Mimicking disorder on a clean graph: Interference-induced inhibition of spread in a cyclic quantum random walk

Quantum interference is one of the key aspects of quantum mechanics that leads to a plethora of interesting phenomena, such as interference pattern in double-slit experiments and Anderson localization of electron wave packets. It also leads to the qualitatively different behavior of quantum random walks (QRWs), first introduced by Aharonov, Davidovich, and Zagury [1] in 1993, from their classical counterparts —the classical random walks (CRWs). Quantum random walks have been studied both in discrete-time [2,3] and continuous-time scenarios [4,5]. In discrete-time QRWs, superposition over walker-coin states where the walker has pursued different paths as instructed by the different coin states, and the resulting entanglement between the coin and position degrees of freedom in a discrete-time QRW are at the root of its differences with respect to a walker in the classical case. In particular, there is inherent interference between the left and the right propagating components in the dynamics of a QRW on a line, and the clockwise and anti-clockwise components on a circle (cyclic graph). Based on the chosen initial conditions or by varying the coin parameters, interference may affect the symmetry of the probability distribution, of the walker on both the line and the cyclic graph, in position space [6]. QRWs have been shown to be useful in realizing quantum memory [7], in search algorithms [8], for simulating dynamics of physical systems [9], etc. Successful experimental implementation of QRWs have been reported in various physical systems, such as optical Galton board, nuclear magnetic resonance systems, atoms trapped in an optical lattice, photons, and trapped ions. Introduction of disorder in the system, e.g., by randomizing the "coin parameter" of a discrete-time quantum random walk [10], or by introducing static disorder in a continuous-time quantum random walk on the glued trees graph [11], or by inserting other imperfections on graphs where the distances between the vertices vary slightly from edge to edge [12] may lead to inhibition of spread of the wave packet in position space. See also [13–18] in this regard. Furthermore, ref. [19] provides a brief review on the effect of disorder and the consequent restriction of spread in continuous-time quantum random walks. See also [20] in this respect.

For a discrete-time quantum random walk on a line, the spread of the probability distribution, as quantified by the standard deviation, scales linearly with the number of steps of the walker, referred to as "ballistic spread". The corresponding "speed" of the classical walker is only the square root of the number of steps, where by "speed", we mean the standard deviation in position of the walker [6,21]. It is possible that for quantum and classical random walks on a cyclic graph, the scaling behaviors with respect to steps or system size may hide interesting information. The long-time properties of the time-averaged probability distributions in both the cases have already been studied [22–24]. Quantum walks on a cyclic graph having the additional feature of "one-step memory" was investigated in [25]. By allowing the coin operation to change at every step according to a sequence or by random means, the associated probability distribution is seen to converge to a uniform distribution over the nodes of a cyclic graph [26]. In another work, the periodicity of the evolution matrix of a Szegedy walk, a type of discrete-time quantum walk, on various types of finite graphs have been discussed [27]. A special feature of QRW on a cyclic graph is that it "mixes" almost quadratically faster than the corresponding classical case [28]. QRWs on cyclic graphs have been implemented experimentally, and further proposals thereof have also been given. In an experimental implementation using an arrangement of linear optical elements, clockwise and anti-clockwise cyclic walks have been realized [29]. In another experimental implementation, continuous-time quantum walks on cyclic graphs using quantum circuits have been realized [30]. Travaglione et al. have proposed a scheme to implement QRW on a line and on a circle in an ion trap quantum computer [31]. Another proposal for experimental implementation of QRW on a cyclic graph, using a quantum quincunx, which may be realized with cavity quantum electrodynamics, is also known [32]. QRWs of non-interacting and interacting electrons on a cyclic graph, with the graph being formed of semiconductor quantum dots, have been studied in [33].

In this paper, we analyze the behaviors of quantum and classical random walks on cyclic graphs. We find that the "collision"-like interference effects on the cyclic lattice, occurring due to the topology of the lattice, leads to the inhibition of spread of the wave packet in a Hadamard quantum walk. The inhibitions of spread, with respect to both number of steps and number of sites, are inferred by analyzing the patterns of Shannon entropies of the corresponding position probability distributions. As mentioned before, such phenomena in QRWs have usually been resultant of an insertion of disorder into the system (cf. [ [34–37]). Here we provide an instance where inhibition of spread occurs in a QRW even when there is lack of disorder. We then show that this behavior is generic, in that it appears also for non-Hadamard walks, and that it is independent of the initial state. Additionally, we find that the scaling exponents of the entropies with respect to the system size, when the number of steps is sufficiently large, are different for classical and quantum random walks. Furthermore, the ratio between these entropies again implies that the quantum system has a restricted spread with respect to the classical one. Finally, we introduce a distance measure using the l¹-norm, for measuring the closeness of two distributions, specifically, the time series data of Shannon entropies for a QRW and a CRW on cyclic graphs of increasing size. Using this measure, we quantify the difference in the spreads of QRW and CRW on cyclic lattices of different size.

The paper is structured as follows. The next section introduces the general aspects of a CRW on a cyclic graph and its operational formalism. The third section introduces the broad facets of a QRW on a cyclic graph, the operational formalism, and describes the inhibition of spread with respect to steps in Hadamard quantum walks. In the fourth section, we study the behavior of CRW and QRW on cyclic graphs with respect to the system size. We present a summary in the last section.

We begin this section by giving a short introduction to CRW on the infinite line. We next introduce the cyclic graph and discuss about the features of the probability distribution of the walk after a large number of steps. We also briefly indicate the mathematical formalism that is used to deal with CRWs on the cyclic graph.

The mathematical setting for random walks are graphs G(V, E) with a vertex set V and an edge set E. Classical random walks consist of a walker localized at a given vertex v who moves by means of randomly choosing one of the two directed edges, for example, of an infinite line or a cyclic graph, with probabilities p and q $(p+q=1)$ at every step. We are restricting ourselves to situations where there are exactly two edges emanating from every vertex, and the graph is connected. This motion is dictated by the result of toss of a coin which could be unbiased or not. Let us however assume that the coin is unbiased, so that $p=q=1/2$. A natural question of interest is the following: What is the pattern of probability distribution of the walker over all the vertices after a given number of coin tosses? It turns out that for a relatively small number of coin tosses or steps, the binomial distribution, a discrete probability distribution, characterizes the probability of finding the walker at each vertex. In the limit of a large number of steps, a Gaussian distribution provides the vertex-wise probabilities.

We mention here that a cyclic graph is one which is like a necklace with the beads representing vertices and the strings between the beads, the edges. In other words, a cyclic graph consists of a single cycle. We note that the line and the cyclic graphs are both connected as well as two-regular, assuming the line to be infinite. A "bipartite" graph consists of a graph in which the vertices can be colored with two different colors, and where each edge connects vertices of different colors. In a non-bipartite graph, such a coloring scheme is not possible. For a cyclic graph with an even number of sites $({=}N_1)$ —a bipartite graph— the probability for the walker to be at the (2i)-th site is $2/N_1$, and the same for the $(2i+1)$-th site is vanishing, for $i=0,1,\ldots,N_{1}-1$, after a large even number of steps. The roles of the even and odd sites get reversed if the number of steps is a large odd number. For a cyclic graph with odd number of sites $({=}N_2)$ —a non-bipartite graph— in the limit of large number of steps, we get a uniform probability distribution with all the sites having a probability of $1/N_2$. This is the distinction between CRWs on a bipartite and a non-bipartite cyclic graph. Consequently the Shannon entropy reaches its maximum possible value in the limit of large number of steps, given by $\log_2 (\frac{N_1}{2})$, for a bipartite graph, and $\log_2 (N_2)$, for a non-bipartite cyclic graph. These are respectively the maximal possible Shannon entropies in the two cases.

Let us now discuss the action of the CRW on a cyclic graph in a bit more detail. It consists of a classical walker moving on sites $x \in \{0,1,\ldots,N-1\}$, based on the outcome of a coin toss. The jump is only to the adjacent vertices. Here the random variable is the position of the walker, whose values are determined by the values of another random variable, the coin toss. We associate a "probability vector" corresponding to the site-wise probability distribution at every step. Now we look at the discrete-step evolution of the probability vector facilitated by a transition matrix (stochastic matrix). For a cyclic graph with N sites ${\in}V$, the probability vector, $\vec{P}(n)$, will have N elements. Here, n denotes the number of steps. Let $\vec{P}(n)=(P_{0}(n), P_{1}(n),\ldots, P_{N-1}(n))^{T}\in \mathbb{R}^N$. The elements of the transition matrix are given by $T_{ij} =\frac{1}{2}$ if j is the nearest neighbor of i, and vanishing otherwise, where i, $j \in \{0,1,\ldots,N-1\}$. From a given vertex i, the walker randomly chooses to move along either of the two edges to reach the nearest neighbors j with a uniform probability of $1/2$. This is because the degree of every vertex of a cyclic graph is two and we are assuming an unbiased coin. Note that $T_{ij}=T_{ji}$. $\vec{P}(n)$ is obtained by the action of the transition matrix $T{:}~\mathbb{R}^N\rightarrow\mathbb{R}^N$ on the probability vector after n –1 steps, $\vec{P}(n-1){:}~\vec{P}(n)=T \vec{P}(n-1)$. The elements of $\vec{P}(n)$ are given by $P_{i}(n)=\sum_{j=0}^{N-1}T_{ij}P_{j}(n-1).$

In this section, we discuss about QRWs in general and then move on to its features on a cyclic graph. We investigate the case of symmetric and initial coin states for our analysis. The associated inhibition of spread is studied in some detail.

An initially localized wave packet of the walker evolves as per the assigned local transition rules from a given vertex through either of the two edges to the corresponding two neighboring vertices based on the outcome of a coin toss at every discrete step. This kind of discrete evolution is facilitated by local action at the respective vertices and modeled by a unitary operator without the conventional Hamiltonian.

We are interested in studying the global properties of the walk, such as the spread, as quantified, e.g., by standard deviation or Shannon entropy in position space. In this paper, we will use the latter quantity to measure the spread. It is defined as $-\sum_{i}p_i\log_2p_i$, for a probability distribution $\{p_i\}$. We use logarithms of base 2 in all calculations of entropy, so that they are measured in bits. Considering a random variable X, the Shannon entropy, H(X), quantifies the amount of uncertainty present in the different values assumed by X before knowing them. Equivalently, it also quantifies the amount of information gained after knowing the values of X. H(X) depends on the probability distribution, $\{p_i\}$, that describes the probability of occurrence of every value, i, of the random variable after a given number of trails. The Shannon entropy is also a measure to quantify the amount of resources needed to store information. Due to the occurrence of bi-modal probability distributions on using a symmetric initial coin state in a QRW and the topology being a cyclic graph, we use the entropy instead of standard deviation to capture the spread of the quantum walker.

For a discrete-time quantum walk (DTQW) on a line, the vertex set V forms the position basis $\{|x\rangle{:}~x \in \mathbb{Z}\}$ that spans the position Hilbert space, $\mathcal{H}_p$. The two basis states, $|0\rangle_{c}$ and $|1\rangle_{c}$ of the coin, which label the two directed edges at any given vertex, span the coin Hilbert space $\mathcal{H}_c$. The unitary evolution of the walk at every step is performed through a coin operation acting on the coin basis states followed by a conditional shift operation on the position basis states - conditioned on the states of the coin. The coin operation is parametrized by a "coin parameter" θ. We focus on the DTQW on a cyclic graph for our study. We will henceforth be denoting DTQW as QRW.

For a QRW on a cyclic graph, the dimension of the position Hilbert space, $\mathcal{H}_p$, is fixed to N, the total number of sites (vertices), and $\mathcal{H}_p= \{|x\rangle{:}~x \in \mathbb{Z} \cap [0, N-1]\}$. Here, the basis states of the coin instruct whether the next step of the walker will be in the clockwise or anti-clockwise direction. On a line, a significant portion of the quantum wave function —"significant" in terms of the corresponding position probabilities—moves away from the point of the initial position of the walker, and so, although there is in principle room for interference between the different parts of the significant portion, it does not happen in a substantial way. The situation is completely different in the case of the QRW on a cyclic graph, where the different parts of the significant portion are forced to "collide" (interfere) with each other due to the topology of the graph, provided N is about the same order as n or smaller than that, where n is the number of steps. This is one of the reasons that makes the cyclic graph an interesting class of graphs on which to study QRWs. For a cyclic graph with an odd number of sites, N₂, in the limit of a large number of steps, the time-averaged probability distribution becomes a uniform distribution with a uniform probability of $1/N_2$. In the case of an even number of sites, the time-averaged probability distribution is not uniform. This is an interesting distinction between a bipartite and a non-bipartite cyclic graph in a QRW.

We now study the QRW on a cycle in more detail, and discuss the corresponding inhibition of spread. To begin, the "coin operator", $\hat{C}_{\theta}{:}~\mathcal{H}_c\rightarrow\mathcal{H}_c$, is given by $\hat{C}_{\theta}=\cos\theta(|0\rangle\langle0|)_c+\sin\theta(|0\rangle\langle1|)_c +\sin\theta(|1\rangle\langle0|)_c-\cos\theta(|1\rangle\langle1|)_c,$ with θ ∈ $(0,\frac{\pi}{2})$. We also consider the conditional shift operator, $\hat{S}_{x}{:}~\mathcal{H}_c\otimes\mathcal{H}_p\rightarrow\mathcal{H}_c\otimes\mathcal{H}_p$, which shifts the position of the walker by a signed (i.e., directed) step length $\Delta x= +1$ if the coin is in the state $|0\rangle_c$, and $\Delta x= -1$ if the same is in $|1\rangle_c$. More precisely, $\hat{S}_{x}=\sum_{x=0}^{N-1}[|0\rangle_c\langle0|_c\otimes|x+1\;(\bmod\; N)\rangle_p\langle x|_p +|1\rangle_c\langle1|_c\otimes|x-1\;(\bmod\; N)\rangle_p\langle x|_p]$. The state after n steps, $|\psi(n)\rangle$, is obtained by the action of the unitary operator $\hat{U}(\theta)=\hat{S}_{x}.~(\hat{C}_{\theta}\otimes I):\mathcal{H}_c\otimes\mathcal{H}_p\rightarrow\mathcal{H}_c\otimes\mathcal{H}_p$ on the state after n − 1 steps, $|\psi(n-1)\rangle{:}~|\psi(n)\rangle=\hat{U}(\theta)|\psi(n-1)\rangle.$ The classical version of this equation is given towards the end of the second section, where the unitary is replaced by a transition matrix. Due to the action of the unitary, the position and coin states become entangled already after the first step of the walk, and the general state of the walker after n steps, for fixed number of sites N, takes the form $|\psi(n)\rangle=\sum_{x=0}^{N-1}[(a_{x}(n)|0\rangle_c+b_{x}(n)|1\rangle_c)\otimes|x\rangle_p],$ where a_x(n), b_x(n) are the probability amplitudes corresponding to the clockwise and anti-clockwise directions. From the aforementioned equation, the probability distribution over the sites x after n steps of the walk is given by $P(x,n)=| a_x(n) |^2 +| b_x(n) |^2.$

Hadamard walk on a cyclic graph

The "Hadamard" walk on a cyclic graph is the case when the coin parameter $\theta=\frac{\pi}{4}$. The Hadamard coin operator $\hat{H}$ is given by $H|0\rangle_{c}=|{+}\rangle_{c}$, $H|1\rangle_{c}=|{-}\rangle_{c}$, where $|\pm\rangle_{c}=\frac{1}{\sqrt{2}}(|0\rangle_{c}\pm|1\rangle_{c})$.

Symmetric initial coin state, enhanced interference and the ensuing inhibition of spread: We choose a symmetric initial state for the coin, viz. $\frac{|0\rangle_c\pm i|1\rangle_c}{\sqrt{2}}$. The sign in front of i is unimportant for further calculations. We begin the discussion for N = 10 and $\theta=\frac{\pi}{4}$, for different numbers of steps. The joint initial state is taken to be $|\psi(0)\rangle=\frac{|0\rangle_c\pm i|1\rangle_c}{\sqrt{2}}\otimes|{6}\rangle_p$. The choice of the initial state of the walker is of course arbitrary. The position probability at the site 6 begins with unity at the initial time, gradually diminishes with increase of the number of steps, reaches a minimum (a local minimum at site 6 with respect to number of steps) and then again increases to reach a maximum (a local maximum at site 6 with respect to number of steps) before diminishing once more. Each such maximum is called "meeting point", arguably of the clockwise and anti-clockwise components of the joint state. The corresponding number of steps are denoted by $n_{\textit{meet}}$. Figure S1 of the Supplementary Material Supplementarymaterial.pdf (SM) exhibits the symmetric site probability distribution for N = 10 and $\theta=\frac{\pi}{4}$, for different numbers of steps. The prefix "S" for figure and section numbers indicate that they belong to the SM. The first few meeting points are marked in black dots on the horizontal axis in the figure. The first meeting happens after 16 steps of the walk, the second meeting after 42 steps, and so on. Due to the oscillatory nature (with respect to number of steps) of the entropy of the probability distribution P(x, n) for fixed n, we take the average entropy, $H^{Q}_{\textit{meet}}$ up to every $n_{\textit{meet}}$ to measure the fluctuation in the probability distribution, instead of considering the entropy itself:

$$\begin{equation} H^{Q}_{\textit{meet}} (n_{\textit{meet}}) = \frac{1}{[n_{\textit{meet}}]}\sum_n\sum_{x=0}^{N-1}(-P(x,n)\log_2 P(x,n)), \end{equation} \tag{ 1 }$$

where the sum over n runs up to $n_{\textit{meet}}$ from just after the previous meeting point, and where $[n_{\textit{meet}}]$ denotes the number of steps in that interval between the meeting points.

The plot of $H^{Q}_{\textit{meet}}$ against $n_{\textit{meet}}$, in fig. 1, captures an important feature of the interference-induced dynamics in a QRW on a clean cyclic graph. The plot is done up to a sufficiently large number of meeting points so that no further appreciable change occurs in the value of $H^{Q}_{\textit{meet}}$. The plot of the same quantity in the classical case is denoted by $H^{C}_{\textit{meet}}$ and depicted by red crosses in fig. 1. See fig. S2 for the associated probability distribution of the CRW. $H^{C}_{\textit{meet}}$ exhibits very little fluctuations with the number of steps and saturates approximately around the value of 2.32 bits. It may be noted here that the classical walker does not encounter a "collision" between the anti-clockwisely and clockwisely traversing components. The corresponding plot, for a QRW (blue dots in fig. 1), displays significantly higher fluctuations (than in the classical case) and saturates approximately around the value of 2 bits, and being lower than the corresponding classical value of 2.32 bits, indicates an inhibition of spread.

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We now compute the entropies themselves (instead of the average entropies), in the classical and the quantum cases, of the position probability distributions, for every n up to a value of n after which we do not envisage any further significant change of behavior, for a fixed number of sites N. We perform the analysis on cyclic graphs for N = 10, 20, and 30 (see fig. 2). See fig. S3 for the same analysis done for N = 500, 600, and 700. We refer to these entropies as "site" entropies to differentiate them from the "average entropies", $H^{Q}_{\textit{meet}}$. We find that irrespective and in spite of the fluctuations present, the entropy with respect to number of steps for the QRW saturates to a value that is lower than the corresponding value in the classical case, for cyclic graphs of various sizes, indicating a certain amount of inhibition of spread in the quantum case. We also search for the asymptotic behavior of the Shannon entropy by calculating its value for cyclic graphs of size N as high as 1000, and observe that the entropy still saturates. We notice that as we increase the number of sites, the fluctuations in the entropy with number of steps in the quantum case is reduced. The long-time averaged values of the site entropy in the "saturation region" for cyclic graphs of size of the order of 1000 are depicted in fig. 3 for both CRW and QRW. For an analysis of the step entropies for the case of asymmetric initial coin states and non-Hadamard walks, see sects. S4 and S5, respectively.

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Until now, we have mainly been looking at the behavior of site entropy as a function of the number of steps for a given number of sites. We now do a role reversal and study the patterns of site entropy with respect to system size for quantum and classical random walks on cyclic graphs. Note that only bipartite graphs are considered for our calculations. After fitting a suitable function to the plot in the quantum case, we find out the limiting behavior of the ratio of the respective site Shannon entropies for the classical and the quantum cases as the system size grows to infinity.

Site entropy with respect to system size for a QRW on a cyclic graph: We have already observed that in general, the site entropy for a fixed number of sites registers a steep moderate monotonic increase up to a certain number of steps, after which saturation to a certain extent occurs. There may be very small amount of fluctuations present in this regime. See fig. 2(a). Due to the fact that significant fluctuations may be present even in the "saturated" regime, a characteristic "converged" value can be obtained, for a given system-size (number of sites), only by performing an average, over steps, in this saturated regime. We present these converged values in fig. 3 against system size for a quantum random walk on a cyclic graph, with the initial state of the coin being again chosen to be $\frac{1}{\sqrt{2}}(|0\rangle \pm i|1\rangle)$. For a precise indication of the range taken while averaging over steps in the saturated regime, see sect. S7. For a system size of N, we denote the converged site Shannon entropy by H^Q(N). We fit the function

$$\begin{equation} \alpha\log_2(1-\beta+\beta N^{\nu}), \end{equation} \tag{ 2 }$$

for $N\geq 1$, to the obtained values of H^Q(N) against N. We use the method of least squares to find α, β, and ν. We find that $\alpha=0.1609$, $\beta=0.0064$, and $\nu=5.9088$. The corresponding least squares error is 0.0330. Note that the term, $1-\beta$, in the argument of the logarithm in (2) is specifically chosen so that H^Q(N) vanishes for N = 1, the single-site system.

Site entropy with respect to system size for a CRW on a cyclic graph: We perform a parallel set of calculations for the classical walker. Again, the site entropy for a given system size has a steep monotonic increase up to a certain number of steps, after which it saturates. Unlike in the quantum case, there are no fluctuations in the saturated region. See fig. 2(b). The corresponding saturated value, H^C(N), behave as $H^C(N)=\log_2(\frac{N}{2}).$

Quantum vs. classical in the large-system-size limit: The ratio $H^C(N)/H^Q(N)$ in the large-N limit is $1/(\alpha \nu)$, so that it is 1.0515 for the above mentioned values of the fitting parameters. The ratio of the entropies in classical to the quantum case is slightly greater than one, which indicates that the CRW spreads out more compared to the QRW for a given system size. This characterizes the spreading behavior in both the scenarios implying a slight slowdown or inhibition of spread in the quantum case. We also note that the two curves corresponding, respectively, to the classical and quantum walkers, never meet for any (non-zero) value of N.

See sect. S6 for an analysis of the closeness of the two time series data distributions corresponding to the CRW and the QRW for each N by inspecting the l¹-norm.

To conclude, we investigated the spreading behavior of a quantum random walker and compared the situation with that for a classical one, on a cyclic graph. The walk of a quantum entity dictated by a quantum coin is distinct from that of a classical one commanded by a classical coin, due to the superposition of different positions of the quantum walker and due to entanglement between the quantum walker and the quantum coin during the evolution. For the quantum walker on the infinite line, e.g., when using a symmetric coin, the position probability distribution dissociates into a bi-modal distribution, and the spread of the position is qualitatively higher than the corresponding classical walker on the same graph. Replacing the infinite line by a cyclic graph, there is a role reversal between the quantum and classical walkers with respect to their spreads. This shift in behavior of the quantum walker for a switching of the lattice is potentially attributable to an additional interference in the quantum walker wave function. For the initial few steps, the quantum walkers on the infinite line and the cyclic graph are not different. The situation changes when the number of steps is large enough, and we see that the initial bi-modal position probability distributions have the possibility to "collide" in the case of a cyclic graph. A single classical walker on any graph does not have this option. A quantum walker on an infinite line, in principle, has this option, but the dynamics of the system does not let this happen. A quantum walker on a cyclic graph, however, is made to collide by the very dynamics due to the topology of the graph. The classical random walker keeps spreading on a cycle and reaches a steady state where the time-averaged probability distribution becomes a uniform distribution over all the sites of the cyclic graph. The quantum walker on a cyclic graph however has a slightly lesser spread, and we refer to this as an instance of inhibition of spread. Thereby, we have mimicked the role of disorder on an infinite line by using a clean cyclic graph, as insertion of disorder usually leads to inhibition of spread of a quantum walker on an infinite line. The spreads were captured by the Shannon entropies of the position probability distributions of the walkers. While we began with the case of the quantum coin operator being chosen so that the corresponding quantum walk is the "Hadamard" walk, we have later on also altered the coin parameter (to consider non-Hadamard walks), which introduces different degrees of interference in the dynamics, and have analyzed the consequent nature of the inhibition of spread. We found that for all the different values of the coin parameter considered, inhibition of spread persists, albeit with varying degrees of fluctuations in the entropy with respect to the number of steps. We saw that the symmetry of the initial coin state has minimal effect on the spreading behavior but has significant effect on the symmetry of the probability distribution. Subsequently, we compared the quantum and classical walkers by considering the behaviors of their entropies with respect to system size. Finally, we have used the l¹-norm as a distance measure between the variations of entropies with respect to number of steps in the quantum and the classical scenarios.