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# Quantum walks with spatiotemporal fractal disorder

⋆Marcelo A. Pires1 ⋆Caio B. Naves2,3 Diogo O. Soares-Pinto2 Sílvio M. Duarte
Queirós4,5 1Universidade Federal de Alagoas, 57480-000, Delmiro Gouveia - AL,
Brazil
2Instituto de Física de São Carlos, Universidade de São Paulo, CP 369,
13560-970, São Carlos - SP, Brazil
3Department of Physics, Stockholm University, AlbaNova University Center, 106
91 Stockholm, Sweden
4Centro Brasileiro de Pesquisas Físicas, Rua Dr Xavier Sigaud 150, 22290-180,
Rio de Janeiro - RJ, Brazil
5National Institute of Science and Technology for Complex Systems, Brazil

###### Abstract

We investigate the transport and entanglement properties exhibited by quantum
walks with coin operators concatenated in a space-time fractal structure.
Inspired by recent developments in photonics, we choose the paradigmatic
Sierpinski gasket. The 0-1 pattern of the fractal is mapped into an
alternation of the generalized Hadamard-Fourier operators. In fulfilling the
blank space on the analysis of the impact of disorder in quantum walk
properties – specifically, fractal deterministic disorder –, our results show
a robust effect of entanglement enhancement as well as an interesting novel
road to superdiffusive spreading with a tunable scaling exponent attaining
effective ballistic diffusion. Namely, with this fractal approach it is
possible to obtain an increase in quantum entanglement without jeopardizing
spreading. Alongside those features, we analyze further properties such as the
degree of interference and visibility. The present model corresponds to a new
application of fractals in an experimentally feasible setting, namely the
building block for the construction of photonic patterned structures.

00footnotetext: ⋆These authors contributed equally to this work.

##  I Introduction

Scale-invariance, i.e. the intuitive sense of indistinguishability between the
overall shape of several natural systems and a portion of them, has marveled
humankind since a long time. Ultimately, this paved the way to the concept of
fractals – either self-similar or self-affine –, i.e. objects whose capacity
and Lebesgue covering dimension are different from one another. Besides its
inherent beauty, fractality has found its place at the first tier of science
and technology [1, 2, 3]. The endeavor to set forth fractal theory
applications in the development of new scale-invariant structures can be
gauged by the amount of literature that has been produced in recent years. For
instance, it was shown it is possible to use molecular self-assembly to
engineer an artificial nanometer-scale Sierpinski hexagonal gasket [4],
structures such as Peano, Greek cross and the Vicsek constructions to build
stretchable electronic devices with unusual mechanics [5]. On the other hand,
fractals went deep into the quantum realm by the assembling of artificial
architectures of electrons with fractal geometry [6, 7].

Such studies allowed the experimental investigation of subjects that have been
studied only from a theoretical perspective for long. In this work, we present
a theoretical investigation of position- and time-dependent quantum walks with
fractals that are feasible to experiments with newly developed photonic
architectures.

Quantum Walks (QW for short) [8] are the celebrated proxy lattice models for
studying wavepacket spreading. Within the scope of disorder-free QWs, it is
well-known the emergence of interesting properties such as ballistic spreading
with a non-gaussian bimodal probability distribution that are both very
distinct from the classical random walk. Still, a richer phenomenology comes
up when new ingredients are introduced in the coin [9, 10, 11, 12, 13, 14, 15]
or step [16, 17, 18, 19, 20, 21, 22] operators.

On the one hand, temporal disorder in QWs can be implemented by breaking the
time independence of the walk operators whilst keeping their spatial
translation invariance [13, 23, 24, 25, 26, 27, 28]. On the other hand,
spatial disorder in QWs can break the lattice constancy of the walk operators
without changing their time invariance [29, 30, 31, 32, 33, 34]. A third type
of disorder – the focus of our work – involves breaking both space and time
constancy of the walk operators. In a comprehensive study [35], it was shown
that spatio-temporal randomness embedded in the coin operator promotes the
transition to a classical-like spreading in quantum walks. Other works –
namely Refs. [10, 36] – confirmed such findings, where it was noticed that
random spatio-temporal disorder leads to a slowing down in the wavepacket
spreading. In Ref. [37], it was found that carefully devised nonrandom space-
time disorder can be used as a probability distributions universal generator.
Particularly, in Ref. [38], it was introduced a method to create spatio-
temporal dependence in the coin operator that is able to display an exact
classical-like binomial distribution without randomness. Those works naturally
lead to the question of what dynamics should emerge if the coin operator is
embedded in a deterministic space-time disorder tailored with nontrivial
patterns such as fractals.

Although fractals can emerge as the output of a QW dynamics [39, 40], we are
going to deal with fractals the other way round, namely as the input in the
modelling process and relate it to previous works as follows: in Ref. [41, 42,
43, 44, 45] the focus was the problem of quantum search and in Ref. [46] it
was conducted an experimental work on continuous-time quantum walks in fractal
photonic lattices whereas in Ref. [47] the goal was to study the scaling of
the spreading by considering a flip-flop shift operator with a four-
dimensional Grover coin. Complementary, the authors of Ref. [48] studied a
discrete-time quantum walk (DTQW) with a two-dimensional coin operator with
space dependence prescribed by the Cantor set and Sierpinski fractals were
considered in a protocol of a continuous-time quantum walks (CTQWs) [49, 50].
Explicitly, we take a different road by considering a theoretical proposal
very close to recent experimental setups by considering a discrete-time QW
with two-state coins where the fractals model a spatio-temporal alternation
between generalized Hadamard and Fourier coin operators.

##  II Model

###  II.1 1D+1 quantum walk

At a given time $t\in\mathbb{N}$ we can write the full wave function
$\Psi_{t}$ as

$\displaystyle\Psi_{t}=\sum_{x\in\mathbb{Z}}\ket{x}\otimes\psi_{t}(x)$ (1)
$\displaystyle\psi_{t}(x)=\psi_{t}^{U}(x)\ket{U}+\psi_{t}^{D}(x)\ket{D},$ (2)

where $\psi_{t}^{U,D}(x)$ are the time- and site-dependent amplitudes of
probability associated with the internal degree of freedom $c=\\{U,D\\}$.

The temporal evolution proceeds with the iterative application of the operator
$\widehat{W}$ as

$\displaystyle\Psi_{t}\xrightarrow{\widehat{W}_{t}}\Psi_{t+1},$ (3)
$\displaystyle\widehat{W}_{t}=\widehat{T}(\widehat{R}_{t}\otimes\mathcal{I}_{\mathbb{Z}}),$
(4)

with the identity operator
$\mathcal{I}_{\mathbb{Z}}=\sum_{x\in\mathbb{Z}}\outerproduct{x}{x}$. In
addition:

  * •

the coin operator that leaves each internal state $\\{U\text{ or }D\\}$ into a
weighted superposition
$\widehat{R}_{t}=\sum_{x}\outerproduct{x}{x}\otimes\widehat{R}_{t}(x)$, with

$\displaystyle\widehat{R}_{t}(x):\begin{cases}|x,U\rangle\rightarrow
c_{UU}(x,t)|x,U\rangle+c_{DU}(x,t)|x,D\rangle\\\ |x,D\rangle\rightarrow
c_{UD}(x,t)|x,U\rangle+c_{DD}(x,t)|x,D\rangle\end{cases},$ (5)

where $c_{ij}(x,t)$ are the elements of the rotation matrices playing the role
of quantum walk coins used in our study: the Hadamard coin, $\widehat{C}_{H}$
, and the Fourier coin, $\widehat{C}_{F}$, mathematically described by

$\widehat{C}_{H}=\begin{pmatrix}\cos\theta_{H}&\sin\theta_{H}\\\
\sin\theta_{H}&-\cos\theta_{H}\end{pmatrix},\quad\quad\widehat{C}_{F}=\begin{pmatrix}\cos\theta_{F}&i\sin\theta_{F}\\\
i\sin\theta_{F}&\cos\theta_{F}\end{pmatrix};$ (6)

  * •

the spin-dependent displacement operator that splits the wavepacket towards
$\pm x$

$\displaystyle\widehat{T}:\begin{cases}|x,U\rangle\rightarrow|x+1,U\rangle\\\
|x,D\rangle\rightarrow|x-1,D\rangle\end{cases}.$ (7)

###  II.2 The Sierpinski gasket concatenation of optical elements

As already mentioned, herein we centre our attention on the impact of one of
the canonical instances of self-similarity, the Sierpinski gasket (SG). It
corresponds to a fractal with (equilateral) triangle external contour shape
obtained by means of the recursive division of a first triangle into ever
smaller triangles. Geometrically, it corresponds to starting with an
equilateral triangle that is divided into four congruent smaller equilateral
triangles with half the size and the removal of the central triangle.
Analyzing from a zooming out perspective, we understand that in doubling the
size of a triangle the Sierpinski generator creates three replicas of the
previous triangle. That allows finding the fractal dimension of the pattern,

$d_{f}=\frac{\ln 3}{\ln 2}=1.58\ldots,$ (8)

which is less than the Lesbegue covering dimension $D=2$ but greater than the
dimension of a line, $d=1$.

Scale invariance mainly splits into two classes: when we are dealing with
homogeneous dimensions and when the fractal spans through different dimensions
such as space and time. In the former case we refer to self-invariance whereas
in the latter we are dealing with self-affinity. In respect thereof, the
Sierpinski gasket is obtained not only using geometrical arguments, as we have
mentioned, but it is the space-time outcome of several dynamical models as
well. Besides the celebrated Rule 90 of cellular automata [51], as well as a
plethora of other algorithms [52], the space-time SG can be obtained by means
of the next fast modular arithmetic rule [53]

$b_{t}(x)=\left[b_{t-1}(x-1)+b_{t-1}(x+1)\right]\mod 2,$ (9)

where $b_{t}(x)$ is a site- and time-dependent binary $\\{0,1\\}$ variable. At
first glance, the Eq. (9) seems to be linear, but a closer look reveals that
the modular arithmetic that constrains $b_{t}(x)$ to either $0$ or $1$
introduces nonlinearity. In Fig. 1, we present $t=50$ generations of the SG.
From Eq. (9) it is clear that inside the cone $-t\leq x\leq t$ there is only
$0-1$.

Figure 1: Sierpinski gasket (SG). In this binary carpet yellow indicates the
value $1$ that corresponds to the application of the operator
$\widehat{C}_{H}$. The other sites inside the largest yellow triangle indicate
the value $0$ that corresponds to the application of the operator
$\widehat{C}_{F}$. In this way it is possible to build a fractal assembly of
optical elements.

We are now in the condition of using the self-similarity seasoning of our work
into the quantum walk dynamics by defining the quantum operator

$\widehat{R}_{t}(x)\equiv
b_{t}(x)\widehat{C}_{H}+(1-b_{t}(x))\widehat{C}_{F}.$ (10)

This equation reads as whenever a given site has $b_{t}(x)=0$ we set
$\widehat{C}_{F}$ and whenever $b_{t}(x)=1$ we set $\widehat{C}_{H}$. That
establishes an important difference between our model and other photonic
transport work; specifically, in the system we study the fractal element
resides within the coin operator selector whereas in other cases the fractal
element has to do with the physical geometry of the grid (see e.g. Ref. [54]).
Overall, we understand the present model as a space-time disordered model and
such traits will be taken into consideration in the interpretation of the
results.

With respect to the initial condition, we have placed the starting point of
the QW at the same initial location of the seed in the fractals. For the
boundary conditions, we have established that for each $t$ we worked with an
augmented chain of positions so that the quantum walker was not able to reach
the boundaries. Instances of the quantum walk as established by our dynamics
are provided in Fig. 2.

Figure 2: Space-time evolution of the normalized probability distribution
$P_{t}(x)/P_{t}^{max}$ where the brightness denotes the magnitude of this
quantity. Quantum carpets for the disorder-free setting (upper) disordered
cases (bottom). We set with $t_{max}=100$ and
$\theta=\\{15^{o},45^{o},75^{o}\\}$.

Taking a friendly experimental setup, we consider the proposals in Refs. [55,
56], where the role of the coin operators is played by beam splitters and the
internal degree of freedom $c=\\{U,D\\}$ is the polarization of the photons.
While $\widehat{C}_{F}$ leaves each state in a superposition with an
additional phase, $\widehat{C}_{H}$ produces a superposition of states without
such extra phase. If $\theta=\pi/4$, we recover the standard Hadamard and
Fourier coin operators that can be implemented with an unbiased beam splitter.

Figure 3: Temporal behaviour of the spreading measure $m_{2}$ in scale
$\log_{10}x\log_{10}$. Data points are the results from simulations and the
lines are linear fittings. Figure 4: Diagram $\alpha$ vs $\theta$. We use
$t_{max}=5\times 10^{5}$ to estimate $\alpha$ from $m_{2}\sim t^{\alpha}$. The
case $\theta=m\pi/2$ with $m=\\{1,3,5\\}$ is not shown because it leads to a
bounded behaviour that leads to $\alpha=0$ (non-spreading) corresponding to
specious localization.

##  III Results

In this section, we will present our results regarding the transport
properties of the quantum walk, as well as the degree of interference,
entanglement entropy, and trace distance between time-consecutive coin states.

###  III.1 Spreading

At a given instant $t$, we compute the probability

$P_{t}(x)=|\psi_{t}^{D}(x)|^{2}+|\psi_{t}^{U}(x)|^{2}\mbox{ .}$ (11)

That is of experimental interest for it allows obtaining preliminary
assessment of the wavepacket transport properties. For each $t$, we also have
computed $P_{t}^{max}$ that is the maximum of $P_{t}(x)$ over the chain.

In Fig. 2, we plot the normalized profile $P_{t}(x)/P_{t}^{\max}$ for the
typical angles $\theta=\\{15^{o},45^{o},75^{o}\\}$, where we immediately note
that the standard disorder free QW (upper panel) already displays a triangle-
like shape. That signals the overwhelming majority of the flux of probability
is near the edges. Such a feature accommodates well our proposal for
implementing QW through the Sierpinski triangle (ST). In the disorder-free
setting, we also see that the smaller the value of $\theta$, the broader the
wavepacket. That is because the mathematical structure of the coin operators
favors the terms $c_{UU}$ and $c_{DD}$ in Eq. (10), which are related to
spreading. However, this property is modified when the flux of probability
passes through the SG. The profiles of $P_{t}(x)$ are neither bimodal-like
distribution nor a Gaussian-like shape. In other words, the intrinsic
nonlinearities of the fractals leaves peculiar fingerprints in the wavepacket.
Even though the SG is symmetric, there are clear asymmetries in the $P_{t}(x)$
due to the alternation between the coin operators that give different phases
to the local spinor Eq. (2).

The qualitative information obtained by the visual inspection of the quantum
carpets presented Fig. 2 is quantitatively boosted by computing the second
statistical moment

$m_{2}(t)=\overline{x^{2}}_{t}=\sum_{x}x^{2}P_{t}(x),$ (12)

whence we compute the scaling exponent

$\alpha=\lim_{t\to\infty}\frac{\log m_{2}(t)}{\log t},$ (13)

that defines the sort of diffusion the system presents. For $0<\alpha<1$, a
system is sub-diffusive; $\alpha=1$, corresponds to standard diffusion
(similar to the classical random walk); when $1<\alpha<2$, it exhibits
superdiffusion; for $\alpha=2$, we have ballistic diffusion like the standard
QW and for $\alpha>2$ a system runs in a superballistic regime with the
particular value $\alpha=3$ corresponding to hiperballistic diffusion [17].

In Fig. 3, we understand that the nonrandom fractal disorder is not able to
decrease the scaling exponent to the level of the diffusive behavior
$\alpha=1$, as happens for random spatiotemporal assembly of coin operations
[35]. Therefrom, we perceive a nonmonotonic and nonsmooth dependence of
$\alpha$ with $\theta$ that arises from substantial enhancement of the
interference between the paths along the time evolution caused by the fractal
infrastructure as previously shown in Fig. 2. This nonmonotonicity is an
intriguing feature if we take into account the overall regularity in the
patterns through the SG.

The behavior of $\alpha$ with $\theta$ is shown in Fig. 4. In the standard QW
it is well-known that changing $\theta$ alters the particular features of
$m_{2}(t)$ whilst keeping the scaling exponent $\alpha$ invariant. Here, our
results show that by tuning $\theta$ it is possible to slightly change (and
with statistical significance) the level of superdiffusivity of the quantum
walker, still below the ballistic regime though. This novel result – the
dependence of $\alpha$ with $\theta$ – is absent in all the previous endeavors
working with coined QWs on fractals [41, 42, 43, 44, 45, 47, 48, 49, 50].
Indeed, with our proposal it is possible to adjust
$\theta=m\,\pi\,\,\,(m\in\mathbb{N})$ in order to obtain a ballistic
spreading, $\alpha=2$. On the other hand, minimal superdiffusion of the
wavepacket is achieved when $\theta=m\,\pi\pm\pi/12.$

Our setup can be interpreted from the perspective of the physics of disordered
systems [57, 3]. Space-time disorder is not only of theoretical interest, but
it has been shown in optical systems that temporally fluctuating spatial
disorder is able to produce a remarkably new phenomenology as well [58]. The
results in Figs. 3 and 4 highlight the emergence of a new mechanism to access
the superdiffusive regime, which is an important class of anomalous diffusion
on its own [59].

On the whole, the results depicted in Figs. 2-4 show that a fractal assembly
of coin operators leads to a rich phenomenology in terms of wavepacket
spreading, even when the coin operators differ only by mere phase factors as
explicit in the operators $\widehat{C}_{H}$ and $\widehat{C}_{F}$ shown in Eq.
(6).

###  III.2 Degree of interference

As a means to study with more detail the interference effects between the
different paths that come up with different dynamics of the walker in this
setup, we have also investigated the degree of interference. The degree of
interference is defined as the norm of the quantity responsible for the
interference in the visibility pattern. The visibility, in turn, suitable for
probability waves is defined as

$\mathcal{V}(x,t)\equiv\frac{\mbox{max}(P_{t}(x))-\mbox{min}(P_{t}(x))}{\mbox{max}(P_{t}(x))+\mbox{min}(P_{t}(x))}\mbox{
.}$ (14)

Then, the degree of interference at each position and time step is defined as
the norm of the numerator of the visibility. For further details on how we
calculated this quantity in the quantum walk with fractal disorder see
Appendix A.1.

$\mu(x,t)=|\mbox{max}(P_{t}(x))-\mbox{min}(P_{t}(x))|\mbox{ .}$ (15)

In Fig. 5, we have the degree of interference at each position as a function
of time for the same evolution represented at Fig. 2 in the second row, i.e.
the disordered case. There, it is possible to see that the interference
pattern evolution closely follows that of the probability distribution; in
other words, the plot tells us that at each site the interference occurs
between its immediate neighbours, as one would expect from the short-range
coupling of the quantum walk step evolution.

Figure 5: Degree of interference spatial-temporal evolution in the walks
following the same parameters of the lower panels (disordered cases) in Fig.
2, (a) $\theta_{H}=\theta_{F}=15^{\circ}$, (b)
$\theta_{H}=\theta_{F}=45^{\circ}$ and (c) $\theta_{H}=\theta_{F}=75^{\circ}$.

By investigating how the degree of interference behaves as a function of the
coin operators Eq. (6), we also found some interesting patterns that resembles
the fractal structure used to implement the disordered coin operator Eq. (10)
as shown in Fig. 6. These patterns arise when we set the Fourier coin operator
to be equal to the identity, i.e. $\theta_{F}=0$, and we start with a coin
initial state parameterized by the angles on the Bloch sphere

$\psi_{0}(0)=\cos\frac{\gamma}{2}\ket{U}+e^{i\phi}\sin\frac{\gamma}{2}\ket{D}\mbox{
,}$ (16)

with angles $\gamma=\pi/2$ and $\phi=0$.

One can understand why the pattern resembles a SG fractal by recalling the
definition of the degree of interference, Eq. (A.1) and Eq. (32), with the
arithmetic rule used to generate the pattern Eq. (9). From Eq. A.1 we note
that for the degree of interference to be non-zero at a given position at
least one of the coin operators applied in its immediate neighbours positions
in the previous time instant must be different than the identity operator
$C_{F}(\theta_{F}=0)$, i.e. $R^{UD}_{t-1}\neq 0$ and/or $R^{DU}_{t-1}\neq 0$.
This matches the sum modulus two where, if $b_{t-1}(x\pm 1)$ are both zero
then $b_{t}(x)=0$. Moreover, one must have a coin in a superposition state,
that is $c_{U}(x+1,t-1),c_{D}(x+1,t-1)\neq 0$ and/or
$c_{U}(x-1,t-1),c_{D}(x-1,t-1)\neq 0$. The lack of a superposition can come
from the application of an identity operator in a state previous state or the
application of $C_{H}$ in one coin at $x+1$ or $x-1$. That means when we have
$C_{H}$ acting on both the coin states at $x\pm 1$ the interference degree can
be zero or non-zero, based on the prior states at these positions, and when
$C_{H}$ acts on either of them the outcome is the same. That differs from the
sum modulus two rule Eq. 9 that states when both $b_{t-1}(x\pm 1)$ are equal
to one then $b_{t}(x)=0$ and when only one of them is equal to one,
$b_{t}(x)=1$. Still, we can affirm that the pattern closely resembles the SG
fractal for the reason that the majority of points are zero, imposing that the
intermediate points in the future are also zero. We also emphasize the
dependence of the pattern on the coin initial state (compare Fig. 6(a) with
fig. 6(f)). Given that one has a small $\theta_{H}$, then one of the
components of the coin state in a given position can be made smaller resulting
in a smaller interference for the points that should be zero and a not for the
ones that should not. At Appendix A.2 we show an analytical calculation of the
initial steps of the degree of interference evolution at Fig. 6 for any
$\theta_{H}$.

Figure 6: Degree of interference spatial-temporal evolution in walks with
$\theta_{F}=0^{\circ}$ initially localized in the origin with coin initial
state parameters $\gamma=\pi/2$, $\phi=0$. In (a) $\theta_{H}=5^{\circ}$, (b)
$\theta_{H}=15^{\circ}$, (c) $\theta_{H}=30^{\circ}$, (d)
$\theta_{H}=45^{\circ}$, (e) $\theta_{H}=60^{\circ}$ and (f)
$\theta_{H}=85^{\circ}$.

###  III.3 Entanglement entropy

To quantify the amount of entanglement generated through the position-coin
system we have employed the entanglement entropy,

$S_{E}\equiv-\mbox{tr}(\rho\log\rho)\mbox{ ,}$ (17)

where $\rho$ is the density operator of the system. Here, we focus only on the
entropy of the coin subsystem. As we are treating a two-level system, the
entanglement entropy varies between $0\leq S_{E}\leq 1$, with zero
corresponding to a separable state and with one to a fully entangled state,
and it is given by

$S_{E}=-\lambda_{+}\log\lambda_{+}-\lambda_{-}\log\lambda_{-}\mbox{ ,}$ (18)

with $\lambda_{\pm}$ being the eigenvalues of the coin density operator
$\rho_{c}$.

Since the evolution of the coin entanglement entropy in a quantum walk often
presents an initial increase to then stabilize around a given saturate value,
we consider the average entanglement entropy in the asymptotic regime, $t\geq
t_{0}\gg 1$. The time evolution of the entanglement entropy also depends on
the coin initial state and coin operator used. Therefore, we take the initial
time after which we can consider the regime as being a quasi-stationary one,
$t_{0}$, based on each evolution.

We start by analyzing the coin entanglement entropy as a function of the coin
operator parameters Eq. (6), $\theta_{H}$ and $\theta_{F}$, at Fig. 7. The
coin initial state considered is the one with $\phi=\gamma=\pi/2$ in the qubit
Bloch-sphere Eq. (16) and localized at the origin. Figure 7 show us that the
entanglement entropy reaches its maximum value for the set of parameters with
$\theta_{F}\lesssim 85^{\circ}$, which is almost the entire set of parameters.
For $\theta_{F}=90^{\circ}$, the maximum value is obtained with
$\theta_{H}=0^{\circ}$, after which its start to decay and reach the minimum
value of $\langle S_{E}\rangle_{t}\approx 0.498$ with $\theta_{H}=90^{\circ}$.
This is an interesting feature since it does not happen in the standard
quantum walk evolution. Moreover, a similar effect was reported for other
types of disordered quantum walk, with the disorder also in the coin operation
and in the shift operator [9, 36, 18, 33]. However, all of them impose a
random quantum walk evolution whereas in the present model the evolution is
deterministic. The presence of temporal disorder in the evolution of those
walks is the key factor leading to the same phenomenon, as was asserted in
[36, 22].

Figure 7: Average coin entanglement entropy in the asymptotic regime as a
function of the parameters of the two possible coin operators Eq. 6,
$\theta_{H}(\mbox{deg})$ and $\theta_{F}(\mbox{deg})$ respectively. The
initial state considered was the one with $\phi=\gamma=\pi/2$ with the initial
position in the origin, $x_{0}=0$.

When we have fixed the coin operators assuming $\theta_{H}=\theta_{F}=\pi/4$,
we observed that by varying the coin initial state angles the average
entanglement entropy does not change for the entire set of parameters. That is
indicative of the robustness of the generation of entanglement between the
coin and position states with regard to changes in the initial state [19].

###  III.4 Asymptotic regime

Next, we probe how the coin state evolves towards its time asymptotic regime.
To that, we employ the trace distance measure

$D(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_{1}\mbox{ ,}$ (19)

where $\|A\|_{1}=\mbox{tr}(\sqrt{A^{\dagger}A})$. If a quantum system is
evolving in a stationary regime, its density operator does not changes with
time, $\rho(t+\Delta t)=\rho(t)$, therefore the trace distance between
$\rho(t+\Delta t)$ and $\rho(t)$ would return zero. In this way we see how the
trace distance between two time-consecutive states evolves.

Figure 8: Trace distance between time-consecutive coin states as a function of
time for (a) different $\theta_{H}$, with $\theta_{F}=\pi/4$, and (b)
different $\theta_{F}$ with $\theta_{H}=\pi/4$. The different angles used in
each plot follows the colors and patterns $0$ (blue dotted), $\pi/6$ (black
circled), $\pi/4$ (red crossed) and $\pi/2$ (orange up triangle). In each
evolution a localized initial state was considered with coin initial state
parameterized by $\phi=\gamma=\pi/2$. The insets indicates the log-log graph
of the same plots with linear fittings of the form $-\beta\ln(t)+\alpha$. For
(a) we have the following angular and linear coefficients: $\beta=0.00\pm
0.00$, $\alpha=0.00\pm 0.00$ (blue dotted); $\beta=0.45\pm 0.04$,
$\alpha=-0.4\pm 0.3$ (black circled); $\beta=0.57\pm 0.04$, $\alpha=0.5\pm
0.3$ (red crossed); $\beta=0.49\pm 0.04$, $\alpha=-0.1\pm 0.3$ (orange up
triangle). For (b) we have the following ones: $\beta=0.59\pm 0.05$,
$\alpha=0.2\pm 0.03$ (blue dotted); $\beta=0.39\pm 0.04$, $\alpha=-1.0\pm 0.3$
(black circled); $\beta=0.57\pm 0.04$, $\alpha=0.5\pm 0.3$ (red crossed);
$\beta=0.08\pm 0.04$, $\alpha=-0.39\pm 0.06$ (orange up triangle).

Initially, we fix one of the parameters of the coin operator
$\theta_{H}=\pi/4$ in Fig. 8(a). There, we can see that for all $\theta_{F}$
but $\theta_{F}=0$, the trace distance decays essentially in the same way. The
inset gives us the log-log graph of the same curves with the respective linear
fittings yielding the decay exponents if $D(\rho_{c}(t+1),\rho_{c}(t))\propto
t^{-\beta}$, confirming this observation.

When we fix the other coin operator with $\theta_{F}=\pi/4$ Fig. 8(b), the
evolution with $\theta_{H}=0$ does not decay as fast as in the case where
$\theta_{F}=0$ in Fig. 8(a), with $\beta\approx 0.59$. For the remaining
angles, the decay rates are essentially the same with the exception of
$\theta_{H}=\pi/2$ that gives us $\beta\approx 0.08$. In this angle something
interesting happens, the trace distance does not decay, on average,
monotonically, having intervals of constant evolution with some increases in
between.

Fig. 9 shows the time evolution of the trace distance between two consecutive
coin states when we set the coin operators to be the those with
$\theta_{H}=\theta_{F}=\pi/4$ and change the coin initial state. In Fig. 9(a),
the polar angle is set $\gamma=\pi/2$ while we change the phase angle, $\phi$.
We see that changes in this parameter does not changes the trace distance
decay behavior, on average, giving us essentially the same decay exponent
$\beta\approx 0.66$ for $\phi=\\{0,\pi/6,\pi/4\\}$ with the most divergent,
but still very close, $\beta\approx 0.57$ for $\phi=\pi/2$. Setting
$\phi=\pi/2$ and changing $\gamma$, Fig. 9(b) demonstrates that changes in
this parameter also does not change the trace distance decay significantly,
now even when we set $\gamma=\pi/2$.

Figure 9: Trace distance between time-consecutive coin states as a function of
time for different angles of the coin initial state Bloch sphere $\phi$ (a),
with $\gamma=\pi/2$, and different $\gamma$(b) with $\phi=\pi/2$ (see eq. 16).
The different angles used in each plot follows the colors and patterns
$0$(blue dotted), $\pi/6$(black circled), $\pi/4$(red crossed) and
$\pi/2$(orange up triangle). In each evolution a localized initial state was
considered with coin operators given by $\theta_{H}=\theta_{F}=\pi/4$. The
insets indicates the log-log graph of the same plots with linear fittings of
the form $-\beta\ln(t)+\alpha$. For (a) we have the following angular and
linear coefficients: $\beta=0.67\pm 0.04$, $\alpha=1.1\pm 0.3$ (blue dotted);
$\beta=0.66\pm 0.04$, $\alpha=1.0\pm 0.3$ (black circled); $\beta=0.65\pm
0.04$, $\alpha=1.0\pm 0.2$ (red crossed); $\beta=0.57\pm 0.04$, $\alpha=0.5\pm
0.3$ (orange up triangle). For (b) we have the following ones: $\beta=0.48\pm
0.05$, $\alpha=-0.1\pm 0.3$ (blue dotted); $\beta=0.51\pm 0.04$,
$\alpha=0.0\pm 0.3$ (black circled); $\beta=0.54\pm 0.04$, $\alpha=0.2\pm 0.3$
(red crossed); $\beta=0.57\pm 0.04$, $\alpha=0.5\pm 0.3$ (orange up triangle).

These results tell us that despite our quantum walk model being disordered, it
goes to the quasi-stationary regime essentially in the same way as if we had
fixed the coin operators used and changed the coin initial state and vice-
versa, excluding the cases where the parameters are near to $\pi/2$. This is
strongly indicative that this property derives from the quantum walk protocol
and only has a small limited dependence on the initial conditions. In the
standard Hadamard walk, the pace at which the coin state converges to the
stationary regime depends on the initial state and coin operator used. For the
initially localized and balanced coin state the decay exponent is $\beta=3/2$,
as opposed to the average decay exponent that we obtained $\beta\approx 0.55$.
Comparing with other types of disordered quantum walks, we see that the
introduction of a fractal disorder in the coin operator induces a faster
transition to the stationary regime than when one introduces a dynamically and
fluctuating randomness [9, 36], where the decay exponent $\beta=1/4$. The
decay exponent of the walk with fractal disorder is also greater than the ones
in the generalized elephant quantum walk, where the randomness is on the step
sizes and selected according to the $q$-exponential, when its parameter $q$ is
less than one but greater than half, $\beta(q)\approx 1/4$, but in the uniform
distribution limit $q\rightarrow\infty$ it becomes smaller, where
$\beta(q)\approx 0.76$.

##  IV Discussion

Within the context of Quantum Walks, the disorder of our SG case befalls
twofold: every site of the grid is associated to a given coin protocol that is
updated at each time step according to the dynamics Eq. (9). On the one hand,
it is know that the slightest spatial disorder induces an absence of diffusion
and the subsequent localization of the quantum walker wave; on the other hand,
temporal randomness can actually enhance spreading of the wavepacket,
especially when such changes occur at typically long time scales [60]. Thus,
taking into consideration this model has got spatial disorder, we would expect
a dramatic decrease of the exponent $\alpha$, which is not the case. As shown,
the system preserves a level of diffusion not far from the standard ballistic
behavior with mild dependence on the angle of the coin. In order to understand
that, we have inspected the local statistics of coin operator updates.
Excluding the central grid site, $x=0$, we have verified that the average time
for coin change, $\tau_{x}$, is never less than a two-digit figure, with that
average time increasing with $|x|$, namely $\tau_{5}=74$, $\tau_{30}=146$,
$\tau_{100}=219$. Combining both elements, we understand the long
characteristic scale of the coin dynamics manages to zero out to a large
extent the likely localization that would be induced by static disorder. At
the same time, we have verified the effect of spatial disorder in augmenting
quantum entanglement [33] (excluding the particular coin angles such as
$\theta=90$), which is lessened by the temporal disorder effects of the
dynamics Eq. (9). In other words, we have verified that deterministic fractal
disorder enhances quantum entanglement without achieving the maximal value
though. It is precisely that deterministic nature of the self-similar fractal
dynamics that reins quantum entanglement strengthening in, as previously
hinted [9]. That said, we can particularly envisage this type of architecture
as a strong candidate for tailored problems for which one aims at increasing
entanglement whilst ensuring high level of spreading, e.g., search algorithms
in a quantum cryptography environment [61].

##  V Final remarks

We have studied quantum walks with spatiotemporal fractal disorder, employing
the paradigmatic Sierpinski gasket as a prototype. Throughout our
investigation, we have explored multiple facets of the quantum walk, providing
insights into its transport properties, degree of interference, entanglement
entropy, and trace distance between time-consecutive coin states.

The employment of fractal principles in the creation of patterned structures
is an avenue of research to be studied from both fundamental and applied
perspectives. In this manuscript, we have provided a novel application of
fractal geometry, namely, we showed that the Sierpinski gasket can be employed
to build photonic architectures aimed at realizing highly nontrivial site- and
time-dependent quantum dynamics. While fractal structures were previously
addressed in the scope of QWs [41, 42, 43, 44, 45, 46, 47, 48, 49, 50] those
efforts had different frameworks and goals than what we presented here.
Accordingly, with this work where the disorder is deterministically
established, we have fulfilled a gap on the characterization of the impact of
different sorts of disorder on the properties of a quantum walk.

An important advantage of this work in relation to previous proposals [42, 43,
47, 44, 45] lies in the fact that the versions of the discrete-time QWs on
fractals studied in those works require the use of high-dimensional coin
operators that are harder to implement. Our protocol simply considers a two-
state coin. Specifically, in Ref. [42] it was used a $d$-dimensional coin
Hilbert space. In Refs. [43, 47, 44, 45] their flip-flop walk requires a four-
dimensional Grover coin. Thus, the experimental feasibility of the present QW
dynamics presented a clear implementable advantage; furthermore, it can be
realized with the state-of-art photonics technology. Still, with minor changes
in the apparatus described in Ref. [56], it is possible to implement the
fractal concatenation of optical elements where the role of time $t$ is played
by the propagation direction. The nonrandom character of our proposal has the
additional feasibility of not requiring an extra sampling processing over
random disorder.

The present work offers new alternatives for bringing the fields of fractal
and photonics even more closer [62, 63]. Still, this contribution brings fresh
insights into the physics of disordered photonics [64]. An interesting open
problem to study is how the wavepacket properties behave when a multifractal
structure is employed to incorporate disorder in the system.

###### Acknowledgements.

CBN acknowledges the prospective funding from the Department of Physics of
Stockholm University. SMDQ thanks CNPq (Grant 302348/2022-0) for financial
support. DOSP acknowledges the support by the Brazilian funding agencies CNPq
(Grant No. 304891/2022-3), FAPESP (Grant No. 2017/03727-0), and the Brazilian
National Institute of Science and Technology of Quantum Information (INCT/IQ).

##  Appendix A The degree of interference

###  A.1 Definition

In the following, we present the way we have calculated the degree of
interference in connection with the proposed quantum walk model. As we
mentioned previously, the degree of interference is defined as the norm of the
quantity responsible for the interference in the visibility. The visibility,
suitable for probability waves, is

$\mathcal{V}=\frac{P_{max}-P_{min}}{P_{max}+P_{min}}\mbox{ .}$ (20)

Then, the degree of interference at each position and time step is defined as
the norm of the numerator of the visibility

$\mu(x,t)=|P_{max}(x,t)-P_{min}(x,t)|\mbox{ .}$ (21)

Let $\ket{\psi(t)}$ be the walker state at time $t$ expanded as

$\Psi(t)=\sum_{x}\ket{x}\otimes(c_{U}(x,t)\ket{U}+c_{D}(x,t)\ket{D})\mbox{ .}$
(22)

The coin operator $\hat{R}_{t}$ in the quantum walk is defined following the
Sierpinski Gasket fractal rule so that

$\hat{R}_{t}=\sum_{x}\outerproduct{x}{x}\otimes\hat{R}_{t}(x)\mbox{ ,}$ (23)

with $\hat{R}_{t}(x)$ given by eq. (10).

Therefore, the recursive maps that provides the walker state coefficients’
time evolution now are going to include the space-time dependent coin operator
factors. Let

$\hat{R}_{t}(x)=R_{t}^{UU}(x)\outerproduct{U}{U}+R_{t}^{UD}(x)\outerproduct{U}{D}+R_{t}^{DU}(x)\outerproduct{D}{U}+R_{t}^{DD}(x)\outerproduct{D}{D}\mbox{
,}$ (24)

consequently, the recursive map for the state coefficients is going to be,
assigning the up state to a displacement to the right

$\displaystyle c_{U}(x,t+1)$
$\displaystyle=R_{t}^{UU}(x-1)c_{U}(x-1,t)+R_{t}^{UD}(x-1)c_{D}(x-1,t)$ (25)
$\displaystyle c_{D}(x,t+1)$
$\displaystyle=R_{t}^{DU}(x+1)c_{U}(x+1,t)+R_{t}^{DD}(x+1)c_{D}(x+1,t)\mbox{
.}$ (26)

The position probability distribution is given by the square norm of the
walker state’s coefficients

$P(x,t)=|c_{U}(x,t)|^{2}+|c_{D}(x,t)|^{2}\mbox{ .}$ (27)

Using the recursive relations Eqs.(25) (and) (26), we find

$\displaystyle P(x,t)$
$\displaystyle=|R_{t-1}^{UU}(x-1)c_{U}(x-1,t-1)+R_{t-1}^{UD}(x-1)c_{D}(x-1,t-1)|^{2}+$
$\displaystyle+|R_{t-1}^{DU}(x+1)c_{U}(x+1,t-1)+R_{t-1}^{DD}(x+1)c_{D}(x+1,t-1)|^{2}$
(28)
$\displaystyle=|R_{t-1}^{UU}(x-1)c_{U}(x-1,t-1)|^{2}+|R_{t-1}^{DD}(x+1)c_{D}(x+1,t-1)|^{2}|R_{t-1}^{UD}(x-1)c_{D}(x-1,t-1)|^{2}+$
$\displaystyle+|R_{t-1}^{DU}(x+1)c_{U}(x+1,t-1)|^{2}+|R_{t-1}^{UD}(x-1)c_{D}(x-1,t-1)|^{2}+$
$\displaystyle+R_{t-1}^{UU}(x-1)(R_{t-1}^{UD}(x-1))^{*}c_{U}(x-1,t-1)c_{D}(x-1,t-1)^{*}+$
$\displaystyle+R_{t-1}^{DU}(x+1)(R_{t-1}^{DD}(x+1))^{*}c_{U}(x+1,t-1)c_{D}(x+1,t-1)^{*}+\mbox{c.c.}$
(29)

Defining the complex part as a function $f(x,t)$:

$\displaystyle
f(x,t)=R_{t-1}^{UU}(x-1)(R_{t-1}^{UD}(x-1))^{*}c_{U}(x-1,t-1)c_{D}(x-1,t-1)^{*}+$
$\displaystyle+R_{t-1}^{DU}(x+1)(R_{t-1}^{DD}(x+1))^{*}c_{U}(x+1,t-1)c_{D}(x+1,t-1)^{*}$

the maximum probability and the minimum probability are going to be

$\displaystyle P_{\max}(x,t)$
$\displaystyle=|R_{t-1}^{UU}(x-1)c_{U}(x-1,t-1)|^{2}+|R_{t-1}^{UD}(x-1)c_{D}(x-1,t-1)|^{2}+$
$\displaystyle+|R_{t-1}^{DU}(x+1)c_{U}(x+1,t-1)|^{2}+|R_{t-1}^{DD}(x+1)c_{D}(x+1,t-1)|^{2}+$
$\displaystyle+f(x,t)+f^{*}(x,t)\mbox{ ,}$ (30)

$\displaystyle P_{\min}(x,t)$
$\displaystyle=|R_{t-1}^{UU}(x-1)c_{U}(x-1,t-1)|^{2}+|R_{t-1}^{UD}(x-1)c_{D}(x-1,t-1)|^{2}+$
$\displaystyle+|R_{t-1}^{DU}(x+1)c_{U}(x+1,t-1)|^{2}+|R_{t-1}^{DD}(x+1)c_{D}(x+1,t-1)|^{2}-f(x,t)-f^{*}(x,t)\mbox{
.}$ (31)

Therefore, the degree of interference will be given by

$\mu_{x,t}=|4\mbox{Re}(f(x,t))|\mbox{.}$ (32)

###  A.2 Degree of interference evolution

In this section, we move on to calculate the initial time steps of the
evolution used to plot the degree of interference patterns in fig. 6. In these
evolution the coin operators used are $\hat{C}_{H}(\theta_{H})$ and the
identity operator, $\hat{C}_{F}(\theta_{F}=0)=\mathbb{I}_{2\times 2}$.

The initial state used is:

$\Psi_{0}=\ket{0}\otimes\frac{\ket{\uparrow}+\ket{\downarrow}}{\sqrt{2}}\mbox{
.}$ (33)

At time $t=0$ the fractal pattern tells us that the coin operator is equal to
$\hat{C}_{H}$ at $x=0$ and the identity for $x\neq 0$. Therefore, after the
application of the unitary operator we are going to have

$\Psi_{1}=\frac{(\cos\theta_{H}+\sin\theta_{H})}{\sqrt{2}}\ket{+1,\uparrow}+\frac{(\sin\theta_{H}-\cos\theta_{H})}{\sqrt{2}}\ket{-1,\downarrow}\mbox{
.}$ (34)

For $t=1$, the coin operator follows $\hat{R}_{1}=\\{\hat{C}_{H},x=\pm
1;\mathbb{I},x\neq\pm 1\\}$. Then, by applying it to the state following with
the application of the shift operator:

$\displaystyle\Psi_{2}$
$\displaystyle=\frac{\cos\theta_{H}}{\sqrt{2}}(\cos\theta_{H}+\sin\theta_{H})\ket{+2,\uparrow}+\frac{\sin\theta_{H}}{\sqrt{2}}\ket{0}\otimes[(\sin\theta_{H}-\cos\theta_{H})\ket{\uparrow}+(\cos\theta_{H}+\sin\theta_{H})\ket{\downarrow}]-$
$\displaystyle-\frac{\cos\theta_{H}}{\sqrt{2}}(\sin\theta_{H}-\cos\theta_{H})\ket{-2,\downarrow}\mbox{
.}$

When $t=2$ and $t=3$, $\hat{R}_{2}=\\{\hat{C}_{H},x=\pm 2;\mathbb{I},x\neq\pm
2\\}$ and $\hat{R}_{3}=\\{\hat{C}_{H},x=\pm 1,\pm 3;\mathbb{I},x\neq\pm 1,\pm
3\\}$. The states at these time are, respectively:

$\displaystyle\Psi_{3}$
$\displaystyle=\frac{\cos^{2}\theta_{H}}{\sqrt{2}}(\cos\theta_{H}+\sin\theta_{H})\ket{+3,\uparrow}+\frac{\cos^{2}\theta_{H}}{\sqrt{2}}(\sin\theta_{H}-\cos\theta_{H})\ket{-3,\downarrow}+$
$\displaystyle+\frac{\sin\theta_{H}}{\sqrt{2}}\ket{+1}\otimes[(\sin\theta_{H}-\cos\theta_{H})\ket{\uparrow}+\cos\theta_{H}(\cos\theta_{H}+\sin\theta_{H})\ket{\downarrow}]-$
$\displaystyle-\frac{\sin\theta_{H}}{\sqrt{2}}\ket{-1}\otimes[\cos\theta(\sin\theta_{H}-\cos\theta_{H})\ket{\uparrow}-(\cos\theta_{H}+\sin\theta_{H})\ket{\downarrow}]\mbox{
, }$

$\displaystyle\Psi_{4}$
$\displaystyle=\frac{\cos^{3}\theta_{H}}{\sqrt{2}}(\cos\theta_{H}+\sin\theta_{H})\ket{+4,\uparrow}-\frac{\cos^{3}\theta_{H}}{\sqrt{2}}(\sin\theta_{H}-\cos\theta_{H})\ket{-4,\downarrow}+$
$\displaystyle+\frac{\sin\theta_{H}}{\sqrt{2}}\cos\theta_{H}\ket{+2}\otimes\\{[\sin\theta_{H}(1+\sin\theta_{H})+\cos\theta_{H}(\sin\theta_{H}-1)]\ket{\uparrow}+\cos\theta_{H}(\cos\theta_{H}+\sin\theta_{H})\ket{\downarrow}\\}+$
$\displaystyle+\frac{\sin\theta_{H}}{\sqrt{2}}\ket{0}\otimes\\{[\sin\theta_{H}(\cos\theta_{H}+\sin\theta_{H})-\cos^{2}\theta_{H}(\sin\theta_{H}-\cos\theta_{H})]\ket{\uparrow}+$
$\displaystyle\qquad\qquad\qquad+[\sin\theta_{H}(\sin\theta_{H}-\cos\theta_{H})-\cos^{2}\theta_{H}(\cos\theta_{H}+\sin\theta_{H})]\ket{\downarrow}\\}+$
$\displaystyle+\frac{\sin\theta_{H}}{\sqrt{2}}\cos\theta_{H}\ket{-2}\otimes\\{\cos\theta_{H}(\sin\theta_{H}-\cos\theta_{H})\ket{\uparrow}-[\sin\theta_{H}(\sin\theta_{H}+1)+\cos\theta_{H}(\sin\theta_{H}-1)]\ket{\downarrow}\\}\mbox{.}$

Now we are going to use these states and the coin operators defined for each
time step to calculate the degree of interference. At $t=0$, as we do not have
a previous time step, we set the interference degree to be equal to one at the
origin:

$\mu(x,0)=\begin{cases}1,\;x=0\\\ 0,\;x\neq 0\end{cases}$ (35)

At $t=1$, we recall that for the degree of interference to be non-zero at a
given position the coin operator applied in its immediate neighbours must be
$\hat{C}_{H}$ and the coin has to be in a superposition as well. For $x=\pm 1$
these requirements are satisfied since the initial state located at the origin
is in a superposition and $\hat{C}_{H}$ was applied. Then, we obtain:

$\mu(x,1)=\begin{cases}2\sin\theta_{H}\cos\theta_{H},\;x=\pm 1\\\ 0,\;x\neq\pm
1\end{cases}$ (36)

In the next time step, $t=2$, given that the state is an entangled one, the
requirement of having a superposition of coin states in a position is not
fulfilled. Therefore, the interference degree is zero for every position:

$\mu(x,2)=0,\;\forall x\mbox{ .}$ (37)

Here we can already notice a difference between the degree of interference
pattern and the Sierpinski-Gasket fractal.

For $t=3$ the same happens, however for the reason that at $x=0$ the coin
operator used is the identity, while at $x=\pm 2$ $\hat{C}_{H}$ is used but
the coin states are not in superposition:

$\mu(x,3)=0,\;\forall x\mbox{ .}$ (38)

At $t=4$ we have to look at the walker state and the coin operator at $t=3$.
We notice that at $x=\pm 1$ the coin state is in a superposition and the
identity is not applied at these positions. Therefore, the only points that
the interference degree can be non-zero are at $x=0$ and $x=\pm 2$. Performing
the calculations, we find:

$\mu(x,4)=\begin{cases}4\cos^{2}\theta_{H}\sin^{3}\theta_{H}\cos
2\theta_{H},\;x=0\\\ 2\cos^{2}\theta_{H}\sin^{3}\theta_{H}\cos
2\theta_{H},\;x=\pm 2\\\ 0,\;x\neq 0,\pm 2\end{cases}$ (39)

Taking one more final step, $t=5$, we find once again that the interference
degree is zero for every position given that identity operator is applied to
every position but at $x=\pm 4$, however the coin state at these positions are
not in superposition:

$\mu(x,5)=0,\;\forall x\mbox{ .}$ (40)

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