The Simplest 2D Quantum Walk Detects Chaoticity

Abstract

Quantum walks are, at present, an active field of study in mathematics, with important applications in quantum information and statistical physics. In this paper, we determine the influence of basic chaotic features on the walker behavior. For this purpose, we consider an extremely simple model consisting of alternating one-dimensional walks along the two spatial coordinates in bidimensional closed domains (hard wall billiards). The chaotic or regular behavior induced by the boundary shape in the deterministic classical motion translates into chaotic signatures for the quantized problem, resulting in sharp differences in the spectral statistics and morphology of the eigenfunctions of the quantum walker. Indeed, we found, for the Bunimovich stadium—a chaotic billiard—level statistics described by a Brody distribution with parameter $\delta \simeq 0.1$. This indicates a weak level repulsion, and also enhanced eigenfunction localization, with an average participation ratio $(\mathrm{PR})\simeq 1150$ compared to the rectangular billiard (regular) case, where the average $\mathrm{PR}\simeq 1500$. Furthermore, scarring on unstable periodic orbits is observed. The fact that our simple model exhibits such key signatures of quantum chaos, e.g., non-Poissonian level statistics and scarring, that are sensitive to the underlying classical dynamics in the free particle billiard system is utterly surprising, especially when taking into account that quantum walks are diffusive models, which are not direct quantizations of a Hamiltonian.

1. Introduction

Quantum walks (QWs), originally proposed by Aharonov [1], have since their inception constituted an important and active area of mathematics, with relevant applications in physics. This interest comes from their efficiency in capturing diffusive properties, which, in turn, provided many applications in quantum information [2,3,4] and quantum optics [5,6]. In particular, they are very suitable to explain and build efficient search algorithms [7], for example, in 2D grids [8], and also, possibly, to improve deep learning [9]. Moreover, QWs have been implemented in several experiments [10,11,12]. In their simplest form, QWs are quantum counterparts of the well-known classical problem of the random walker on the line, taking a step to the right or left depending on the outcome of a coin toss. The straightforward quantum-mechanical counterpart can be thought of as a spin particle that moves to the right or left, with its displacement depending on its spin state, whose evolution is given by the so-called quantum coin, the analogue of the classical coin toss.

Given the relevance of QWs in finite systems applications, studying the effects of boundaries is of the utmost importance [13,14], especially the behavior with respect to the complexity of the landscape in which the walker is moving. Relevant questions are raised at this point: For instance, could a minimal QW model reveal quantum signatures of chaos in the same way as the quantized models directly derived from the Hamilton’s classical equations of motion do? Can the simplest 2D QW serve as a probe for classical chaoticity? Considering the different nature of the dynamics involved, these are challenging questions that we address in this work. In fact, QWs are not direct quantizations, in the usual sense, of deterministic classical models and may show unexpected properties [15]. Detecting chaos with the most simple QW model not only bridges the gap between Hamiltonian chaos and random QWs, but it can also be related to understanding localization from different sources [16,17].

In this respect, quantum chaos [18] is the perfect arena where the quantum signatures of classical chaos have been traditionally studied. This area of physics has witnessed many important results, with the so-called Bohigas–Giannoni–Schmit (BGS) [19] conjecture probably being the most celebrated. It prescribes a random matrix behavior in quantized chaotic Hamiltonian systems. Notably, this subject is very active today, with many questions still open. In particular, the well known level repulsion, derived from BGS, was generalized to open systems under the name of the Grobe–Haake–Sommers (GHS) conjecture [20], but it has been recently challenged in [21]. Another well known phenomenon in quantum chaos is the localization on marginally stable—for example, on bouncing balls and other trajectories—and unstable orbits, also known as scarring; it constitutes a hallmark of quantum chaos which always gives rise to surprising new results [22].

Accordingly, we plan to borrow some concepts and tools from quantum chaos in order to unveil the consequences that a domain producing chaotic dynamics has on the QWs. For that purpose, we consider a very simple model consisting of a 2D QW constrained by a hard wall. A deterministic classical free particle moving inside will show regular or chaotic dynamics depending on the domain shape [18]. Here, we take the rectangle and the paradigmatic Bunimovich stadium billiards as benchmark examples of both behaviors, respectively. We conclude that a streamlined QW model consisting only of a single spin and alternate and independent movements along each coordinate perceives the different nature that these boundaries have in the dynamics at the classical level. This is reflected by the presence of scarring on unstable periodic orbits and more localization in the Bunimovich stadium case. Moreover, a non-Poissonian level statistics shows that the QW is sensitive to the corresponding classical dynamics of a free particle inside a billiard. This is a remarkable result, since QWs are diffusive models not directly attached to the quantization of a Hamiltonian.

The paper is organized as follows: In Section 2, we describe our QW model, and in Section 3, we show the results, where we mainly focus on the spectral behavior and the morphology of the spatial part of the eigenfunctions of the evolution operator. Finally, in Section 4, we present our conclusions and also outline some possible future developments.

2. QW Model

The simple QW model that we have chosen to study is defined as a spin $1/2$ particle moving inside a 2D billiard, whose state at any (discrete) time is

$$|\Psi (t)\rangle =\sum _{m,n}{U}_m^n(t)|m,n,u\rangle +D_m^n(t)|m,n,d\rangle ,$$

(1)

where $U_m^n$ and $D_m^n$ represent the probability amplitudes of the particle located at position $(m,n)$, with $(m,n)$ integers defining the position in a grid inside the billiard. Such particle can have either spin up, u, or down, d. The evolution is given by unitary operators that act on the spin space (a coin operator reflecting the effect of a coin toss) and on the position space (a walk operator reflecting the step taken by the walker conditional on the spin state). Additionally, there is a spin flip each time the walker reaches the border of the billiard. (The justification of this choice is given below, at the beginning of Section 3). As an explicit motivation for our model, we can view the previous evolution as that corresponding to the motion of an electron inside a cavity. We can think of this model as consisting of two independent QWs, except for the fact that both share the same spin. The unbounded version is usually called Alternate QW (AQW) [23,24].

Furthermore, we consider a bounded QW model inside the rectangular billiard and also the Bunimovich stadium depicted in Figure 1. For comparison purposes, the rectangular billiard is taken as the rectangle in which the Bunimovich stadium is contained, both having a horizontal length two times that of the vertical one. Recall that the classical dynamics in the rectangular domain are regular.

The evolution of our walker can be summarized algorithmically by the following iterative sequence:

(Quantum) Coin toss →

Horizontal step (with eventual boundary spin flip) →

(Quantum) Coin toss →

Vertical walk (with eventual boundary spin flip) →

(see inner part of the Bunimovich stadium in Figure 1). The corresponding operator for one time step is given by ${\hat{Q}}_w={\hat{W}}_n{\hat{C}}_2{\hat{W}}_m{\hat{C}}_1$, where ${\hat{W}}_m$ and ${\hat{W}}_n$ are horizontal and vertical displacement operators, respectively, which include reflection at the boundaries (by means of the previously mentioned spin flip), and ${\hat{C}}_2$ and ${\hat{C}}_1$ are the coin operators. See more details in the first part of Appendix A.1.

To conclude this section, we discuss the time evolution under these dynamics of a representative position eigenstate on the two spatial domains (rectangular and Bunimovitch stadium billiards) considered in our calculation. In Figure 2, we show some snapshots of the evolution of the corresponding probability density distribution for the rectangular billiard case. A grid of size $m_R=150$ and $n_U=75$ and an initial position eigenstate at the center ($m=75,n=37$) are taken in our calculations; for spin up, the amplitude is $1/\sqrt{2}$, while for spin down, it is $i/\sqrt{2}$. Four representative snapshot times have been chosen so as to (approximately) cover the lapse of time taking the ‘border’ of the wavefunction to return the to center of the billiard, after bouncing once at the walls. Similar results are presented for the Bunimovich stadium in Figure 3. The results clearly show that before the time corresponding to the bounce on the circular part of the Bunimovich stadium, both probability distributions coincide, as expected. However, after the main reflection on the circle at $t=76$, they become markedly different, this showing the regularity and strong symmetry existing in the rectangular billiard (aside from minor details due to the grid dimensions and coin shape) and the lack of them in the stadium. In fact, the probability peak at the center in the case of the rectangular domain is gradually destroyed in the Bunimovich case for later times, as can be checked by comparing the lower panels of both figures.

3. Chaotic Signatures: Spectral Behavior and Morphology of the Eigenfunctions

An important point concerning our QW model is worth discussing here. An unbounded one-dimensional QW is translationally invariant, and as a consequence, a simple Dispersion Relation (DR) can be obtained. A similar situation arises for the unbounded AQW in two dimensions [25], leading to a generalized DR, which is a straightforward extension of the 1D solution. In the case of a cylinder, i.e., a compact domain with periodic boundary conditions in one direction, the same situation effectively happens, and a DR has been derived for this case [26]. On the other hand, a very interesting class of billiards consists of a completely bounded domain with reflective boundaries as the case considered in this work. In this case, the effect of the boundaries in QWs can be implemented simply by means of spin flips.

3.1. Spectral Statistics

Since, in our case, there is no analytical DR, in what follows, we will proceed using numerical explorations of the behavior of our system, considering spectral statistics and the morphology of the corresponding eigenfunctions.

For this purpose, we directly diagonalize the evolution operator for one time step, i.e., ${\hat{Q}}_w$, in the spin-position basis set, and then study its (unfolded) spectrum, for cases considered here. To speed up the computations (which are heavy), we here take a (smaller) grid of size $(m_R,n_U)=(50,25)$. (Notice that this grid implies a smaller position basis for the Bunimovich stadium than for the rectangular billiard.) Moreover, though this choice generates finite-size effects, we have found that it represents a fair compromise between computational cost and physical significance. It is also important to emphasize that desymmetrizing the Bunimovich stadium is a customary procedure to avoid unwanted symmetries and, then, unveil the eventual Wigner surmise (or other statistics in different systems) for the eigenvalues associated to the corresponding Hamiltonian. Failing to do so allows the unwanted overlapping of uncorrelated spectra corresponding to different symmetry classes. Keeping our line of reasoning, we will proceed in this way below.

In Figure 4, we show the spectral statistics for our model, where the eigenphases (i.e., the phases of the complex unimodular eigenvalues) of the evolution operator have been considered in the construction of the histograms, $P(s)$, for the unfolded level spacing s. Indeed, after ordering the eigenphases between 0 and $2\pi$, the unfolding of these values is performed by simply dividing the distances between nearest neighbors by the corresponding mean (arithmetic) distance. Obviously, more sophisticated unfolding procedures, such as, for example, fitting a polynomial to the cumulative level density, can be used. However, for our purposes, i.e., a straightforward detection of spectral behavior differences, the simple method used is plenty enough. The results for two different pairs of $\alpha$ and $\beta$ values (coin phases) are shown in the different panels of Figure 4; being equal (symmetrical coins) in (a) and (c), but different (asymmetrical coins) in (b) and (d); see caption for details. Although not directly applicable, in principle, to our QW scenario, we plot in each panel, together with the histograms for comparison, the so-called Wigner surmise [27]

$$P_{\mathrm{W}}(s)={\displaystyle \frac{\pi }{2}}sexp\left({\displaystyle \frac{-\pi s^2}{4}}\right),$$

(2)

as a blue (black) dashed line. This surmise is known to describe the spectral behavior of chaotic Hamiltonians. In the other extreme of level statistics, we have that for regular systems, spectral spacings are known to satisfy the Poisson distribution, given by

$$P_{\mathrm{P}}(s)=exp(-s).$$

(3)

Given that the QW dynamics are different from the ones corresponding to the quantum Hamiltonian of a free particle inside a billiard, deviations from these two analytical expressions are expected in our results. Hence, we have also fitted the data for the Bunimovich case, using least squares and the root mean square (RMS) error, to the Brody distribution [28], a very simple function that interpolates between Poisson and the Wigner surmise, given by the expression

$$P_{\mathrm{B}}(s)=a{s}^{\delta }exp(-b{s}^{\delta +1}),$$

(4)

where $a=(\delta +1)b$, $b={\Gamma ((\delta +2)/(\delta +1))}^{\delta +1}$, $\delta$ being a fitting parameter between 0 (Poisson distribution) and 1 (Wigner surmise). The corresponding results are displayed by means of red (gray) solid lines in Figure 4a,b, and the fitting values are reported in

Table 1. Best-fitting Brody parameter and errors of the Brody and Wigner spectral distributions for the results of our QW model in the Bunimovich stadium billiard.

Coins			RMS Error
$\alpha$	$\beta$	$\delta$	$P_{\mathrm{B}}$	$P_{\mathrm{W}}$
$\pi /4$	$\pi /4$	0.07	0.044	0.131
$\pi /4$	$\pi /3$	0.15	0.069	0.154

. We notice by direct eye examination that our results behave very differently from those of the Wigner surmise, something also ascertained by the values of $\delta =0.07$ and $0.15$ and the fact that the error is less than 0.5 (see Table 1). By inspection, it is also evident that the Poisson ($\delta =0$) curve is closer to the histogram bars, though there is some relevant level repulsion. This is a remarkable result in that it suggests a novel spectral behavior, not previously described, to the best of our knowledge, in the literature. The reason for this can be that more symmetries besides those already broken exist (almost immune to the asymmetrical coins choice) and uncorrelate the spectrum leading to this peculiar Poisson-like statistics with level repulsion. In the future, we will investigate a model that can explain this kind of spectral behavior, perhaps related to the Anderson transition or other properties of disordered systems.

In the case of the rectangular billiard, whose associated Hamiltonian is regular, we have evaluated the error of the Poisson distribution (see Figure 4c,d). The results are reported in

Table 2. Poisson distribution errors for the QW in the rectangular billiard.

Coins		RMS Error
$\alpha$	$\beta$	$P_{\mathrm{P}}$
$\pi /4$	$\pi /4$	0.140
$\pi /4$	$\pi /3$	0.090

.

As can be clearly seen, the behavior is not strictly Poissonian, although it resembles it with the exception of the very large first bar, something which signals a particularly high lack of level repulsion, which, again, is almost immune to the asymmetrical coins choice.

Summarizing, the shape of the histograms in the Bunimovich and the rectangular billiards strongly differs in the small spacing region. Though the QW dynamics are clearly different from those of a quantum free particle moving inside a hard wall billiard, the statistical behavior of the spectra in these two cases allows to distinguish between them.

At this point, another interesting question arises: Can we also identify other differences of this kind, reminiscent of the ones found in the Hamiltonian dynamics, at the system eigenfunction level?

3.2. Eigenfunctions Morphology

To answer this question, we resort to analyzing the morphology of the eigenfunctions. For that purpose, we use the Participation Ratio (PR), a commonly used measure of localization in quantum chaos [18]. The PR is calculated by first normalizing the eigenfunctions $|\Phi \rangle$, and then evaluating

$$\mathrm{PR}={\left(\sum _{m,n}|{U_{\Phi }}_m^n{|}^4+{|{D_{\Phi }}_m^n|}^4\right)}^{-1},$$

(5)

where ${U_{\Phi }}_m^n$ and ${D_{\Phi }}_m^n$ are the components of $|\Phi \rangle$). For example, in the case of the rectangular billiard, the values of the PR range from 1 to $2m_R{n}_U$ ($2\times 50\times 25$), this roughly indicating how many position basis elements participate into a given eigenfunction (besides the factor 2 coming from the spin part of the Hilbert space). As a consequence, larger PR values indicate less localized states in this basis. In Figure 5, we show the histogram corresponding to $P(\mathrm{PR})$ vs. PR.

It can be directly observed that localization on ∼1150 basis elements is the approximate typical behavior for the Bunimovich stadium, while this value amounts to ∼1500 and 1400 for the case of the rectangular billiard. Recall that this is partly due to the different effective dimensions of the position basis in the two cases; the Bunimovich stadium basis size being approximately $0.91$ times that of the rectangular billiard. This is around half of the bases sizes (again, take into account the spin space dimension). However, not only are the PRs in the Bunimovich stadium billiard more localized than in the rectangle, on average, but also, their distribution is more biased towards lower PR values in a meaningful way (PR approximately $\in [600,800]$) range, where those for the rectangular billiard have smaller values). In what follows, we are going to deepen into these features, by analyzing some representative cases.

We next display and analyze some examples of eigenfunctions that help to understand the behavior of wavefunction localization in our QW model. States are labeled with their corresponding values of the PR and eigenphase in the figure captions. In Figure 6, some examples of the most delocalized (left column) and most localized (right column) eigenfunctions in the Bunimovich stadium (upper row) and in the rectangular billiard (lower row) are shown; both correspond to QWs with symmetrical coins. In this figure, we notice that the two delocalized states (left column) are very different, with that corresponding to the Bunimovich stadium looking more irregular (or chaotic, in the sense of the quantum chaos theory [18]), and that for the rectangular billiard looking markedly more regular. For these maximally delocalized states, the analogy with the Hamiltonian system is obvious, and the different nature of the dynamics clearly manifests itself. However, the situation is completely different for the most localized states presented in the right column. As a matter of fact, here, we observe eigenfunctions extremely peaked near the boundaries, which are not usually found in quantum billiards. In our opinion, the nature and origin of this states deserve further study, which is beyond the scope of the present work.

The case of asymmetrical coins is considered in Figure 7. In this case, both systems apparently behave in an irregular fashion, but a closer examination reveals that the only change from the previous results is the tilting induced by the asymmetry of the coins, which clearly biases one of the 1D translations with respect to the other. This results in a probability density distributed more along diagonal lines. A remark about the effect of not using an unbiased Hadamard-like coin is in order here. When considering $\beta \in (\pi /4;\pi /2]$, we anticorrelate more and more the horizontal and vertical walk directions, thus reaching a displacement along the main diagonal of the position domain at $\beta =\pi /2$. The same happens for $\beta \in [0;\pi /4)$, but in this case, the two walking directions become more and more correlated towards $\beta =0$, where a displacement along the other diagonal happens. In any case, the differences between regular and irregular behavior due to the shape of the boundaries in the corresponding Hamiltonian dynamics can still be detected given that we are not in these limits when taking $\beta =\pi /3$; this value represents well the typical biased, but still not completely correlated, scenario.

More interestingly, in the case of the Bunimovich stadium billiard, we have also found localization on structures similar to those found in the corresponding Hamiltonian system (by using an automated search for intermediate PR states followed by visual inspection). This effect can be responsible for the greater localization found in this case (notice their typical PR values). In fact, we have found bouncing ball states [18] that closely resemble those which are ubiquitous in the quantum chaos literature [29,30,31]. A representative example is shown in Figure 8a for the symmetrical coins case. In the standard quantum chaotic model, this family of eigenfunctions is localized on marginally stable orbits that form a continuous family. In the QW scenario, their manifestation is remarkable, having in mind that a purely diffusive behavior is expected. The rest panels of Figure 8 correspond to scarred states [18] on short periodic orbits (POs) of the Hamiltonian classical system, namely, the “rectangle” orbit in (b), a whispering gallery mode (which is a special case similar but not equal to the bouncing ball family) in (c), and a scar on the “bow tie” trajectory in (d). See also a representation of these POs in Figure 9.

Scarring is a well-described phenomenon in quantum chaos. The term scar was first coined by Heller [32,33] to define an unexpected localization along unstable POs of the probability density of certain eigenstates of Hamiltonian systems, whose classical behavior was chaotic. Since then, it became one of the cornerstones in quantum chaos, since it demonstrated the existence of the quantum correspondence of classical chaos.

Next, making a quantitative evaluation of the scarring effect present in the eigenfunctions of our QW model evolution operator is in order. For this purpose, we resort to the method of the semiclassical construction of resonant (or scar) functions along POs. This method will be briefly described next, but we refer the reader to reference [29] and references therein for more details. These resonances can essentially be seen as the product of a plane wave in the direction along a given scarring PO, using the semiclassical approximation for the unidimensional motion along the orbit, and a Gaussian wave packet in the transverse coordinate, which follows a dynamics without dilation–contraction along the unstable and stable manifolds of the unstable trajectory. The wavenumber k is approximately given by the Bohr–Sommerfeld quantization rule $kL=2\pi n$, L being the length of the PO. For simplicity, we ignore here, but not in the calculations, topological contributions [18], such as Maslov indices and boundary conditions, which guarantee continuity along the PO.) In Figure 9, we show the relevant POs needed to evaluate the localization of the eigenfunctions in Figure 8. We notice that the POs in Figure 9a,c are suitable to build scar functions that well approximate bouncing ball and whispering gallery kind of eigenfunctions, but also that they are not the only possible choice since they originate in singular families of POs.

To ascertain the scarring character of the eigenstates in Figure 8, we have evaluated the overlap between the probability distributions corresponding to the scar functions computed on these POs (that are shown in Figure 10) and those corresponding to the bouncing ball and scarred eigenfunctions in Figure 8. The results are given in the caption of Figure 10. It is interesting to note that it is possible to obtain $60\%$ or more of the eigenfunctions with such a simple and semiclassical approximation, and this figure increases to almost $80\%$ in the case of the bouncing ball, but this is due to the extreme localization on a given region of the whole domain existing in this case.

Finally, it is noticed that the typical sequence of horizontal excitations of the bouncing ball states is also present in the QW for both, symmetric and asymmetric, coin cases. The only difference happens when considering the asymmetrical coin which just tilts them. One example of this sequence is shown in Figure 11.

Localization on POs of the analogue classical Hamiltonian system by a diffusive quantum model that is not a direct quantization of it is a very striking result. We think that our very simple diffusion model along orthogonal directions in configuration space is enough to mimic the evolution provided by the Helmholtz equation inside a billiard. But if all the standard scarring can be reproduced with it, this still remains an open question that certainly deserves further investigation. Finally, we present

Table 3. Summary of results for both billiard shapes considered in this paper.

	Rectangular	Bunimovich
Character	regular	chaotic
Poisson vs. Brody	RMSE = 0.09 ∼ 0.14	$\delta =0.07\sim 0.15$
Average PR	1500	1150
Scarring	no	yes

, which summarizes our main obtained QW results, especially pointing out the differences found between the regular and the chaotic billiards considered in this work.

4. Conclusions

In these paper, we have studied the properties of a QW taking place inside compact bidimensional domains with different boundary shapes, which are characterized by the regular and irregular behaviors of their corresponding classical and quantum Hamiltonian dynamics. The simple diffusive evolution given by our QW is able to detect these two regimes which, in principle, have only been associated to the classical behavior of a free particle inside these cavities and its corresponding quantization. We also find the presence of a strong scarring phenomena, which, in our opinion, is a striking result. This allows to conclude that the simplest generalization of the QW on the line to 2D billiards constitutes a new paradigmatic example of quantum chaos.

We should emphasize that our simplest 2D QW detects chaoticity not through the well-known Wigner–Dyson statistics, but through a combination of weak level repulsion, altered localization properties, and the striking presence of scarring. The obtained spectra in our calculations are not fully random matrix-like, this suggesting that the model captures a different aspect of quantum chaos that is not present in Hamiltonian systems.

Indeed, a deeper theoretical explanation of this behavior is a promising avenue of research, but this is a limitation of our work at present. One of the most interesting questions that arises from it is how diffusive systems, such as QWs, are able to detect chaoticity, and in particular, which combinations of coin properties and grids do allow for localization on POs and which do not. Although our work does not yet permit convincingly concluding it, we foresee that small perturbations to the Hadamard-like coin keep localization on short orbits, making them become deformed. On the other hand, the asymmetrical coin case considered breaks all localization, except that on the bouncing balls, which exhibits a marginally stable orbit. Diffusion directions compared to the combined domain shape and coin symmetries should be relevant; the concrete details of their influence will be studied in the future. Moreover, the underlying position grids automatically associated with discrete QWs could induce different behaviors when compared to the usual billiard models. In fact, a hopping Hamiltonian on lattice billiards has been investigated in [34], showing the same spectral properties as the continuum counterparts, although this has been conducted in the open system scenario; there, new lattice scars were found. Then, an interesting question arises: Is the connection to latticed Hamiltonian scars direct?

To conclude this section, we indicate some relevant possibilities for future research. In the first place, more complex situations, such as having two particles inside the domain or using other coins, should be considered in the future, since they would serve as different (more realistic?) models for electrons inside a cavity [35]. Another interesting point worth studying is if our chaotic QW can be used to improve 2D grid searches, as an alternative to the nonlinear QW model studied in [8], for instance. Finally, studying the phase space structure of the QW dynamics, with Husimi functions, for example, should be considered as an interesting aspect in future research. Also, the sensitivity of these results to different coin operations and grid geometries is an intriguing question.