# Bounds on an effective thermalization beyond the Zeno limit

Guilherme Zambon<EMAIL_ADDRESS>Diogo O. Soares-Pinto
<EMAIL_ADDRESS>Instituto de Física de São Carlos, Universidade de São Paulo,
CP 369, 13560-970, São Carlos, SP, Brasil.

###### Abstract

Developing protocols for preserving information in quantum systems is a
central quest for implementing realistic quantum computation. In this regard,
the quantum Zeno effect has emerged as a widely utilized technique to
safeguard classical information stored in quantum systems. However, existing
results pertaining to this method often assume operations performed infinitely
fast on the system of interest, which only serves as an approximation to real-
world scenarios where the temporal resolution of any experimental apparatus is
inherently finite. In this study, we go beyond this conventional assumption
and derive the effective Zeno dynamics for any time interval between
operations. Our analysis considers a qubit undergoing thermalization, as
described by a generalized amplitude damping channel, while the operations
performed consist of projections onto an orthonormal basis that may or may not
coincide with the pointer basis to which the system is thermalizing. By
obtaining the probability of successfully storing a bit of information after a
given time, we investigate the performance of the protocol in two important
scenarios: the limit of many interventions, with a first-order correction to
the Zeno limit, and the limit of very few interventions. In doing so, we
provide valuable insights into the protocol’s performance by establishing
bounds on its efficacy. These findings enhance our understanding of the
practical applicability of the quantum Zeno effect in preserving classical
information stored in quantum systems, allowing for better design and
optimization of quantum information processing protocols.

##  I Introduction

Building a universal quantum computer has been so far one of the most
challenging tasks of the twenty-first century Physics. Since Feynman’s famous
insight on employing controlled quantum systems to simulate other quantum
systems found in nature [1], physicists have been developing theoretical and
experimental tools to make these simulations possible. Ideally, everything
that is needed to perform useful quantum computation is a set of physical
qubits and the ability to experimentally implement any sequence of logical
gates on this set. In reality, however, this might not be enough.

As no quantum system is ever truly isolated from its surroundings, during the
computation the evolution of the experimentally controlled physical systems
might be affected by the interactions between them and their environment [2].
Such effects, generally known as noise, may continuously modify the quantum
state stored in the qubits, which is problematic, as it is essential to
preserve this state during and in between computational steps to reliably
implement any algorithm on a real quantum computer.

To address these challenges, techniques such as quantum error correction and
dynamical decoupling can be employed. While these methods offer valuable
approaches for protecting quantum states, they have inherent drawbacks.
Quantum error correction requires storing an informational qubit in multiple
physical qubits, necessitating substantial overhead and resources [3, 4, 5, 6,
7, 8, 9, 10, 11, 12]. On the other hand, dynamical decoupling relies on
precise knowledge of the qubit-environment dynamics, which may be challenging
to ascertain in realistic scenarios [13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26].

Alternatively, if our goal is solely to preserve classical information stored
in a quantum system, the quantum Zeno effect presents a compelling approach,
as it circumvents the drawbacks of the aforementioned techniques [27, 28, 29,
30, 31, 32, 33]. It consists in performing repeated projective measurements on
the system of interest, which in the limit of infinitely frequent operations
(Zeno limit) is known to freeze the dynamics at some subspaces and to generate
an effective Zeno dynamics at others [34, 35, 36].

Given a free dynamics and a set of operations to be periodically performed,
one could obtain the effective dynamics in the Zeno limit using a suitable
result, like those found in Refs. [30, 31, 32, 33]. However, in practical
experimental scenarios there is always a limitation on the frequency at which
external operations can be realized. This limitation implies that while the
Zeno limit serves as a useful approximation in certain cases, it is ultimately
unattainable.

Here, we address this limitation by explicitly determining the effective
dynamics for any time interval between operations for the relevant case of a
qubit undergoing thermalization, as described by a generalized amplitude
damping channel. The operations performed are non-selective projective
measurements in as basis that may or may not coincide with the pointer basis
to which the qubit is thermalizing. We particularly explore the Zeno limit, a
first order correction to it and also the limit of very few operations
performed. Finally, these results are used to establish bounds on the
protocol’s efficiency, as quantified by the probability of successfully
storing one bit of classical information on a thermalizing qubit for some
given period of time.

##  II Setup

Consider a qubit used to store a classical bit of information over some period
of time. Given its interaction with the environment, it must be treated as an
open quantum system, whose state evolves non-unitarily [2]. A common approach
to this situation is to model the effects of the environment with a quantum
channel $\mathcal{E}_{t}$, such that the state of the qubit at some time $t$
will be given by

$\displaystyle\rho_{t}=$ $\displaystyle~{}\mathcal{E}_{t}(\rho_{\mathrm{in}})$
(1) $\displaystyle=$
$\displaystyle\sum_{i=0}^{3}K_{i}(t)\rho_{\mathrm{in}}K_{i}^{\dagger}(t),$

where $K_{i}(t)$ are the Kraus operators associated to the channel
$\mathcal{E}_{t}$ and $\rho_{\mathrm{in}}$ is the initial state of the qubit
[37].

For many relevant situations, the environment can be simply modeled as a
thermal bath at some fixed temperature $T$, so the qubit is expected to
thermalize at this temperature and asymptotically approach a thermal state.
This type of evolution is achieved by using the amplitude damping channel
$\mathcal{E}_{AD}$ [38], whose Kraus operators are given by

$K_{0}=\sqrt{p}\quantity(\ket{e_{0}}\bra{e_{0}}+\sqrt{1-\gamma}\ket{e_{1}}\bra{e_{1}}),$
(2a) $K_{1}=\sqrt{p\gamma}\ket{e_{0}}\bra{e_{1}},$ (2b)
$K_{2}=\sqrt{1-p}\quantity(\sqrt{1-\gamma}\ket{e_{0}}\bra{e_{0}}+\ket{e_{1}}\bra{e_{1}}),$
(2c) $K_{3}=\sqrt{(1-p)\gamma}\ket{e_{1}}\bra{e_{0}}.$ (2d)

In the above equations, $\gamma:=\gamma_{t}$ is a temporal parameter
monotonically increasing in $t$, satisfying $\gamma_{0}=0$ and $\gamma_{t}\to
1$ as $t\to\infty$. It can be verified that $\gamma_{t}=0$ implies
$\mathcal{E}_{AD}(\rho)=\rho$, as expected, while $\gamma_{t}=1$ implies
$\mathcal{E}_{AD}(\rho)=p\ket{e_{0}}\bra{e_{0}}+(1-p)\ket{e_{1}}\bra{e_{1}}$,
which is a thermal state in the basis $\\{\ket{e_{0}},\ket{e_{1}}\\}$. It
follows that $p=p(T)$ is the population of the ground state in thermal
equilibrium. Setting its energy to be zero and the energy of the excited state
to be $E$, yields

$p(T)=\frac{1}{1+e^{-E/k_{B}T}},$ (3)

implying $p\to 1$ as $T\to 0$ and $p\to 1/2$ as $T\to\infty$.

Notice that in thermal equilibrium the system will be diagonal in the pointer
basis $\\{\ket{e_{0}},\ket{e_{1}}\\}$, determined by the system-environment
dynamics [26]. Nevertheless, we assume the experimenter could choose any other
basis to be the computational basis $\\{\ket{0},\ket{1}\\}$ in which the
classical bit is initially stored, either as
$\rho_{\mathrm{in}}=\ket{0}\bra{0}$ or as $\rho_{\mathrm{in}}=\ket{1}\bra{1}$.
Each $\ket{j}$ can still be written in the pointer basis using a convenient
parametrization,

$\ket{0}=\cos\theta\ket{e_{0}}+e^{i\phi}\sin\theta\ket{e_{1}},$ (4a)
$\ket{1}=\sin\theta\ket{e_{0}}-e^{i\phi}\cos\theta\ket{e_{1}},$ (4b)

for $\theta\in[0,\pi/2]$ and $\phi\in[0,2\pi)$. In Appendix A we show that
this is indeed the most general form of the computational basis, given the
orthogonality constraint $\bra{0}\ket{1}=0$.

To analyze the dynamics of the system we must propose an explicit expression
for $\gamma_{t}$, which depends on the specific physical situation that is
being represented by the amplitude damping channel. Here, we assume that the
bath is much larger than the qubit and also that their coupling is
sufficiently weak, implying that there are no memory effects due to
information backflow, i.e., the dynamics of the qubit is Markovian [39, 40,
41, 42, 43]. This is done by imposing that the family of quantum channels is
CP-divisible and homogeneous in time

$\mathcal{E}_{t+t^{\prime}}=\mathcal{E}_{t^{\prime}}\mathcal{E}_{t}.$ (5)

In Appendix B it is shown that this condition implies an exponential decay for
$\gamma_{t}$, that is

$\gamma_{t}=1-e^{-t/\tau},$ (6)

where $\tau$ is some characteristic relaxation time determined by the system-
environment interaction dynamics [38].

Consider the qubit starts in a pure state in the computational basis
$\rho_{\mathrm{in}}=\ket{j}\bra{j},~{}j=0,1$. It is then subject to non-
selective projective measurements in this basis after every interval $\Delta$
of time, which can be seen as the temporal resolution of the experimental
apparatus. In the Zeno limit we have $\Delta\to 0$, while in the limit of few
measurements after some time $t$ we have $\Delta\sim t$. We define the ground
state population after some time $t$ as

$a(t):=\bra{0}\rho_{t}\ket{0},$ (7)

with $a_{0}:=a(0)=1-j$. Now we determine how this population evolves in time.

##  III Results and Discussion

###  III.1 Effective dynamics

Between measurements, the state of the qubit will simply evolve according to
the amplitude damping channel, exponentially approaching the thermal state on
the pointer basis. This implies the population $a(t)$ will change and the
coherences $\bra{0}\rho_{t}\ket{1}$ and $\bra{1}\rho_{t}\ket{0}$ will become
non-zero. However, when the next measurement is performed these coherences
will vanish, while the populations will remain unchanged. Therefore, the time
dependence of $a(t)$ will be determined by the probabilities of the system
transitioning from one state to another within an interval $\Delta$ of time.
We use the conditional probability rule to define the transitioning
probabilities

$\displaystyle a(t+\Delta)=$ $\displaystyle~{}P(0,t+\Delta)$ (8)
$\displaystyle=$ $\displaystyle~{}P(0,t)P(0,t+\Delta|0,t)$ $\displaystyle+$
$\displaystyle~{}P(1,t)P(0,t+\Delta|1,t)$ $\displaystyle=$
$\displaystyle~{}a(t)P_{0|0}+(1-a(t))P_{0|1},$

where we have defined $P_{0|1}$ as the probability of the qubit transitioning
from $1$ to $0$ within an interval $\Delta$ of time, and the normalization
implies $P_{0|0}=1-P_{1|0}$.

Notice that due to the Markov hypothesis, the transition probabilities
$P_{1|0}$ and $P_{0|1}$ are only functions of $\Delta$ and not of $t$. Eq. (8)
then yields

$\displaystyle a(t+\Delta)=$
$\displaystyle~{}P_{0|1}+a(t)\quantity[1-\quantity(P_{0|1}+P_{1|0})].$ (9)

In Appendix C we recurrently expand the above equation from $a_{0}$, resulting
in the following ground state population for any $t=n\Delta$

$a(t)=a_{\infty}-\quantity(a_{\infty}-a_{0})\quantity[1-\quantity(P_{0|1}+P_{1|0})]^{t/\Delta},$
(10)

where

$\displaystyle a_{\infty}:=$ $\displaystyle~{}\lim_{t\to\infty}a(t)$ (11)
$\displaystyle=$ $\displaystyle~{}\frac{P_{0|1}}{P_{0|1}+P_{1|0}}$

is the (computational basis) ground state population at thermal equilibrium.

In Appendix D, we explicitly determine the transition probabilities as
functions of $\Delta$ and $\theta$, which allows us to write

$\displaystyle a(t,\Delta,\theta)=$
$\displaystyle~{}a_{\infty}-\quantity(a_{\infty}-a_{0})\Big{[}\sin^{2}(2\theta)e^{-\Delta/2\tau}$
(12) $\displaystyle+$
$\displaystyle~{}\cos^{2}(2\theta)e^{-\Delta/\tau}\Big{]}^{t/\Delta}$

and

$a_{\infty}(\Delta,\theta)=\frac{1}{2}\quantity[1+\frac{(2p-1)\cos(2\theta)}{\cos^{2}(2\theta)+\sin^{2}(2\theta)\frac{1-e^{-\Delta/2\tau}}{1-e^{-\Delta/\tau}}}].$
(13)

It should be pointed out that this expression for the ground state population
is only exact for instants right after the measurement is performed, that is,
for $t=n\Delta$. With this result in hand, we are able to explore some
relevant limits regarding $\Delta$.

####  III.1.1 Zeno limit

We first define dimensionless quantity $\delta:=\Delta/\tau$. The Zeno limit
is then given by $\delta\to 0$, which means the interval between measurements
is much smaller than the characteristic decay time of the thermalization,

$\displaystyle a^{\mathrm{Zeno}}(t,\theta):=$ $\displaystyle\lim_{\delta\to
0}a(t,\delta,\theta)$ (14) $\displaystyle=$
$\displaystyle~{}a_{\infty}^{0}-e^{-t/\tau^{\mathrm{eff}}}(a_{\infty}^{0}-a_{0}),$

where

$\displaystyle a_{\infty}^{0}(\theta):=$ $\displaystyle\lim_{\delta\to
0}a_{\infty}(\delta,\theta)$ (15) $\displaystyle=$
$\displaystyle~{}\frac{1}{2}+\frac{(2p-1)\cos(2\theta)}{1+\cos^{2}(2\theta)}$

is the ground state population at thermal equilibrium in the Zeno limit (or
the zeroth order expansion of $a_{\infty}$ in $\delta$), and

$\tau^{\mathrm{eff}}(\theta):=\frac{4\tau}{3+\cos(4\theta)},$ (16)

is the effective characteristic time of the thermalization.

As in this limit the system is never able to create coherences in the
computational basis, the dynamics is fully described
$a^{\mathrm{Zeno}}(t,\theta)$. Also, this expression is valid for any $t$. As
we can see in Fig. 1, the dynamics is a single exponential starting in $a_{0}$
and asymptotically approaching $a_{\infty}$, with characteristic time
$\tau^{\mathrm{eff}}$. Such behavior can be described as an effective
thermalization, as the state is constrained by the measurements to remain
diagonal in the computational basis and exponentially approaches an effective
thermal state. As in general $a_{\infty}\neq p$, we might have an effective
temperature which differs from the original one. Importantly, given that
$\tau^{\mathrm{eff}}\geq\tau$, the effective thermalization is always slower
than the original one, implying that the information about the initial state
is kept for longer times in the Zeno effective dynamics.

Figure 1: Computational basis ground state population as a function of time.
The graph shows the specific case in which the angle $\theta$ between
$\ket{e_{0}}$ and $\ket{0}$ is $\pi/6$ and the $\ket{e_{0}}$ state population
at thermal equilibrium is $p=0.9$, implying the temperature of the environment
is relatively low. The $\ket{0}$ state population population exponentially
approaches its equilibrium value, which is different from $p$. This happens
for both $\rho_{\mathrm{in}}=\ket{0}\bra{0}$ ($a_{0}=1$) and
$\rho_{\mathrm{in}}=\ket{1}\bra{1}$ ($a_{0}=0$), that is, independently of the
classical bit being initially stored.

####  III.1.2 First order correction

In situations where the approximation $\delta\sim 0$ is not a good one, a
first order correction to the Zeno limit might be useful. We define $c^{1}$ as
the first order correction in $\delta$ to $a(t,\delta,\theta)$,

$a(t,\delta,\theta)=a^{\mathrm{Zeno}}(t,\theta)+\delta~{}c^{1}(t,\theta)+\mathcal{O}(\delta^{2}).$
(17)

Performing the expansion of (12) up to first order in $\delta$ yields the
correction

$c^{1}(t,\theta)=c_{\infty}^{1}-e^{-t/\tau^{\mathrm{eff}}}(c_{\infty}^{1}-c_{0}^{1}),$
(18)

with

$c_{\infty}^{1}(\theta)=-\frac{(2p-1)\cos(2\theta)\sin^{2}(2\theta)}{\quantity[3+\cos(4\theta)]^{2}}$
(19)

and

$c_{0}^{1}(t,\theta)=a_{0}\frac{\sin^{2}(4\theta)}{32}t.$ (20)

Alternatively, we could explicitly write $a(t,\delta,\theta)$ up to first
order in $\delta$ as

$\displaystyle
a^{1}(t,\delta,\theta)=a_{\infty}^{1}-e^{-t/\tau^{\mathrm{eff}}}(a_{\infty}^{1}-a_{0}^{1}),$
(21)

where

$\displaystyle a_{\infty}^{1}(\theta,\delta)=$
$\displaystyle~{}\frac{1}{2}+\frac{2(2p-1)\cos(2\theta)}{3+\cos(4\theta)}$
(22) $\displaystyle-$
$\displaystyle~{}\delta\frac{(2p-1)\cos(2\theta)\sin^{2}(2\theta)}{\quantity[3+\cos(4\theta)]^{2}}$

is the first order expansion of $a_{\infty}$ in $\delta$, and

$a_{0}^{1}(t,\delta,\theta):=a_{0}\quantity(1+\delta\frac{\sin^{2}(4\theta)}{32}t).$
(23)

Interestingly, the first order correction to $a(t,\delta,\theta)$ does not
change the characteristic thermalization time $\tau^{\mathrm{eff}}$ and
provides a first order correction to the equilibrium value $a_{\infty}$.
However, the dynamics will not be described by a simple exponential anymore,
as there will be also a correction to the initial state which is not constant,
but linear in time, as shown in Eq. (23). The comparison between the Zeno
limit and the first order dynamics is shown in Fig. 2.

Figure 2: Comparison between the $\ket{0}$ state population in the Zeno limit
and its expansion up to first order in $\delta$ for the case $\delta=0.1$.
This shows how the system evolves differently when the approximation
$\delta\sim 0$ ceases to be a good one. In this case, the characteristic time
$\tau$ of the thermalization is only $10$ times greater than the time $\Delta$
between measurements.

####  III.1.3 Free limit

On the other side of the $\delta$ limits we have the nearly-free dynamics,
where very few measurements are performed within a period of time. Since our
general expression for the ground state population is only exact for
$t=n\Delta$, and we are treating the case $\Delta\sim t$, it is convenient to
consider $\Delta=t$, which is the case in which we leave the qubit evolve
freely during the whole time $t$, and only measure it in the end of the
storage time. Eq. (12) then yields

$\displaystyle a^{\mathrm{Free}}(t,\theta)=$
$\displaystyle~{}a_{\infty}^{F}-\quantity(a_{\infty}^{F}-a_{0})\Big{[}\sin^{2}(2\theta)e^{-t/2\tau}$
(24) $\displaystyle+$ $\displaystyle~{}\cos^{2}(2\theta)e^{-t/\tau}\Big{]},$

where

$a_{\infty}^{F}(\theta)=\frac{1}{2}\quantity[1+\quantity(2p-1)\cos(2\theta)].$
(25)

Notice that in this case the dynamics is described by a combination of two
exponentials, one decaying with $\tau$ and the other with $2\tau$. The
comparison between the free limit and the Zeno limit is shown in Fig. 3.

Figure 3: Comparison between the Zeno limit and the Free limit of the
evolution. In this particular case they differ substantially. However,
comparing Eqs. (14) and (24) we can see that they coincide when the angle
$\theta$ is $0$, $\pi/4$ or $\pi/2$.

###  III.2 Success probability

Now that we have determined how the ground state population of the pointer
basis evolve in several situations, we proceed to the calculation of the
probability of successfully storing the bit of classical information within a
period of time. First, notice that the probability of a successful storage
given that we started with $0$ is simply the ground state population with
$a_{0}=1$. On the other hand, the probability of a successful storage given
that we started with $1$ is the excited state population, which is given by
$(1-a)$ with $a_{0}=0$. Then, assuming the bits $0$ and $1$ are equally likely
to be stored at this point of the computation, we obtain

$P_{\mathrm{suc}}=~{}\frac{1}{2}\quantity[a_{a_{0}=1}+(1-a_{a_{0}=0})].$ (26)

Combining this equation with the expressions for the ground state population
we obtain for a general $\Delta$

$\displaystyle P_{\mathrm{suc}}=$
$\displaystyle~{}\frac{1}{2}\Bigg{\\{}1+\Big{[}\sin^{2}(2\theta)e^{-\Delta/2\tau}+\cos^{2}(2\theta)e^{-\Delta/\tau}\Big{]}^{t/\Delta}\Bigg{\\}},$
(27)

while for the Zeno limit

$\displaystyle P_{\mathrm{suc}}^{\mathrm{Zeno}}=$
$\displaystyle~{}\frac{1}{2}\quantity[1+e^{-t/\tau^{\mathrm{eff}}}],$ (28)

the first order expansion

$\displaystyle P_{\mathrm{suc}}^{1}=$
$\displaystyle~{}\frac{1}{2}\quantity[1+e^{-t/\tau^{\mathrm{eff}}}\quantity(1+\delta\frac{\sin^{2}(4\theta)}{32}t)],$
(29)

and the free limit

$\displaystyle P_{\mathrm{suc}}^{\mathrm{Free}}=$
$\displaystyle~{}\frac{1}{2}\quantity[1+\sin^{2}(2\theta)e^{-t/2\tau}+\cos^{2}(2\theta)e^{-t/\tau}].$
(30)

Interestingly, as we assume both initial states to be equally likely, the
expressions for the success probability do not depend on the pointer basis
ground state population $p$, and therefore do not depend on the temperature of
the environment. In Fig. 4 we see that $P_{\mathrm{suc}}^{\mathrm{Zeno}}$ and
$P_{\mathrm{suc}}^{\mathrm{Free}}$ provide, respectively, lower and upper
bounds to the general $P_{\mathrm{suc}}$.

Figure 4: Success probability as a function of time in the Zeno and Free
limits. Since the results are symmetrical with respect to $\pi/4$, the three
angles were chosen to be distributed within the interval $0<\theta<\pi/4$,
depicting the bounds in the limits $\theta\sim\pi/4$, $\theta\sim 0$ and
within an intermediate band. Any finite $\Delta$ will place the evolution in
the shaded area between the two curves. These areas are better observed in the
zoomed insets inside the graphs. The curves coincide when the angle $\theta$
is $0$, $\pi/4$ or $\pi/2$.

##  IV Conclusions

We have shown that the quantum Zeno effect might be useful for preserving
classical information stored in quantum systems even beyond the Zeno limit.
After explicitly determining the effective dynamics for any time interval
between operations, we have established bounds on the protocol’s efficiency,
as quantified by the probability of successfully storing classical information
over a given period of time. This way, we have demonstrated the practical
applicability and limitations of the quantum Zeno effect in our model. We
expect these insights to contribute to the design and optimization of quantum
information processing protocols, enabling better strategies for preserving
information in the presence of noise. Further research in this direction may
help advance the field of quantum information processing and pave the way for
the realization of useful quantum computation in a near future.

###### Acknowledgements.

GZ acknowledges financial support from FAPESP (Grant No. 2022/00993-9) and
DOSP acknowledges the support from the funding agencies CNPq (Grant No.
307028/2019-4), FAPESP (Grant No. 2017/03727-0), and the Brazilian National
Institute of Science and Technology of Quantum Information (INCT-IQ) Grant No.
465469/2014-0.

##  Appendix A Computational basis orthogonality constraint

Using a parametrization reminiscent of the Bloch sphere for each vector of the
computational basis we obtain

$\ket{0}=\cos\theta_{0}\ket{e_{0}}+e^{i\phi_{0}}\sin\theta_{0}\ket{e_{1}},$
(31a)
$\ket{1}=\cos\theta_{1}\ket{e_{0}}+e^{i\phi_{1}}\sin\theta_{1}\ket{e_{1}},$
(31b)

for $\theta_{0},\theta_{1}\in[0,\pi/2]$ and $\phi_{0},\phi_{1}\in[0,2\pi)$.
Despite being parameterized by four angles, they cannot vary freely as the
orthogonality between $\ket{0}$ and $\ket{1}$ must be satisfied. Writing
explicitly

$\displaystyle 0=$ $\displaystyle\bra{0}\ket{1}$ (32) $\displaystyle=$
$\displaystyle\cos\theta_{0}\cos\theta_{1}+e^{i(\phi_{1}-\phi_{0})}\sin\theta_{0}\sin\theta_{1}.$

As the left hand side is real, so must be the right hand side, then

$e^{i(\phi_{1}-\phi_{0})}=\pm 1,$ (33)

but since both $\sin$ and $\cos$ of $\theta_{0}$ and $\theta_{1}$ are positive
in the considered intervals, we must have

$e^{i(\phi_{1}-\phi_{0})}=-1,$ (34)

as it yields

$\sin\theta_{0}\sin\theta_{1}=\cos\theta_{0}\cos\theta_{1}.$ (35)

Therefore, we have

$e^{i\phi_{0}}=-e^{i\phi_{1}}:=e^{i\phi}.$ (36)

Also,

$\tan\theta_{0}\tan\theta_{1}=1,$ (37)

which in the considered intervals imply

$\sin\theta_{1}=\cos\theta_{0}:=\cos\theta,$ (38)

yielding the expressions as presented in the main text.

##  Appendix B Exponential decay

Given a general initial state for the qubit

$\rho_{\mathrm{in}}=\begin{bmatrix}a&b\\\ b^{*}&1-a\end{bmatrix}$ (39)

the effects of the Kraus operators are calculated from Eq. (2):

$K_{0}\rho_{\mathrm{in}}K_{0}^{\dagger}=p\begin{bmatrix}a&b\sqrt{1-\gamma}\\\
b^{*}\sqrt{1-\gamma}&1-a(1-\gamma)\end{bmatrix},$ (40a)
$K_{1}\rho_{\mathrm{in}}K_{1}^{\dagger}=p\gamma\begin{bmatrix}0&0\\\
0&1-a\end{bmatrix},$ (40b)
$K_{2}\rho_{\mathrm{in}}K_{2}^{\dagger}=(1-p)\begin{bmatrix}a(1-\gamma)&b\sqrt{1-\gamma}\\\
b^{*}\sqrt{1-\gamma}&1-a\end{bmatrix},$ (40c)
$K_{3}\rho_{\mathrm{in}}K_{3}^{\dagger}=(1-p)\gamma\begin{bmatrix}a&0\\\
0&0\end{bmatrix},$ (40d)

and from Eq. (1)

$\displaystyle\rho_{t}=$ $\displaystyle~{}\mathcal{E}_{t}(\rho_{\mathrm{in}})$
(41) $\displaystyle=$
$\displaystyle\begin{bmatrix}p-(1-\gamma_{t})(p-a)&b\sqrt{1-\gamma_{t}}\\\
b^{*}\sqrt{1-\gamma_{t}}&(1-p)+(1-\gamma_{t})(p-a)\end{bmatrix}.$

Consider that we take the state $\rho_{t}$ and apply
$\mathcal{E}_{t^{\prime}}$. Using the last result, we can set $a\to
p-(1-\gamma_{t})(p-a)$ and obtain for the $\ket{e_{0}}\bra{e_{0}}$ term

$\displaystyle\bra{e_{0}}\quantity[\mathcal{E}_{t^{\prime}}(\rho_{t})]\ket{e_{0}}=$
$\displaystyle~{}p-(1-\gamma_{t^{\prime}})(p-[p-(1-\gamma_{t})(p-a)])$ (42)
$\displaystyle=$
$\displaystyle~{}p-(1-\gamma_{t})(1-\gamma_{t^{\prime}})(p-a)$
$\displaystyle=$ $\displaystyle~{}p-f_{t}f_{t^{\prime}}(p-a),$

where we defined

$f_{t}:=1-\gamma_{t},$ (43)

which leads to

$\displaystyle\mathcal{E}_{t^{\prime}}(\rho_{t})=$
$\displaystyle\begin{bmatrix}p-f_{t}f_{t^{\prime}}(p-a)&b\sqrt{f_{t}f_{t^{\prime}}}\\\
b^{*}\sqrt{f_{t}f_{t^{\prime}}}&(1-p)+f_{t}f_{t^{\prime}}(p-a)\end{bmatrix}.$
(44)

On the other hand, if we apply $\mathcal{E}_{t+t^{\prime}}$ on
$\rho_{\mathrm{in}}$ we obtain

$\displaystyle\mathcal{E}_{t+t^{\prime}}(\rho_{\mathrm{in}})=$
$\displaystyle\begin{bmatrix}p-f_{t+t^{\prime}}(p-a)&b\sqrt{f_{t+t^{\prime}}}\\\
b^{*}\sqrt{f_{t+t^{\prime}}}&(1-p)+f_{t+t^{\prime}}(p-a)\end{bmatrix}.$ (45)

Notice that requiring Eq. (5) to hold implies

$\mathcal{E}_{t^{\prime}}\quantity[\mathcal{E}_{t}(\rho_{\mathrm{in}})]=\mathcal{E}_{t+t^{\prime}}(\rho_{\mathrm{in}})$
(46)

for any $\rho_{\mathrm{in}}$, meaning that for any $a,b\in\mathbb{C}$ we
should have

$\displaystyle\begin{bmatrix}p-f_{t+t^{\prime}}(p-a)&b\sqrt{f_{t+t^{\prime}}}\\\
b^{*}\sqrt{f_{t+t^{\prime}}}&(1-p)+f_{t+t^{\prime}}(p-a)\end{bmatrix}=$ (47)
$\displaystyle\begin{bmatrix}p-f_{t}f_{t^{\prime}}(p-a)&b\sqrt{f_{t}f_{t^{\prime}}}\\\
b^{*}\sqrt{f_{t}f_{t^{\prime}}}&(1-p)+f_{t}f_{t^{\prime}}(p-a)\end{bmatrix},$

which is equivalent to the condition

$f_{t}f_{t^{\prime}}=f_{t+t^{\prime}}.$ (48)

Now, we show that the only function that satisfies this condition is the
exponential. Consider the derivative

$\displaystyle f^{\prime}_{t}=$ $\displaystyle\lim_{h\to
0}\frac{f_{t+h}-f_{t}}{h}$ (49) $\displaystyle=$ $\displaystyle\lim_{h\to
0}\frac{f_{t}f_{h}-f_{t}}{h}$ $\displaystyle=$
$\displaystyle~{}f_{t}\lim_{h\to 0}\frac{f_{h}-1}{h}.$

The limit only exists if we further assume $\lim_{t\to 0}f_{t}=1$, or simply
$f_{0}=1$ for a continuous function, in which case the L’Hôpital rule applies

$\displaystyle f^{\prime}_{t}=$ $\displaystyle~{}f_{t}\lim_{h\to
0}\frac{\derivative{h}[f_{h}-1]}{\derivative{h}h}$ (50) $\displaystyle=$
$\displaystyle~{}f_{t}\lim_{h\to 0}f^{\prime}_{h}$ $\displaystyle=$
$\displaystyle~{}f_{t}f^{\prime}_{0}.$

This simple differential equation has solutions of the form

$f_{t}=e^{\alpha t},~{}\alpha\in\mathbb{R},$ (51)

while returning to $\gamma_{t}$ yields

$\gamma_{t}=1-e^{\alpha t}.$ (52)

Given that by hypothesis we must have $\dot{\gamma}>0$, then $\alpha$ is
necessarily negative, which allows us to define a positive characteristic time
for the dynamics

$\tau:=-\frac{1}{\alpha},$ (53)

finally obtaining

$\gamma_{t}=1-e^{-\frac{t}{\tau}}.$ (54)

##  Appendix C Ground state population

We simplify Eq. (9) by defining

$\alpha:=P_{0|1},$ (55) $\beta:=1-\quantity(P_{0|1}+P_{1|0}),$ (56)

and rewrite it as a discrete evolution

$a_{n+1}=\alpha+\beta a_{n}.$ (57)

Notice that expanding for a few terms yields

$a_{1}=\alpha+\beta a_{0},$ (58a) $a_{2}=\alpha+\alpha\beta+\beta^{2}a_{0},$
(58b) $a_{3}=\alpha+\alpha\beta+\alpha\beta^{2}+\beta^{3}a_{0},$ (58c)

so for a general term we have

$a_{n}=\alpha\sum_{i=0}^{n-1}\beta^{i}+\beta^{n}a_{0},$ (59)

and using the known result for the sum of a geometrical progression

$\displaystyle a_{n}=$
$\displaystyle~{}\alpha\frac{\beta^{n}-1}{\beta-1}+\beta^{n}a_{0}$ (60)
$\displaystyle=$
$\displaystyle~{}\frac{\alpha}{1-\beta}-\quantity(\frac{\alpha}{1-\beta}-a_{0})\beta^{n}.$

Then we notice that

$\frac{\alpha}{1-\beta}=\frac{P_{0|1}}{P_{0|1}+P_{1|0}}=a_{\infty}$ (61)

and return to the original variables

$a(t)=a_{\infty}-\quantity(a_{\infty}-a_{0})\quantity[1-\quantity(P_{0|1}+P_{1|0})]^{t/\Delta}.$
(62)

##  Appendix D Transition probabilities

For simplicity, we calculate $P_{0|0}$ from the Kraus representation of the
amplitude damping channel and use $P_{1|0}=1-P_{0|0}$,

$\displaystyle P_{0|0}=$ $\displaystyle\bra{0}\rho_{\Delta}\ket{0}$ (63)
$\displaystyle=$
$\displaystyle\bra{0}\quantity[\sum_{i=0}^{3}K_{i}\rho_{\mathrm{in}}K_{i}^{\dagger}]\ket{0}$
$\displaystyle=$
$\displaystyle\bra{0}\quantity[\sum_{i=0}^{3}K_{i}\quantity(\ket{0}\bra{0})K_{i}^{\dagger}]\ket{0}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{3}\bra{0}K_{i}\ket{0}\bra{0}K_{i}^{\dagger}\ket{0},$

requiring the calculation of $\bra{0}K_{i}\ket{0}$ for each $K_{i}$. Eq. (2)
yields

$\bra{0}K_{0}\ket{0}=\sqrt{p}\quantity(\cos^{2}\theta+\sqrt{1-\gamma}\sin^{2}\theta),$
(64a) $\bra{0}K_{1}\ket{0}=\sqrt{p}\sqrt{\gamma}\cos\theta\sin\theta,$ (64b)
$\bra{0}K_{2}\ket{0}=\sqrt{1-p}\quantity(\sqrt{1-\gamma}\cos^{2}\theta+\sin^{2}\theta),$
(64c) $\bra{0}K_{3}\ket{0}=\sqrt{1-p}\sqrt{\gamma}\cos\theta\sin\theta,$ (64d)

and notice that in all the four cases
$\bra{0}K_{i}^{\dagger}\ket{0}=\bra{0}K_{i}\ket{0}$, implying that
$\bra{0}K_{i}\ket{0}\bra{0}K_{i}^{\dagger}\ket{0}=\quantity(\bra{0}K_{i}\ket{0})^{2}$.
These results may be inserted into Eq. (63)

$\displaystyle P_{0|0}=$
$\displaystyle\sum_{i=0}^{3}\quantity(\bra{0}K_{i}\ket{0})^{2}$ (65)
$\displaystyle=$
$\displaystyle~{}p\quantity(\cos^{2}\theta+\sqrt{1-\gamma}\sin^{2}\theta)^{2}$
$\displaystyle+$ $\displaystyle~{}p\gamma\cos^{2}\theta\sin^{2}\theta$
$\displaystyle+$
$\displaystyle~{}(1-p)\quantity(\sqrt{1-\gamma}\cos^{2}\theta+\sin^{2}\theta)^{2}$
$\displaystyle+$ $\displaystyle~{}(1-p)\gamma\cos^{2}\theta\sin^{2}\theta$
$\displaystyle=$
$\displaystyle~{}\quantity(1-\gamma+p\gamma)\cos^{4}\theta+\quantity(1-p\gamma)\sin^{4}\theta$
$\displaystyle+$
$\displaystyle~{}\quantity(\gamma+2\sqrt{1-\gamma})\cos^{2}\theta\sin^{2}\theta,$

and applying trigonometric identities results in

$\displaystyle P_{0|0}=$
$\displaystyle~{}\frac{1}{4}\Big{[}3-\gamma+\sqrt{1-\gamma}+2\gamma(2p-1)\cos(2\theta)$
(66) $\displaystyle+$
$\displaystyle~{}\quantity(1-\gamma-\sqrt{1-\gamma})\cos(4\theta)\Big{]},$

while similar calculations for $P_{1|1}$ yield

$\displaystyle P_{1|1}=$
$\displaystyle~{}\frac{1}{4}\Big{[}3-\gamma+\sqrt{1-\gamma}-2\gamma(2p-1)\cos(2\theta)$
(67) $\displaystyle+$
$\displaystyle~{}\quantity(1-\gamma-\sqrt{1-\gamma})\cos(4\theta)\Big{]}.$

Finally, we use the explicit expression for $\gamma(t)$ and return to the
transition probabilities

$\displaystyle P_{0|1}=$
$\displaystyle~{}\frac{1}{4}\Bigg{[}\Big{(}1-\cos(4\theta)\Big{)}\quantity(1-e^{-\Delta/2\tau})$
(68a) $\displaystyle+$
$\displaystyle~{}\Big{(}1+2(2p-1)\cos(2\theta)+\cos(4\theta)\Big{)}\quantity(1-e^{-\Delta/\tau})\Bigg{]}$
$\displaystyle P_{1|0}=$
$\displaystyle~{}\frac{1}{4}\Bigg{[}\Big{(}1-\cos(4\theta)\Big{)}\quantity(1-e^{-\Delta/2\tau})$
(68b) $\displaystyle+$
$\displaystyle~{}\Big{(}1-2(2p-1)\cos(2\theta)+\cos(4\theta)\Big{)}\quantity(1-e^{-\Delta/\tau})\Bigg{]}.$

Now we can explicitly write

$\displaystyle P_{0|1}+P_{1|0}=$
$\displaystyle~{}1-\frac{1}{2}\Big{[}\quantity(1-\cos(4\theta))e^{-\Delta/2\tau}$
(69) $\displaystyle+$
$\displaystyle~{}\quantity(1+\cos(4\theta))e^{-\Delta/\tau}\Big{]}$
$\displaystyle=$
$\displaystyle~{}1-\quantity[\sin^{2}(2\theta)e^{-\Delta/2\tau}+\cos^{2}(2\theta)e^{-\Delta/\tau}]$

and

$a_{\infty}(\Delta,\theta)=\frac{1}{2}\quantity[1+\frac{(2p-1)\cos(2\theta)}{\cos^{2}(2\theta)+\sin^{2}(2\theta)\frac{1-e^{-\Delta/2\tau}}{1-e^{-\Delta/\tau}}}],$
(70)

as shown in the main text.

## References

  * Feynman [2018] R. P. Feynman, Simulating physics with computers, in  _Feynman and computation_ (CRC Press, 2018) pp. 133–153.
  * Rivas and Huelga [2012] A. Rivas and S. F. Huelga,  _Open Quantum Systems_ , Vol. 10 (Springer, 2012).
  * Terhal [2015] B. M. Terhal, Quantum error correction for quantum memories, Reviews of Modern Physics 87, 307 (2015).
  * Shor [1995] P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Physical Review A 52, R2493 (1995).
  * Laflamme  _et al._ [1996] R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Perfect quantum error correcting code, Physical Review Letters 77, 198 (1996).
  * Calderbank and Shor [1996] A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Physical Review A 54, 1098 (1996).
  * Steane [1996] A. M. Steane, Error correcting codes in quantum theory, Physical Review Letters 77, 793 (1996).
  * Freedman  _et al._ [2003] M. Freedman, A. Kitaev, M. Larsen, and Z. Wang, Topological quantum computation, Bulletin of the American Mathematical Society 40, 31 (2003).
  * Brun  _et al._ [2006] T. Brun, I. Devetak, and M.-H. Hsieh, Correcting quantum errors with entanglement, Science 314, 436 (2006).
  * Pastawski  _et al._ [2015] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, Journal of High Energy Physics 2015, 1 (2015).
  * Fiusa  _et al._ [2023] G. Fiusa, D. O. Soares-Pinto, and D. P. Pires, Fidelity-based distance bounds for $n$-qubit approximate quantum error correction, Physical Review A 107, 032422 (2023).
  * Boyers  _et al._ [2019] E. Boyers, M. Pandey, D. K. Campbell, A. Polkovnikov, D. Sels, and A. O. Sushkov, Floquet-engineered quantum state manipulation in a noisy qubit, Physical Review A 100, 012341 (2019).
  * Viola and Lloyd [1998] L. Viola and S. Lloyd, Dynamical suppression of decoherence in two-state quantum systems, Physical Review A 58, 2733 (1998).
  * Viola  _et al._ [1999] L. Viola, E. Knill, and S. Lloyd, Dynamical decoupling of open quantum systems, Physical Review Letters 82, 2417 (1999).
  * Viola and Knill [2003] L. Viola and E. Knill, Robust dynamical decoupling of quantum systems with bounded controls, Physical Review Letters 90, 037901 (2003).
  * Khodjasteh and Lidar [2005] K. Khodjasteh and D. A. Lidar, Fault-tolerant quantum dynamical decoupling, Physical Review Letters 95, 180501 (2005).
  * Uhrig [2007] G. S. Uhrig, Keeping a quantum bit alive by optimized $\pi$-pulse sequences, Physical Review Letters 98, 100504 (2007).
  * West  _et al._ [2010] J. R. West, D. A. Lidar, B. H. Fong, and M. F. Gyure, High fidelity quantum gates via dynamical decoupling, Physical Review Letters 105, 230503 (2010).
  * Yang  _et al._ [2011] W. Yang, Z.-Y. Wang, and R.-B. Liu, Preserving qubit coherence by dynamical decoupling, Frontiers of Physics in China 6, 2 (2011).
  * Khodjasteh  _et al._ [2013] K. Khodjasteh, J. Sastrawan, D. Hayes, T. J. Green, M. J. Biercuk, and L. Viola, Designing a practical high-fidelity long-time quantum memory, Nature Communications 4, 1 (2013).
  * Du  _et al._ [2009] J. Du, X. Rong, N. Zhao, Y. Wang, J. Yang, and R. Liu, Preserving electron spin coherence in solids by optimal dynamical decoupling, Nature 461, 1265 (2009).
  * Szwer  _et al._ [2010] D. J. Szwer, S. C. Webster, A. M. Steane, and D. M. Lucas, Keeping a single qubit alive by experimental dynamic decoupling, Journal of Physics B: Atomic, Molecular and Optical Physics 44, 025501 (2010).
  * Biercuk  _et al._ [2009] M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, W. M. Itano, and J. J. Bollinger, Experimental uhrig dynamical decoupling using trapped ions, Physical Review A 79, 062324 (2009).
  * Souza  _et al._ [2012] A. M. Souza, G. A. Álvarez, and D. Suter, Robust dynamical decoupling, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, 4748 (2012).
  * Zhang and Suter [2015] J. Zhang and D. Suter, Experimental protection of two-qubit quantum gates against environmental noise by dynamical decoupling, Physical Review Letters 115, 110502 (2015).
  * Zurek [1981] W. H. Zurek, Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?, Physical Review D 24, 1516 (1981).
  * Itano  _et al._ [1990] W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Quantum zeno effect, Physical Review A 41, 2295 (1990).
  * Presilla  _et al._ [1996] C. Presilla, R. Onofrio, and U. Tambini, Measurement quantum mechanics and experiments on quantum zeno effect, annals of physics 248, 95 (1996).
  * Kondo  _et al._ [2016] Y. Kondo, Y. Matsuzaki, K. Matsushima, and J. G. Filgueiras, Using the quantum zeno effect for suppression of decoherence, New Journal of Physics 18, 013033 (2016).
  * Möbus and Wolf [2019] T. Möbus and M. M. Wolf, Quantum zeno effect generalized, Journal of Mathematical Physics 60, 052201 (2019).
  * Burgarth  _et al._ [2019] D. Burgarth, P. Facchi, H. Nakazato, S. Pascazio, and K. Yuasa, Generalized adiabatic theorem and strong-coupling limits, Quantum 3, 152 (2019).
  * Burgarth  _et al._ [2020] D. Burgarth, P. Facchi, H. Nakazato, S. Pascazio, and K. Yuasa, Quantum zeno dynamics from general quantum operations, Quantum 4, 289 (2020).
  * Becker  _et al._ [2021] S. Becker, N. Datta, and R. Salzmann, Quantum zeno effect in open quantum systems, in  _Annales Henri Poincaré_ , Vol. 22 (Springer, 2021) pp. 3795–3840.
  * Álvarez  _et al._ [2006] G. A. Álvarez, E. P. Danieli, P. R. Levstein, and H. M. Pastawski, Environmentally induced quantum dynamical phase transition in the spin swapping operation, The Journal of chemical physics 124, 194507 (2006).
  * Álvarez  _et al._ [2007] G. A. Álvarez, E. P. Danieli, P. R. Levstein, and H. M. Pastawski, Decoherence under many-body system-environment interactions: A stroboscopic representation based on a fictitiously homogenized interaction rate, Physical Review A 75, 062116 (2007).
  * Fernández-Alcázar and Pastawski [2015] L. J. Fernández-Alcázar and H. M. Pastawski, Decoherent time-dependent transport beyond the landauer-büttiker formulation: A quantum-drift alternative to quantum jumps, Physical Review A 91, 022117 (2015).
  * Wilde [2013] M. M. Wilde,  _Quantum Information Theory_ (Cambridge University Press, 2013).
  * Nielsen and Chuang [2000] M. A. Nielsen and I. Chuang,  _Quantum Computation and Quantum Information_ (Cambridge University Press, 2000).
  * Rivas  _et al._ [2014] Á. Rivas, S. F. Huelga, and M. B. Plenio, Quantum non-markovianity: characterization, quantification and detection, Reports on Progress in Physics 77, 094001 (2014).
  * Breuer  _et al._ [2016] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Colloquium: Non-markovian dynamics in open quantum systems, Reviews of Modern Physics 88, 021002 (2016).
  * De Vega and Alonso [2017] I. de Vega and D. Alonso, Dynamics of non-markovian open quantum systems, Reviews of Modern Physics 89, 015001 (2017).
  * Li  _et al._ [2018] L. Li, M. J. Hall, and H. M. Wiseman, Concepts of quantum non-markovianity: A hierarchy, Physics Reports 759, 1 (2018).
  * Chruściński [2022] D. Chruściński, Dynamical maps beyond markovian regime, Physics Reports 992, 1 (2022).