# Quantum Walks in Weak Stochastic Gauge Fields

Jan Wójcik  Institute of Spintronics and Quantum Information, Faculty of
Physics, Adam Mickiewicz University, 61-614 Poznań, Poland

###### Abstract

The behaviour of random quantum walks is known to be diffusive. Here we study
discrete time quantum walks in weak stochastic gauge fields. In the case of
position and spin dependent gauge field, we observe a transition from
ballistic to diffusive motion, with the probability distribution becoming
Gaussian. However, in contradiction to common belief, weak stochastic electric
gauge fields reveal the persistence of Bloch oscillations despite decoherence
which we demonstrate on simulations and prove analytically. The proposed
models provide insights into the interplay between randomness and coherent
dynamics of quantum walks.

Quantum walks introduced by Aharonov, Davidowich and Zagury Aharonov et al.
(1993) are simple platforms allowing studies of complex quantum mechanical
phenomena. They describe the dynamics of quantum particles on a lattice and
thus simulate various physical systems Cedzich et al. (2013, 2019);
Chandrashekar and Laflamme (2008); Crespi et al. (2013); Jolly and Di Molfetta
(2023); Wójcik et al. (2004); Zhang et al. (2016). Due to their simplicity
they are powerful tools while studying for example topological phenomena
Harper et al. (2020); Kitagawa (2012); Kitagawa et al. (2010); Asbóth (2012);
Asbóth et al. (2014); Lam et al. (2016); Obuse and Kawakami (2011); Obuse et
al. (2015); Grudka et al. (2023a).
This paper will focus on the 1D discrete time quantum walks (see Wu et al.
(2019) for review). DTQWs have been studied extensively in various
experimental setups such as ion traps Schmitz et al. (2009); Zähringer et al.
(2010), NMR Du et al. (2003); Ryan et al. (2005), optical lattices Dadras et
al. (2018); Karski et al. (2009) and linear optics Cardano et al. (2015); Do
et al. (2005); Perets et al. (2008); Tang et al. (2018).
The general idea behind devising quantum walk was to quantize the random walk
model. Thus the DTQW consists of two unitary operations namely the Step
operator and the Coin-toss operator. Contrary to diffusive dynamics of
classical random walk, the DTQW spread is ballistic. This effect opened up a
new brunch of quantum algorithms Portugal (2013); Childs (2009); Lovett et al.
(2010); Singh et al. (2021); Buluta and Nori (2009); Shenvi et al. (2003).

Figure 1: Probability distribution after 400 steps of DTQW on a $d=501$ vertex
cycle in strong ($\phi=200\frac{2\pi}{d}$ stochastic gauge field averaged over
1000 realizations. The orange distribution corresponds to DTQW in position and
spin dependent gauge field, and the blue distribution corresponds to DTQW in
only position dependent gauge field. The line of parabolic fit is shown in the
figure which suggests the Gaussian probability distribution of the state.
Figure 2: Averaged over 1000 realizations evolution of DTQW in weak (position
and spin dependent) gauge field defined in Eq. 5 of initial state given by Eq.
6 with $r=0.9$, and $\phi=\frac{2\pi}{d}$.

Despite the enormous attention the algorithmic usage of DTQW got, Buerschaper
and Burnett in 2004 proposed an extension to the model. They introduced the
third unitary operation the Phase operator. This simple extension brought back
treating DTQW as a platform that can be used to explain complex physical
phenomena. This new model of phase quantum walks (PQW) was shortly proved to
be useful. Wójcik et al. Wójcik et al. (2004) using the PQW managed to explain
unintuitive Bouwmeester et al. Bouwmeester et al. (1999) experimental
observations.
Proper setting of the phase operator can lead to correspondence with different
physical setups. Thus the model of PQW sparked the interest of many Arnault
and Debbasch (2016a, b); Boada et al. (2017); Bru et al. (2016); Buarque et
al. (2021); Cedzich and Werner (2016); Cedzich et al. (2019, 2020, 2013); Di
Molfetta and Debbasch (2014); Jay et al. (2021); Márquez-Martín et al. (2018);
Sajid et al. (2019, 2021); Upreti et al. (2020); Yalçınkaya and Gedik (2015).
For example in the Wójcik et al. (2004) Wójcik et al. were investigating the
PQW with phase linearly dependent on the position. This model corresponds to
an electron in a constant electric field and thus what they found was the
phenomenon known as the Bloch osculations. Moreover it was shown that the 1D
PQW simulates the dynamics of Dirac particles in arbitrary electric Debbasch
et al. (2012); Genske et al. (2013a); Cedzich et al. (2013) and gravitational
fields Di Molfetta et al. (2013, 2014); Succi et al. (2015).
In this work, we shall consider DTQWs with random gauge fields where the
randomness is time-like. In models with spatial-like randomness one expects to
observe Anderson localization. However, when the randomness is time-like the
expected effect is decoherence and diffusive behavior of particles. In DTQWs,
it was observed e.g. in models with random (in time) coins Ahlbrecht et al.
(2011) (for more examples see KENDON (2007); Vieira et al. (2014); Brun et al.
(2003); Joye (2011)). The randomness can also be introduced into DTQW by using
gauge fields with varying in time magnitudes. Molfetta and Debbasch introduced
this idea in a 2018 paper Di Molfetta and Debbasch (2014) where they discussed
DTQWs in random electric and gravitational gauge fields. They found that due
to the randomness introduced to the system, the walk loses coherence and the
behavior of the walks becomes diffusive.
In this paper we analyze and compere two models of DTQWs in weak stochastic
gauge fields. Two mentioned models although similar will differ in their gauge
fields. Both gauge fields will be generated by phase factor linearly dependent
on position. The difference is that the first model will have phase factor
dependent not only on the position but also on the spin whereas the second
model will have phase factor only dependent on the position. The second model
is thus describing a particle in electric field and we shall refer to it as an
electric quantum walk. In the strong field regime of the both models the
randomness of the field leads to qualitatively common behavior i.e. the
diffusion (see Fig. 1). However, although in the strong field of those models,
dynamics seem similar, we show a striking difference between them in the weak
field regime.
We investigate the evolution of quantum particle on a $d$ vertex cycle. The
state of the particle we will describe as a $2d$ dimensional vector in the
Hilbert space. Where $d$ comes from the $d$ possible positions and a factor
$2$ comes from the coin degree of freedom. The coin is responsible for
choosing whether the particle goes forward or backward. The most general DTQW
evolution is governed by a unitary evolution operator

$U=(I^{(d)}\otimes C)\ S,$ (1)

where

$S|{x,\pm 1}\rangle=|{x\pm 1,\pm 1}\rangle$ (2)

is the Step operator and

$C=e^{-i\theta\sigma_{y}},$ (3)

Figure 3: Probability distribution after 400 steps of evolution defined in 5
averaged over 1000 realizations. The whole evolution is shown in Fig. 2. The
line of parabolic fit is shown in the figure which suggests the Gaussian
probability distribution of the state.

is the Coin-toss operator which in general can be an arbitrary
$\mathcal{U}(2)$ matrix but we choose this specific one for simplicity
($\sigma_{y}$ being the Pauli matrix). In this form, the Coin-toss operator is
just a rotation of two state coin around the $y$ axis. For the simplicity, we
put $\theta=\pi/4$. As we mentioned we will use two different gauge fields.
The first will be generated by a position and spin dependent phase shift
operator

$\Phi_{B}|{x,\pm}\rangle=e^{\pm i\phi x}|{x,\pm}\rangle,$ (4)

where the $\phi$ is the magnitude of the field. Since we discuss the walk on a
cycle $\phi=\frac{2\pi}{d}q$ but we focus on the weak field regime so we will
stick to $\phi=\frac{2\pi}{d}$ ($q=1$). The PQW with this particular gauge
field has been already discussed Grudka et al. (2023b). However, we propose a
setup in which the field is applied with probability $r$. Thus the state is
evolved in the following way

$|{\Psi(t+1)}\rangle=\left\\{\begin{array}[]{ll}U|{\Psi(t)}\rangle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{with
probability $1-r$},\\\ U\Phi_{B}|{\Psi(t)}\rangle~{}~{}~{}~{}~{}~{}\text{with
probability $r$}.\end{array}\right.$ (5)

Let’s evolve the following initial state

$|{\Psi}\rangle_{\text{init}}=\frac{1}{2}(|{x=d/2}\rangle+|{x=d/2+1}\rangle)\otimes(|{1}\rangle-i|{-1}\rangle),$
(6)

Figure 4: DTQW in weak stochastic electric gauge field evolution defined in
Eq. 8 of initial state
$|{\Psi}\rangle=\frac{1}{\sqrt{2}}|{x=d/2}\rangle\otimes(|{1}\rangle-i|{-1}\rangle)$
with $r=0.9$, $\phi=\frac{2\pi}{d}$ and $d=501$. One can see the Bloch
oscillations in the behavior of the particle even in the case of a random
electric gauge field.

with stochastic evolution defined above with $r=0.9$. The evolution averaged
over 1000 realizations is shown in Fig. 2. One can see that after a short
time, the state stops behaving ballistic and instead becomes diffusive. As a
result of decoherence that led to diffusive behavior of the walk, the
probability distribution after $400$ steps is Gaussian (see Fig. 3). Thus even
in a weak field regime, DTQW with a random (in time) gauge field defined by
the generator given in Eq. 4 has a diffusive behavior. However, this shouldn’t
be surprising as it is common for random DTQW Schreiber et al. (2011).
Let’s now move on to the case of electric quantum walk. The electric gauge
field will be generated by the following operator

$\Phi_{E}|{x,\pm}\rangle=e^{i\phi x}|{x,\pm}\rangle.$ (7)

Note that this phase shift has only spatial dependence. Again the evolution
will be stochastic

$\displaystyle|{\Psi(t+1)}\rangle=\begin{cases}U|{\Psi(t)}\rangle&\text{with
probability }(1-r),\\\ U\Phi_{E}|{\Psi(t)}\rangle&\text{with probability
}r.\end{cases}$ (8)

One could suspect that evolution again should be diffusive but surprisingly it
is not the case. Even in DTQW with a weak stochastic electric gauge field, the
particle undergoes the Bloch oscillations in the regime of $\phi\ll 1$ and
$r\gg 0$ as one can see in the numerical simulations presented in Fig. 4.

Figure 5: On the left side we plotted the evolution of standard electric
quantum walk with $\phi=\frac{2\pi}{d}$ on a $d$ vertex cycle starting from
initial state
$|{\Psi}\rangle=\frac{1}{\sqrt{2}}|{x=d/2}\rangle\otimes(|{1}\rangle-i|{-1}\rangle)$.
As a result, we get standard Bloch oscillations with period
$T=\frac{2\pi}{\phi}=501$ steps. On the right side, we plotted the evolution
of DTQW with stochastic electric gauge field (with $r=0.9$) again with
$\phi=\frac{2\pi}{d}$ on a $d$ vertex cycle averaged over 1000 realizations.
Once again one can see the Bloch oscillations but this time with a longer
period. It can be seen that the period of the oscillations is in agreement
with our theorem $T\approx\frac{2\pi}{\phi(r+2(1-r)^{2})}\approx 545$ steps.

To explain why the Bloch oscillations survive even in the random DTQW we will
refer to the following commutation relations

$\displaystyle[\Phi_{E},S]=i\phi\ \Omega_{z}S\Phi_{E},$ (9)
$\displaystyle[\Phi_{E},C]=0,$ (10)

where

$\Omega_{z}=I^{(d)}\otimes\sigma_{z}$ (11)

Using those relations one can derive

$\displaystyle
U\Phi_{E}=\Phi_{E}U+[U,\Phi_{E}]=\Phi_{E}U+U\Phi_{E}(i\phi\Omega_{Z})=$
$\displaystyle=\Phi_{E}U+\Phi_{E}U(i\phi\Omega_{z})+U\Phi_{E}(-\phi^{2}),$
(12)

which due to the $\phi\ll 1$ we will approximate

$U\Phi_{E}\approx\Phi_{E}Ue^{i\phi\Omega_{Z}}$ (13)

For sufficiently large $r$ we can assume that in the interval $l=(1-r)^{-1}$
steps there is one step at which the field is switched off. We call such a
step defective. For example, let’s consider the case of $r=5/6$. An exemplary
6 step interval could look like this

$U_{6}=U\Phi_{E}U\Phi_{E}U\Phi_{E}UU\Phi_{E}U\Phi_{E}.$ (14)

In this form, it is hard to tell what the evolution would look like. Our main
goal is to get rid off defect on the third step of evolution. We can do that
at the cost of the magnitude of the field by separating the field operators
$\Phi_{E}=\Phi_{E}^{5/6}\Phi_{E}^{1/6}$. Next step is to leave the
$\Phi_{E}^{5/6}$ and move the $\Phi_{E}^{1/6}$ to the defect. This swap is
possible in the weak field regime ($\phi\ll 1$) where we can use Eq. 13.

$\displaystyle
U_{6}=U\Phi_{E}^{5/6}e^{-i\frac{\phi}{6}\Omega_{Z}}U\Phi_{E}^{5/6}e^{-i2\frac{\phi}{6}\Omega_{Z}}U\Phi_{E}^{5/6}e^{-i3\frac{\phi}{6}\Omega_{Z}}\cdot$
$\displaystyle\cdot
U\Phi_{E}^{5/6}U\Phi_{E}^{5/6}e^{i2\frac{\phi}{6}\Omega_{Z}}U\Phi_{E}^{5/6}e^{i\frac{\phi}{6}\Omega_{Z}}.$
(15)

We were able to erase the defected step at the cost of very small random in
time rotations about the $z$ axis of the coin. Note that this rotation does
not depend on position. This randomness is hidden in the prefactors of $\phi$.
However this way easy to spot regularities arise which look promising in terms
of averaging over the realizations. To this end let us rewrite the single step
evolution operator in the following form

$U(t,j)=(I\otimes C)\ S\ \Phi_{E}^{r}\ e^{in_{t,j}\phi\Omega_{Z}},$ (16)

where

$\displaystyle n_{t,j}=\begin{cases}(i+1)(1-r)&\text{for }i<j,\\\ 0&\text{for
}i=j,\\\ 1-i(1-r)&\text{for }i>j.\end{cases}$ (17)

the subscript $t$ denotes which step from the interval is it and $j$ denotes
at which step was the defect. Averaging above over all possible $j$ we get

$\bar{U}(t)=(I\otimes C)\ S\ \Phi_{E}^{r}e^{i\bar{n}_{t}\phi\Omega_{Z}},$ (18)

where

$\bar{n}_{t}=-2(1-r)^{2}\left(t-\frac{(1-r)^{-1}-1}{2}\right).$ (19)

So the perturbation $e^{i\bar{n}_{t}\phi\Omega_{Z}}$ linearly depends on time.
However, as Cedzich and Werner had done in Cedzich and Werner (2016) we can
transform this time dependent evolution operator into only spatial dependent
using the local gauge transformation.

$\tilde{U}=V_{t}^{\dagger}\ \bar{U}(t)\ V_{t-1},$ (20)

with

$V_{t}=\sum_{x}|{x}\rangle\langle{x}|\otimes e^{i\eta xt},$ (21)

where $\eta=-2\phi(1-r)^{2}$. So the time dependent $\bar{U}(t)$ is gauge
equivalent (up to phase that does not influence the evolution) to

$\tilde{U}=(I\otimes C)\ S\ \Phi_{E}^{(r+2(1-r)^{2})\phi}.$ (22)

The above unitary operator is just a standard electric DTQW evolution operator
but with a changed phase. Since the period of Bloch oscillations is given by
$T=\frac{2\pi}{\phi}$ the above described evolution should give rise to Bloch
oscillations with a period

$T=\frac{2\pi}{\phi(r+2(1-r)^{2})}.$ (23)

Thus for $r=0.9$, we expect the period to be longer than in standard electric
quantum walk, and in fact, the simulations we showed in Fig. 5 reproduce that
result.
Getting back to Eq. 22 one can observe that when $r$ goes to $1$ the average
evolution operator becomes

$\tilde{U}=(I\otimes C)\ S\ \Phi_{E}^{r\phi}.$ (24)

Thus one can say that as the intuition suggests in the regime of $\phi\ll 1$
and with parameter $r$ close to $1$ the average evolution operator resembles
the evolution operator of DTQW in an electric gauge field with magnitude
$r\phi$.
Note that the same line of reasoning could not be applied to the model we
studied at the beginning. DTQW in weak stochastic gauge field dependent on
both position and spin has nonvanishing commutator

$[\Phi_{B},C]=-2i\sin\theta\sum_{x}|{x}\rangle\langle{x}|\otimes\sin(\phi
x)\sigma_{x}.$ (25)

Because of the factor $\sin(\phi x)$ in the commutator the swapping procedure
does not work. The crucial point leading to the striking difference in the
weak field regime of both models is hidden in the commutation relations (Eq.
10 and Eq. 25). In the second model, the evolution survives the randomness and
does not turn classical because the field generating operator commutes with
coin-toss and commutator $[\Phi_{E},S]$ is proportional to $\phi$. Due to this
property in the weak field regime where $\phi^{2}$ can be ignored, we were
able to make use of Eq. 13. In the first model even in the weak field it is
not an option. Note that due to the periodic boundary condition, the minimal
value for $\phi$ is $\frac{2\pi}{d}$. So $sin(\phi x)$ spans values from $-1$
to $1$ regardless of $\phi$ being small. Thus we cannot move around the phase
operators to get rid of the defect like we did in the second case Eq.Quantum
Walks in Weak Stochastic Gauge Fields
In summary, we compared two models of DTQWs in stochastic gauge fields. The
first corresponds to a gauge field generated by a position and spin dependent
phase factor, and the second corresponds to an electric quantum walk. Despite
their common behavior in the strong field regime (diffusion), we found a
striking difference in the weak field regime. Unexpectedly, the stochastic
electric quantum walk survives the randomness and does not turn into
diffusion. We managed to solve the model analytically by invoking the
commutation relations of the model. We confirmed our observation in numerical
simulations. Note also that electric quantum walks have been experimentally
studied, for example, in setups like Cs atoms in optical lattices Genske et
al. (2013b) or in coupled fiber loops Regensburger et al. (2011). Thus, we
believe that the presented phenomenon can be realized experimentally in
similar setups.
We want to express our gratitude to Antoni Wójcik for illuminating
discussions. This research is supported by the Polish National Science Centre
(NCN) under the Maestro Grant no. DEC-2019/34/A/ST2/00081.

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