# Parrondo’s paradox in quantum walks with inhomogeneous coins

Vikash Mittal<EMAIL_ADDRESS>Department of Physics, National
Tsing Hua University, Hsinchu 30013, Taiwan  Yi-Ping Huang
<EMAIL_ADDRESS>Department of Physics, National Tsing Hua University,
Hsinchu 30013, Taiwan  Physics Division, National Center for Theoretical
Sciences, Taipei 10617, Taiwan  Institute of Physics, Academia Sinica, Taipei
115, Taiwan

###### Abstract

Parrondo’s paradox, a counterintuitive phenomenon where two losing strategies
combine to produce a winning outcome, has been a subject of interest across
various scientific fields, including quantum mechanics. In this study, we
investigate the manifestation of Parrondo’s paradox in discrete-time quantum
walks. We demonstrate the existence of Parrondo’s paradox using space and
time-dependent coins without the need for a higher-dimensional coin or adding
decoherence to the system. Our results enhance the feasibility of practical
implementations and provide deeper insights into the underlying quantum
dynamics, specifically the propagation constrained by the interference pattern
of quantum walks. The implications of our results suggest the potential for
more accessible and efficient designs in quantum transport, broadening the
scope and application of Parrondo’s paradox beyond conventional frameworks.

##  I Introduction

Parrondo’s paradox [1, 2, 3, 4, 5], a counterintuitive phenomenon where two
losing strategies combine to produce a winning outcome, has been an active
topic of research in physics and the mathematics community [6, 7, 8, 9, 10].
Initially, it was conceptualized in classical game theory [5, 11, 12], but
this paradox has found applications across various fields, including finance
[13, 14], evolutionary biology [15, 16, 17, 18], and statistical physics [19,
20, 21, 22, 12, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. The search for
Parrondo’s paradox has recently extended into the domain of quantum mechanics,
particularly within the framework of quantum walks [33, 34, 35, 36, 37, 38,
39, 40, 41, 42, 43, 9, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. The
first (and only as per our knowledge) experimental implementation of
Parrondo’s paradox in quantum walks was presented in Ref. [46] using an
optical setup.

Quantum walks, the quantum analog of classical random walks [56, 57], have
proven to be a powerful tool for modeling and analyzing quantum systems [58,
59, 60, 61]. Unlike classical random walks, quantum walks are governed by
quantum interference and superposition principles, leading to distinct
behaviors such as faster mixing and hitting times [62]. These quantum effects
offer deeper insights into the propagation of quantum states and their
underlying dynamics, with potential application in quantum computation and
quantum information processing [63, 64, 65, 66, 67, 68, 69]. In particular,
discrete-time quantum walks (DTQW) offer a versatile platform for studying
complex quantum phenomena, such as topological phases [70, 71, 72, 73, 74, 75]
where the walker moves in discrete steps determined by a quantum coin.

In this paper, we investigate the manifestation of Parrondo’s paradox in
discrete-time quantum walks. Earlier studies have demonstrated Parrondo’s
paradox in quantum systems using various approaches. For instance, the authors
in Ref. [41] have employed a three-state coin and in Ref. [76] a four-state
coin to induce the paradox, while in Ref. [51], the authors have introduced
decoherence to achieve similar outcomes. In Ref. [45], the authors have
realized Parrondo’s paradox with time-dependent coins, although in a different
setting than ours, where the coin operator is chosen to be a Hadamard coin and
time dependence is encoded linearly in the coin. Although these methods have
successfully shown the paradox in quantum walks, they add layers of complexity
that may hinder practical implementation.

However, here we report our findings, which reveal that neither of these
conditions is necessary. By employing space and time-dependent coins, we
demonstrate the existence of Parrondo’s paradox in a simpler and more elegant
manner. Our findings not only challenge the conventional understanding of the
requirements for Parrondo’s paradox in quantum walks but also open up new
avenues for exploring quantum strategies and their applications in quantum
transport. The space- and time-dependent coins used in our formalism have been
motivated by past experimental setups where such settings were implemented.
The ability to induce such paradoxical behavior without additional complexity
enhances the feasibility of practical implementations in quantum technologies.

The paper is organized as follows: In Section II, we provide a brief overview
of discrete-time quantum walks and the theoretical framework underlying
Parrondo’s paradox. Section III details our methodology, including the
construction of space and time-dependent coins. In Section IV and Sec V, we
present our results, showcasing the emergence of Parrondo’s paradox in the
quantum walk model. Finally, we discuss the implications of our findings and
potential future research directions and conclude in Section VI.

##  II 1D DTQW

A discrete-time quantum walk is the quantum analogue of a classical random
walk, characterized by the movement of a quantum particle (the walker) on a
lattice, governed by a quantum coin [56, 77, 57, 78]. Due to the quantum
nature of the walker and the coin, the position state of the walker is a
superposition of multiple lattice sites. This provides a quadratically fast
spread of the walker across the lattice compared to its classical counterpart
[77]. As opposed to classical random walks, quantum walks are governed by
quantum superpositions of amplitudes rather than classical probability
distributions. The quantum walk setup has been physically realized using
trapped atoms and trapped ions [79, 80, 81, 82], waveguide arrays [83],
photonic setups [84, 85, 83, 86, 87, 88], nuclear magnetic resonance [89, 90],
Bose-Einstein condensates [91] orbital angular momentum states of light [88]
and photons [92].

One-dimensional (1D) DTQW is defined as a particle hopping over a 1D lattice
with an internal degree of freedom, spin, for example, which is equivalent to
the coin in a classical random walk. The unit step of a quantum walk consists
of a conditional shift operator $\mathcal{T}$ and a coin flip operator
$\mathcal{C}(\theta)$ for a real parameter $\theta$. We can represent these
operators in the position basis of the lattice
$\\{\ket{n}\\}\in\mathcal{H}_{L}$ and spin basis
$\\{\ket{\uparrow},\ket{\downarrow}\\}\in\mathcal{H}_{C}$ of the coin. In this
basis, the conditional translation operator $\mathcal{T}$ translates the
quantum walker with spin $\ket{\uparrow}$ and $\ket{\downarrow}$ to one
lattice site forward and one lattice site backward, respectively. The
composite Hilbert space of the coin-lattice system can be written as
$\mathcal{H}=\mathcal{H}_{C}\otimes\mathcal{H}_{L}$. The conditional shift (or
translation) and coin operators read as follows

$\mathcal{T}=\sum_{x}\outerproduct{\uparrow}{\uparrow}\otimes\outerproduct{x+1}{x}+\outerproduct{\downarrow}{\downarrow}\otimes\outerproduct{x-1}{x},$
(1)

and

$\mathcal{C}(\theta)=e^{-i\theta\sigma_{y}/2}\otimes\mathds{1}_{N},$ (2)

where $-2\pi\leq\theta<2\pi$ is a real parameter and $\sigma_{y}$ is the Pauli
matrix along the $y$-axis. Here, $\mathds{1}_{N}$ represents the identity
operation on the lattice with $N$ sites. The unitary operator, which governs
the time evolution of the walker for a unit time step, reads

$U(\theta)=\mathcal{T}\mathcal{C}(\theta).$ (3)

The state of the walker after $t$ time steps starting from $\ket{\psi(0)}$ is
denoted as

$\displaystyle\ket{\psi(t)}$ $\displaystyle=(U(\theta))^{t}\ket{\psi(0)}$ (4)
$\displaystyle=\sum_{x}\psi_{\uparrow,x}(t)\ket{\uparrow}\otimes\ket{x}+\psi_{\downarrow,x}(t)\ket{\downarrow}\otimes\ket{x},$

and the probability of finding the walker at the $x$th lattice site after $t$
time steps is given by

$\displaystyle P(x,t)$
$\displaystyle=\absolutevalue{\bra{\uparrow}\otimes\innerproduct{x}{\psi(t)}}^{2}+\absolutevalue{\bra{\downarrow}\otimes\innerproduct{x}{\psi(t)}}^{2}$
$\displaystyle=\absolutevalue{\psi_{\uparrow,x}(t)}^{2}+\absolutevalue{\psi_{\downarrow,x}(t)}^{2}.$
(5)

where $\psi_{\uparrow,x}(t)$ and $\psi_{\downarrow,x}(t)$ are the amplitudes
of the lattice state at time $t$ corresponding to the two coin states
$\ket{\uparrow}$ and $\ket{\downarrow}$ respectively. For a given probability
distribution $P(x,t)$ of the quantum walker, the average position of the
wavefunction, or equivalently, the expectation value of the position operator,
$\hat{X}$ defined as

$\expectationvalue*{\hat{X}(t)}=\sum_{x}xP(x,t).$ (6)

This quantity will be of importance when we discuss the winning criterion in
quantum walk games in upcoming sections. In Figures 1, 1, and 1, we present a
detailed analysis of the probability distribution of the walker, as described
by Eq. (5), evolving over 100 time steps across various initial states. Figure
1 specifically compares the probability distributions observed in classical
and quantum walks, highlighting their distinct behaviors. Quantum walks
clearly show the dynamic evolution of probability amplitudes over time,
resulting in notable constructive and destructive interference patterns.
Constructive interference is especially prominent at the edges of the lattice,
while significant destructive interference occurs at the origin. In contrast,
classical random walks follow a predictable Gaussian distribution pattern
without any interference effects.

Figure 1 illustrates the variance of the probability distribution, denoted as
$\sigma^{2}(t)$, within a 1D DTQW with a specific initial state. The variance
increases quadratically with the number of time steps $t$, following the
relationship $\sigma^{2}(t)\propto t^{2}$. This growth rate is twice as fast
as that observed in classical random walks. The distinct behavior of the
variance is further illustrated in Figure 1, which uses a logarithmic scale to
show the variance trends in both classical and quantum walks. Consequently,
the quantum walk and classical walk are often referred to as ballistic and
diffusive processes, respectively, highlighting their inherent dynamics and
variance characteristics.

Figure 1: Probability distribution of a walker for a $1$D DTQW with $N=201$
lattice sites after $t=100$ time steps for different initial states. 1
$\ket{\psi(0)}=\ket{\downarrow}_{C}\otimes\ket{0}_{L}$, 1
$\ket{\psi(0)}=\ket{\uparrow}_{C}\otimes\ket{0}_{L}$, 1
$\ket{\psi(0)}=(\ket{\uparrow}_{C}+i\ket{\downarrow}_{C})/\sqrt{2}\otimes\ket{0}_{L}$
in contrast to classical walk (red). The rotation angle in the coin flip
operator $\mathcal{C}(\theta)$ is chosen to be $\theta=\pi/2$ for all the
plots. 1 The variance, $\sigma^{2}(t)$ for quantum and classical walks as a
function of time steps $t$ on a logarithmic scale. In the subplots (a), (b),
and (c), only the points with non-zero probability are plotted for better
presentation.

##  III Parrondo’s Paradox

Parrondo’s paradox is an intriguing phenomenon where two losing strategies or
games, when alternated or combined, result in a winning outcome. Extending
this paradox to the realm of quantum mechanics, specifically in the setting of
1D DTQW, involves defining analogous strategies using quantum coins as players
and determining the conditions under which a winning outcome is achieved.

In a quantum walk with discrete-time dynamics, the progression of the walker
is determined through the application of a coin operator followed by a shift
operator. The walker’s state comprises a superposition of positional states
influenced by the coin operator, which governs the direction of movement. To
introduce Parrondo’s paradox within the context of quantum walks, it is
necessary to establish multiple quantum games involving a minimum of two
distinct games. Conventionally, quantum games within quantum walk contexts are
characterized by the selection of a specific coin operator. By utilizing two
coin operators, denoted as $\mathcal{C}_{A}$ and $\mathcal{C}_{B}$, we create
two distinct quantum games. The choice of a particular coin operator is
analogous to picking a strategy for engaging in the game and advancing the
quantum walk. The two games individually lead to losing outcomes when applied
repeatedly. Before moving further, we first define the winning criterion in
the games.

To quantify the winning or losing nature of these games or strategies, we use
the expectation value of the position operator defined in Eq. 6. A positive
drift in the expectation value over time indicates a winning strategy, whereas
a negative drift signifies a losing one. In the context of Parrondo’s paradox,
even though $\expectationvalue*{\hat{X}(t)}$ decreases when either
$\mathcal{C}_{A}$ or $\mathcal{C}_{B}$ is applied individually (signifying
losing outcomes), the combined sequence of these coin operators might results
in an increasing $\expectationvalue*{\hat{X}(t)}$, indicating a net winning
outcome. Now, we define the two quantum games as follows:

  1. 1.

Game A (Coin $\mathcal{C}_{A}$): In this game, the coin operator
$\mathcal{C}_{A}$ is applied at each time step of the quantum walk. The coin
operator $\mathcal{C}_{A}$ is a unitary matrix that, when used consistently,
results in a quantum walk that exhibits a drift towards negative
$x$-direction, leading to a net losing outcome.

  2. 2.

Game B (Coin $\mathcal{C}_{B}$): This game uses a different coin operator
$\mathcal{C}_{B}$ at each time step. Similar to $\mathcal{C}_{A}$, the coin
operator $\mathcal{C}_{B}$ also results in a biased quantum walk in a negative
$x$-direction, leading to another losing outcome.

Having defined the terminology for individual games, we now detailed our
notation for combined strategies. We define the combined game or strategy as
follows:

$\mathcal{C}_{AB}^{(m,n)}\equiv\mathcal{C}_{A}^{m}\mathcal{C}_{B}^{n}$ (7)

where $m,n=0,1,2,\dots n$. This translates to the application of
$\mathcal{C}_{A}$ $m$-times, followed by the application of $\mathcal{C}_{B}$
$n$-times in a single time step of a quantum walk. The different pairs of
$(m,n)$ correspond to distinct quantum games or strategies.

The paradox highlights how quantum strategies that are individually
disadvantageous can combine to produce results that are not intuitive,
broadening our understanding of quantum dynamics and offering potential
applications in quantum computation and information processing. In the next
section, we detail our construction of deterministic combined sequences using
the individual games and leave the probabilistic construction for Appendix A.

Figure 2: The expectation value and the probability distribution of the walker
as a function of time for different sequences of coins. The two individual
coins or quantum games $\mathcal{C}_{A}$ and $\mathcal{C}_{B}$ are
characterized by $\theta_{A}=\pi/2$ and
$(\theta_{B-},\theta_{B+})=(-\pi/8,\pi/4)$ respectively. (a) the expectation
value of the position of the quantum walker for individual coins
$\mathcal{C}_{A}$, $\mathcal{C}_{B}$ and a trivial combination of two coins
$\mathcal{C}_{AB}^{(1,1)}$ = $\mathcal{C}_{A}$$\mathcal{C}_{B}$, (b), (c), and
(d) is the probability distribution of the quantum walkers with coins
$\mathcal{C}_{A}$, $\mathcal{C}_{B}$ and $\mathcal{C}_{AB}^{(1,1)}$
respectively. In all the cases, the probability current of the quantum walker
is biased towards negative $x$ at all times, which results in a losing
strategy for these cases. Now we consider the non-trivial (that leads to a
portability current in the opposite direction and hence leads to a winning
strategy) sequence of coins: $\mathcal{C}_{AB}^{(2,1)}$ =
$\mathcal{C}_{A}^{2}$$\mathcal{C}_{B}$ and $\mathcal{C}_{AB}^{(2,2)}$ =
$\mathcal{C}_{A}^{2}\mathcal{C}_{B}^{2}$. (e) the expectation value of the
position of the position operator where we observe a counterintuitive shift in
probability distribution. (f) and (g) represent the probability distribution
of quantum walker with time for coin sequences $\mathcal{C}_{AB}^{(2,1)}$ and
$\mathcal{C}_{AB}^{(2,2)}$ respectively. The system size is taken to be
$N=801$. The initial state of the composite system is considered to be
$\ket{\Psi(0)}=\ket{\psi}_{C}\otimes\ket{\psi}_{L}=\ket{\downarrow}\otimes\ket{0}$.

##  IV Site-Dependent Coin

In this section, we detail our construction to realize Parrondo’s paradox
using the site-dependent coin in discrete-time quantum walks. Let us consider
two distinct quantum coins $\mathcal{C}_{A}$ and $\mathcal{C}_{B}$, each
corresponding to different quantum games. When applied independently, these
coins result in losing strategies, meaning that the expectation value of the
position operator leads to a net negative outcome.

  1. 1.

Game A (Coin $\mathcal{C}_{A}$):
The coin operator $\mathcal{C}_{A}$ represents the first losing strategy. It
is defined as a simple SU(2) rotation along the $y$-axis, but when applied
over several steps, it fails to produce a distribution for the walker with the
positive expectation value of the position operator. It reads

$\mathcal{C}_{A}=e^{-i\theta_{A}\sigma_{y}/2}\otimes\mathds{1}_{N}$ (8)

where $\theta\in[-2\pi,2\pi]$ is chosen such that the resulting quantum walk
exhibits a losing strategy.

  2. 2.

Game B (Coin $\mathcal{C}_{B}$):
The coin operator $\mathcal{C}_{B}$ represents the second losing strategy.
Again, we define this as a simple SU(2) rotation along the $y$-axis, but with
a rotation angle dependent on the lattice sites. It reads

$\mathcal{C}_{B}=\sum_{x}e^{-i\theta_{B}(\theta_{B_{+}},\theta_{B_{-}};x)\sigma_{y}/2}\otimes\outerproduct{x}{x}.$
(9)

where

$\displaystyle\theta_{B}(\theta_{B_{+}},\theta_{B_{-}};x)$ (10)
$\displaystyle=\frac{\theta_{B_{+}}\left(1+\tanh(x)\right)+\theta_{B_{-}}\left(1-\tanh(x)\right)}{2}.$

Here, as before, $\theta_{B}{\pm}$ are chosen such that the resulting quantum
walk exhibits a losing outcome. Such site-dependent rotations have already
been employed experimentally in photonic quantum walk setup [71].

When the walker undergoes a quantum walk with only coin $\mathcal{C}_{A}$
applied at each step, the state evolution is governed by the following time
evolution operator

$U_{A}=\mathcal{T}\mathcal{C}_{A}.$ (11)

With a particular choice of $\theta_{A}$, over time, the probability
distribution of the walker’s position under $U_{A}$ does not result in
positive drift and hence corresponds to a losing situation.

Similarly, for coin $\mathcal{C}_{B}$, the state evolution is governed by the
following time evolution operator

$U_{B}=\mathcal{T}\mathcal{C}_{B}$ (12)

which also leads to negative drift.

We now consider a numerical simulation of the individual quantum walk. The
initial state of lattice and coin combined system, $\ket{\Psi(0)}$ is
localized at the origin and it reads

$\ket{\Psi(0)}=\ket{\downarrow}\otimes\ket{0}.$ (13)

Further, the coin parameters for Games A and B are chosen to be
$\theta_{A}=\pi/2$ and $(\theta_{B-},\theta_{B+})=(-\pi/8,\pi/4)$
respectively. We extended our analysis to examine the effects of various
initial states of the quantum walker in Appendix B, where we show the
existence of several regions in the phase space of coin parameters where the
manifestation of Parrondo’s effect can be observed.

After applying the individual unitary operations defined at the beginning of
this section for a sufficient number of steps, the resulting probability
distribution $P(x,t)$ and the expectation value of the position,
$\expectationvalue*{\hat{X}(t)}$ are plotted in Fig. 2. In the top row of Fig.
2, we plot the expectation value of the position,
$\expectationvalue*{\hat{X}(t)}$, and the probability distribution as a
function of time. As it is evident from the plots, there is a higher
probability of finding the walker at the negative positions compared to the
initial state. We observe that the expectation value of the position operator
keeps on decreasing with time $t$, and it is impossible for us to win this
game independently with the individual coins $\mathcal{C}_{A}$ and
$\mathcal{C}_{B}$. We discuss the probability distribution for individual and
combined strategies for the first few steps in Appendix C.

### Winning Strategy

The central idea of Parrondo’s paradox is that two individually losing
strategies can be combined in a specific sequence to produce a winning
outcome. For our purpose, we define the following deterministic sequence,
which is combined using the individual games with coins $\mathcal{C}_{A}$ and
$\mathcal{C}_{B}$ as

$\mathcal{C}_{AB}^{(2,1)}=\mathcal{C}_{A}^{2}\mathcal{C}_{B}.$ (14)

where $\mathcal{C}_{A}$ and $\mathcal{C}_{B}$ are the same coin operators
defined in Eq. 8 and 9. This notation concisely captures the order of
operations applied in each step, ensuring clarity and avoiding clutter. The
combined evolution operator for this sequence can be expressed as:

$U_{AB}\equiv\mathcal{T}\mathcal{C}_{AB}^{(2,1)}$ (15)

where each time step involves using the coin $\mathcal{C}_{A}$ twice followed
by $\mathcal{C}_{B}$ once. The above sequence introduces constructive
interference patterns that lead to a net positive bias in the walker’s
position. This constructive interference is not present when either
$\mathcal{C}_{A}$ or $\mathcal{C}_{B}$ is applied independently. The effect of
the combined strategy modifies the walker’s probability distribution in a way
that enhances the probability of finding the walker on the positive side of
the origin.

To quantitatively analyze the emergence of a winning strategy, we again
examine the probability distribution $P(x,t)$ and
$\expectationvalue*{\hat{X}(t)}$ of the walker’s position $x$ with time $t$.
The combined strategy is considered winning if $P(x,t)$ shows a significant
bias towards certain positions that are defined as winning states. The results
of the simulations are plotted in the bottom row of Fig. 2. The plot shows the
peak of the probability distribution towards the right, indicating a higher
probability of finding the walker at these positions compared to the initial
state. These peaks represent the winning outcome induced by the combined
strategy. This is evident from the plot of $\expectationvalue*{\hat{X}(t)}$
where we see a linear growth over time and an overall probability current
towards the positive $x$-direction.

We further introduce another sequence of coin operators as

$\mathcal{C}_{AB}^{(2,2)}=\mathcal{C}_{A}^{2}\mathcal{C}_{B}^{2}$ (16)

which also results in the probability current shifting towards the right and
hence a winning scenario, as shown in Fig. 2. Hence, the two individual
strategies with a probability distribution biased toward the left can together
result in a scheme that results in a probability distribution biased toward
the right. This is the essence of Parrondo’s Paradox.

Moreover, the choice of the two parameters $(\theta_{B-},\theta_{B+})$ of
$\mathcal{C}_{B}$ is not restricted to the one which is used till so far and
reported in Fig. 2. We investigate the full phase space comprising of three
variables at hand, namely, $\theta_{A}$, $\theta_{B-}$, and $\theta_{B+}$. To
start with, we take a fixed value of $\theta_{A}$ and plot the probability
bias (expectation value of the position operator) as a function of other
parameters $\theta_{B-}$ and $\theta_{B+}$, for individual and for different
combined strategies, in Fig. 3. In Fig. 4, we plot the expectation value of
the position operator as a function of $\theta_{A}$ and $\theta_{B+}$ by
keeping the $\theta_{B-}$ fixed. The plot uses a color gradient to highlight
the regions of winning (red) and losing (blue) strategies. These plots provide
a clear and intuitive visualization of how different parameters of coin
operators influence the dynamics of the quantum walk. The red-shaded regions
highlight the parameters of the coins where the combined effects of the coin
operators result in constructive interference of positive drifting dynamics,
thereby producing a net positive drift in the expectation value. These
parameters demonstrate the manifestation of Parrondo’s paradox, where
individually losing strategies combine to yield a winning outcome.

Apart from Ref. [71], where such site dependence is encoded in coin operators
in photonic settings, recently, a study has been reported to construct quantum
circuits that realize discrete-time quantum walks with an arbitrary position-
dependent coin operator [93].

We explore further a probabilistic scheme in Appendix A where the qualitative
behavior remains the same, and we have a winning situation using the combined
strategies with appropriate weights for the coins.

Note, we have also considered $\mathcal{C}_{AB}^{(1,1)}$ =
$\mathcal{C}_{A}$$\mathcal{C}_{B}$ and $\mathcal{C}_{AB}^{(1,2)}$ =
$\mathcal{C}_{A}$$\mathcal{C}_{B}^{2}$; however, these two sequences do
provide any advantage over the individual coin operators which are evident
from the results in Fig. 2. The probability current remains biased towards
negative $x$-direction with time.

Figure 3: The expectation value of the position of the walker as a function of
the coin parameters $\theta_{B-}$ and $\theta_{B+}$ for the individual game
(a) $\mathcal{C}_{B}$, and the combined sequences (b)
$\mathcal{C}_{AB}^{(1,1)}=\mathcal{C}_{A}\mathcal{C}_{B}$, (c)
$\mathcal{C}_{AB}^{(2,1)}=\mathcal{C}_{A}^{2}\mathcal{C}_{B}$, and (d)
$\mathcal{C}_{AB}^{(2,2)}=\mathcal{C}_{A}^{2}\mathcal{C}_{B}^{2}$ for $t=200$
time steps. The coin parameter, $\theta_{A}$ for coin $\mathcal{C}_{A}$, is
kept fixed at $\pi/2$. The red and the blue shaded regions correspond to the
winning and losing situation, respectively. The system size is taken to be
$N=501$. The initial state of the composite is considered to be
$\ket{\Psi(0)}=\ket{\psi}_{C}\otimes\ket{\psi}_{L}=\ket{\downarrow}\otimes\ket{0}$.

Figure 4: The expectation value of the position of the walker as a function of
the coin parameters $\theta_{A}$ and $\theta_{B+}$ for the individual game (a)
$\mathcal{C}_{B}$, and the combined sequences (b)
$\mathcal{C}_{AB}^{(1,1)}=\mathcal{C}_{A}\mathcal{C}_{B}$, (c)
$\mathcal{C}_{AB}^{(2,1)}=\mathcal{C}_{A}^{2}\mathcal{C}_{B}$, and (d)
$\mathcal{C}_{AB}^{(2,2)}=\mathcal{C}_{A}^{2}\mathcal{C}_{B}^{2}$ for $t=200$
time steps. The coin parameter, $\theta_{B-1}$ for coin $\mathcal{C}_{B}$ is
kept fixed at $\pi/2$. The red and the blue shaded regions again correspond to
the winning and losing situation, respectively. The system size is taken to be
$N=501$. All the other parameters are the same as in Fig. 3.

##  V Time-Dependent Coin

In Ref. [94], the authors demonstrated a quantum walk with a time-dependent
coin bias in a photonic setup. By linearly ramping the time-dependent coin
flip operation, they observed periodic revivals of the walker distribution.
This work sheds light on practical implementations of time-dependent coins in
quantum walk setups. These experimental studies motivate us to further explore
Parrondo’s paradox-like scenarios with time-dependent coins.

We start by considering the following parameterization of the coin operator

$\mathcal{C}(q(t),\alpha(t),\beta(t))=\begin{pmatrix}\sqrt{q(t)}&\sqrt{1-q(t)}e^{i\alpha(t)}\\\
\sqrt{q(t)}e^{i\beta(t)}&-\sqrt{q(t)}e^{i(\alpha(t)+\beta(t))}\end{pmatrix}$
(17)

where the three coin parameters can, in general, be a function of position $x$
or time $t$ or both, depending upon the setup under consideration. For our
purpose, we consider these parameters to be a function of time $t$ only. These
parameters are defined as follows: $0\leq q(t)\leq 1$, and
$0\leq\alpha(t),\beta(t)\leq 2\pi$. For example, it is straightforward to
recover some of the common coins, like the Hadamard and Fourier coins, by
setting

$\mathcal{C}_{H}(q(t)=1/2,\alpha(t)=0=\beta(t))=\dfrac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}$ (18)

and

$\mathcal{C}_{F}(q(t)=1/2,\alpha(t)=\pi/2=\beta(t))=\dfrac{1}{\sqrt{2}}\begin{pmatrix}1&i\\\
i&1\end{pmatrix}$ (19)

respectively. For our purpose, we start a similar game where the two players,
$A$ and $B$, are provided with two different coins $\mathcal{C}_{A}$ and
$\mathcal{C}_{B}$, which are characterized as follows:

$\mathcal{C}_{A}=C(1/2,\alpha(t),0),\;\;\;\mathcal{C}_{B}=C(1/2,0,\beta(t)).$
(20)

where $\alpha(t)$ and $\beta(t)$ are distributed uniformly in the interval $0$
to $2\pi$. In Fig. 5, we plot the expectation value of the position operator
and the probability distribution for the walker as a function of time steps
for the individual coins given to the two players. As the quantum walk
evolves, the behavior is similar to the case of a site-dependent coin, and we
do not see any revival in the dynamics. Hence, it is not possible to win the
game with individual coins. In this case, the composite system is initialized
in the following state

$\ket{\Psi(0)}=\dfrac{1}{\sqrt{2}}(\ket{\uparrow}+i\ket{\downarrow})\otimes\ket{0}.$
(21)

Figure 5: The probability distribution and the expectation value of the
position of the walker as a function of time for different sequences of coins.
The two different coins $\mathcal{C}_{A}$ and $\mathcal{C}_{B}$ are
characterized by $\alpha(t)$ and $\beta(t)$ respectively. In the top row, we
plot the dynamics for the individual coins $A$ and $B$. In the bottom row, we
consider a strategy where $\mathcal{C}_{A}^{2}$ is used for an even number of
time steps and $\mathcal{C}_{B}^{2}$ otherwise. The system size is taken to be
$N=1001$. The composite system is initialized
$\ket{\Psi(0)}=\ket{\psi}_{C}\otimes\ket{\psi}_{L}=(\ket{\uparrow}+i\ket{\downarrow})/\sqrt{2}\otimes\ket{0}$.

### Winning Strategy

Now, we introduce one of the strategies or a sequence of coin operators, which
is a combination of the individual coins $\mathcal{C}_{A}$ and
$\mathcal{C}_{B}$ such that it results in a probability distribution with a
bias towards the right to the origin. It is given as

$\mathcal{C}_{AB}=\begin{cases}\mathcal{C}_{A}^{2},&\text{if}\;\;t\in\text{even}\\\
\mathcal{C}_{B}^{2},&\text{if}\;\;t\in\text{odd}\\\ \end{cases}$ (22)

i.e., we use the coins twice and alternatively in order to win the game. We
now choose the above coin operator to evolve the quantum walk and plot the
behavior of the expectation value of the position and the probability
distribution as a function of time steps $t$ in Fig. 5. We observe the shift
of probability current towards the right as the quantum walk evolves. Such
strategies [95], which involve alternate actions of different coins, have been
reported recently to recreate Parrondo’s paradox, although with different
choices of coins.

##  VI Conclusion

In conclusion, our investigation into Parrondo’s paradox in discrete-time
quantum walks reveals that the paradox can be induced using space- and time-
dependent coin operators without the need for multi-state coins or
decoherence. This finding challenges the conventional understanding and
demonstrates that a simpler, more elegant approach can achieve the same
paradoxical outcomes previously reported in more complex setups. Unlike
earlier methods that introduced additional layers of complexity, our use of
general SU(2) rotations for the coin operators provides greater flexibility
and control over the quantum walk dynamics, making it more feasible for
practical implementation and experimental verification. Our results not only
simplify the experimental realization of Parrondo’s paradox but also open up
new avenues for exploring quantum strategies and their applications in quantum
transport, computation, and information processing. This study paves the way
for more accessible and versatile quantum algorithms, contributing
significantly to the advancement of quantum technologies. It would be
intriguing to see if Parrrondo’s paradox can be generalized for more than two
games. One of the future extensions of this work is to probe a combined
strategy of the form

$\mathcal{C}_{AB}(m,n)\equiv\mathcal{C}_{A}^{m}\mathcal{C}_{B}^{n}$ (23)

and see what combinations $(m,n)$ leads to a winning situation.

##  VII Acknowledgments

This research has been supported by the MOST Young Scholar Fellowship (Grants
No. 112-2636-M-007-008- and No. 113-2636-M-007-002-), National Center for
Theoretical Sciences (Grants No. 113-2124-M-002-003-) from the Ministry of
Science and Technology (MOST), Taiwan, and the Yushan Young Scholar Program
(as Administrative Support Grant Fellow) from the Ministry of Education,
Taiwan.

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##  Appendix A Probabilistic Parrondo sequence

While a deterministic sequence of alternating quantum coins can demonstrate
Parrondo’s paradox, another intriguing approach is to combine these strategies
probabilistically. In a probabilistic sequence, the quantum walker switches
between different coin operators according to specific probabilities rather
than a fixed pattern. This method introduces an additional layer of complexity
and randomness, which can still yield a winning outcome despite each strategy
being a losing strategy on its own. In a probabilistic sequence, we define a
probability $q$ for choosing the coin $\mathcal{C}_{A}$ and a probability
$1-q$ for choosing the coin $\mathcal{C}_{B}$ at each step, i.e.

$\mathcal{C}_{AB}=q\mathcal{C}_{A}+(1-q)\mathcal{C}_{B}.$ (24)

We plotted the expectation value of the walker’s position in Fig. 6 for
different probabilistic strategies and found that Parrondo’s effect still
occurs for certain coin weights. By adjusting the probabilities, we observed
that some combinations led to a net positive movement of the walker, showing a
winning situation. This means that even with a mix of strategies, we can still
achieve a winning outcome by choosing the right probabilities for the coin
flips.

Figure 6: The expectation value of the position of the walker as a function of
time for the combined probabilistic strategy discussed in Appendix A and is
given in Eq, 24. The plots correspond to different values of $q$ and an
average is taken over $50000$ iterations.

##  Appendix B Dependence on the initial state of the coin

We explore the role of the initial state of the coin on the manifestation of
Parrondo’s paradox in discrete-time quantum walks and report significant
insights. We found that the initial coin state plays a crucial role in
determining the dynamics and eventual outcomes of the quantum walk.
Specifically, a most general state of the coin, which is a superposition of
the basis state $\ket{\uparrow}$ and $\ket{\downarrow}$, characterized by the
parameters $\theta$ and $\phi$, given by

$\ket{\psi}_{C}=\cos(\dfrac{\theta}{2})\ket{\uparrow}+e^{i\phi}\sin(\dfrac{\theta}{2})\ket{\downarrow},$
(25)

directly influence the interference patterns and probability distributions
that emerge during the walk. By systematically varying the parameters $\theta$
and $\phi$, and hence the initial state of the coin, we plotted the
expectation value of the walker’s position using site-dependent and step-
dependent coins for different initial states in Fig. 7 and Fig. 8,
respectively. We observed that certain configurations either enhance or
suppress the paradoxical effect. These results show that the choice of initial
state and coin configuration can significantly impact the outcome, either
strengthening or weakening the paradoxical behavior.

This sensitivity to the initial coin state underscores the intricate interplay
between quantum coherence, superposition, and the probabilistic nature of
quantum walks, highlighting the need for precise control and preparation of
initial states in experimental realizations. Our findings suggest that by
carefully selecting the initial coin state, one can optimize the conditions
for observing Parrondo’s paradox, thereby enhancing the practical
applicability of quantum walks.

Figure 7: The expectation value of the position of the walker at a time
$t=1000$ as a function of the parameters of the initial state of the coin
$\theta$ and $\phi$ given in Eq. 25. The two individual coins
$\mathcal{C}_{A}$ and $\mathcal{C}_{B}$ are characterized by
$\theta_{A}=\pi/2$ and $(\theta_{B-},\theta_{B+})=(-\pi/8,\pi/4)$
respectively. The coin sequences are chosen to be (a) $\mathcal{C}_{A}$, (b)
$\mathcal{C}_{B}$ and (c)
$\mathcal{C}_{AB}^{(2,2)}=\mathcal{C}_{A}^{2}\mathcal{C}_{B}^{2}$. The red and
the blue shaded regions correspond to the winning and losing situation,
respectively. The system size is $2101$ and the walker is localized at
$\ket{x=0}$ at $t=0$.

Figure 8: The expectation value of the position of the walker at a time
$t=250$ as a function of the parameters of the initial state of the coin
$\theta$ and $\phi$ given in Eq. 25. The two individual coins
$\mathcal{C}_{A}$ and $\mathcal{C}_{B}$ are characterized by
$C(1/2,\alpha(t),0)$ and $C(1/2,0,\beta(t))$ respectively. The coin sequences
are chosen to be (a) $\mathcal{C}_{A}$, (b) $\mathcal{C}_{B}$, and (c)
$\mathcal{C}_{A}^{2}$ is used for an even number of time steps and
$\mathcal{C}_{B}^{2}$ for odd one. The red and the blue shaded regions
correspond to the winning and losing situation, respectively. The system size
is $401$ and the walker is localized at $\ket{x=0}$ at $t=0$.

##  Appendix C Dynamics for a first few steps

For a better understanding of Parrondo’s paradox in the context of quantum
walks, we plot the dynamics of the quantum walker over the first few steps in
Fig. 9. These visualizations provide crucial insights into how the application
of individual coin operators $\mathcal{C}_{A}$ and $\mathcal{C}_{B}$ and the
combined operator influence the walker’s probability distribution and overall
trajectory. By examining the position probabilities at each step, we can
observe the distinct interference patterns that emerge from the superposition
of states induced by different coin operators.

Plotting the expectation value of the position operator
$\expectationvalue*{\hat{X}(t)}$ over these initial steps further illustrates
the onset of Parrondo’s paradox. While $\expectationvalue*{\hat{X}(t)}$ may
decrease when a single coin operator is used repeatedly, the combined sequence
results in a noticeable positive drift. These dynamics, captured through our
plots, provide a visual confirmation of the paradox, showcasing the beneficial
effects of combined strategies on the quantum walker’s evolution from the very
beginning.

Figure 9: The probability distribution of the walker for the first few steps
for the coins (a) $\mathcal{C}_{A}$, (b) $\mathcal{C}_{B}$, and
(c)$\mathcal{C}_{AB}^{(2,2)}$ = $\mathcal{C}_{A}^{2}\mathcal{C}_{B}^{2}$. All
the other parameters are the same as Fig. 2.