreement between *n*, whereas when |*n*| approaches to **Figure 1**).

Regarding the expectation value of the position of the quantum walker,

its magnitude does not stem from the location of the largest maximum of

the expectation value of the position of the walker will increase linearly with time:

as it can be checked in **Figure 2**. The converse is not true [40, 41]: in order to get quantum walkers that show an exact symmetry in the parity one has to demand that

but also that [3] -

equations that have only three main families of solutions [38], being the most relevant of them the one corresponding to

## 3. Inhomogeneous QWs

The fact that not only

Consider a general inhomogeneous, time-dependent unitary operator

(24) |

where _{,} and

In this case, the information supplied by the initial state of the system is not so important: Assume that

with

In practice, this means that we can modify

The recursive equations of the wave-function components under the present dynamics induced by are straightforward variations of Eqs. (12) and (13):

and

Since we have a specific interest in revealing a new kind of invariance, we will introduce

(30) |

and

(31) |

Therefore, our task is to find out nontrivial relationships connecting both set of parameters. Regarding this, note that

## 4. Invariance

The properties of the system enumerated up to this point are based on the moduli of the components of the wave function. This means, in particular, that if one has that

and

If we assume the validity of Eqs. (32) and (33) and replace these expressions in Eqs. (28) and (29), the conditions to recover Eqs. (30) and (31) are

These equations lead to the following prescription to modify the phases leaving invariant the moduli of the components of the wave function:

### 4.1. Invariance of global observables

The first conclusion that can be drawn from Eqs. (34)–(36) is that there is an infinite variety of choices for

This assumption simplifies enormously Eqs. (34)–(36):

One particular choice that satisfies the above requirements is *n* and *t*, and the following functional forms for

a possible solution of Eqs. (37) and (38). The above expressions lead to the following homogeneous update rule for

where

We illustrate in **Figure 3** the invariance of _{,} and

We can sketch a complementary picture that may help in the understanding the behavior of

onto the *u*_{t} direction, that is,

where **Figure 2**, when *u*_{t} is not a periodic phenomenon at all. The absence of periodicity implies that vector *u*_{t} defines an everywhere dense but enumerable subset of points in the ring associated with colatitude

### 4.2. Exact invariance

Obviously, we can go further and demand exact invariance in the problem. This can be achieved by setting

As is shown below, these equations can be expressed in terms of finite differences which in turn lead to partial derivatives. In fact, in the expression of _{,} it appears a time derivative, whereas the formulas for

Equations (46) to (48) read, as we have anticipated,

This means, in particular, that we can transform an inhomogeneous coin into a time-dependent one

### 4.3. Continuous limit

Let us express Eqs. (34) to (36) in a slightly different way. Consider the discrete difference operators

and similarly for

Observe how the expression connecting

At this point, it is appropriate to note that we are not taking into account the issue of the parity of indexes *n* and *t*: since the instances of

However, our interest in this Section is to analyze the continuous limit,

We need to relate

where

and *X* and *T* of all the magnitudes is implicitly assumed from now on.

The exact invariance, *e* and masses **A**:

whose respective space-time components must change according to the formulas

Note that

and, when one introduces these relationships into Eqs. (67) and (68), one obtains the standard gauge transformations for the components of the potential **A**,

a transform that keeps invariant the electric field *E*_{X} acting upon the system,

If we reconsider the example introduced at the end of Section 4.2,

we can conclude that it corresponds to a case in which the electric field *A*_{X},

In the most general case, when **A** is

which departs from the gauge invariance of potential **A**. However, if we investigate the change in the electric field induced by Eqs. (75) and (76) we find

Clearly,

that results in the invariance of the electric field. A possible choice is to demand that both

Another alternative solution to Eq. (78) has appeared above, in Section 4.1. The equivalent expressions for Eqs. (37) and (38) in the continuous limit read:

what provides another solution to Eq. (78). Note that in this case Eqs. (65) and (66) show not merely covariance but perfect invariance in the mass-less case,

## 5. Conclusion

Along this Chapter, we have analyzed some interesting aspects of discrete-time QWs on the line, specifically those related with the emergence of invariance. In the first part, we have elaborated a succinct but comprehensive review covering the main features of the most elementary version of this process, when the unitary operator which assumes the function of the coin in the classical analog is kept fixed. We have described the dynamics that determines the evolution of the walker, supplied explicit formulas for assessing the precise state of the system at any time and approximate expressions that capture the main traits of the process in the stationary limit. These equations have been very useful to pinpoint the role played by the different parameters on the solution to the problem and put into context the generalization considered afterward.

The second part of the Chapter contemplates the situation in which the coin is time and site dependent. In particular, we have focused our interest on the phase parameters that define the unitary operator and determined the constraints that must be imposed in these changing phases if one wants to obtain invariance. This invariance can be demanded at two different levels: one can require that the invariance connects states belonging to the same ray of the Hilbert space or a milder condition, that the transformation modifies unevenly the two wave-function components. In this latter case, global properties (e.g., the probability that the particle is in a particular place or in a given spin state) remain unaltered but some other local quantum properties depending on the relative phase of these components can become modified.

The Chapter ends by analyzing the introduced invariance in the continuous limit. This approach unveils that the evolution of a time- and site-inhomogeneous quantum walk can be understood in terms of the dynamics of a particle coupled to an electromagnetic field and that the new symmetry shown by the walker can be interpreted as a manifestation of the well-known gauge invariance of electromagnetism.