# Kaniadakis holographic dark energy in Brans-Dicke cosmology

S<EMAIL_ADDRESS>1 Research Institute for Astronomy
and Astrophysics of Maragha (RIAAM), University of Maragheh, P.O. Box
55136-553, Maragheh, Iran

###### Abstract

By using the holographic hypothesis and Kaniadakis generalized entropy, which
is based on relativistic statistical theory and modified Boltzmann-Gibbs
entrory, we build Kaniadakis holographic dark energy (DE) model in the Brans-
Dicke framework. We drive cosmological parameters of Kaniadakis holographic DE
model, with IR cutoff as the Hubble horizon, in order to investigate its
cosmological consequences. Our study shows that, even in the absence of an
interaction between the dark sectors of cosmos, the Kaniadakis holographic
dark energy model with the Hubble radius as IR cutoff can explain the present
accelerated phase of the universe expansion in the Brans-Dicke theory. The
stability of the model, using the squared of sound speed, has been checked and
it is found that the model is unstable in non-interacting case and can be
stable for some range of model parameters within the interacting case.

##  I Introduction

The possibility of justifying the current accelerated universe using the
energy density of quantum fields in vacuum leads to a well-known model called
holographic dark energy (HDE) HDE3 ; HDE4 ; RevH . In the original version,
Bekenstein entropy ($S_{BH}$) and apparent horizon are employed as the entropy
of a gravitational system and IR cutoff, respectively HDE4 . Its shortcomings
motivate physicists to modify the model for example by considering other IR
cutoffs, or modified entropies RevH ; HDE5 ; HDE17 ; HDE01 ; HDE10 ; HDE12 ;
stab . Motivated by the long range nature of gravity, recently, some new
entropies have been proposed as the alternatives of Bekenstein entropy which
show suitable results by themselves in cosmological setups Tsallis ; morad2016
; morad2017 ; kum ; Sayahian ; Renyi ; Tavayef ; age ; morad2020 ; maj .

These entropies may also be justified by the quantum aspects of gravity epl1 ;
epl2 ; epjplus ; barrow , and additionally, the corresponding HDE models
provide acceptable description for the current phase of the universe expansion
even if the Hubble radius is considered as the IR cutoff Sayahian ; Renyi ;
Tavayef ; morad2020 . The latter is important issue, because this horizon is
thermodynamically the proper causal boundary 66 ; Renyi , a property that
motivates physicists to deem the Hubble horizon as the IR cutoff. It was
recently shown that the Kaniadakis entropy of a black hole is obtained as
morad2020

$S_{\kappa}=\frac{1}{\kappa}\sinh(\kappa S_{BH}),$ (1)

where $\kappa$ is a free parameter. Motivated by the thermodynamic importance
of the apparent horizon (Hubble horizon) 66 , authors in Ref. morad2020
utilized this horizon as the IR cutoff, and proposed a new HDE model called
KHDE (Kaniadakis HDE) which seems to be able to justify the current
accelerated expanding phase of the universe. The applicability of KHDE in non-
flat universe as well as in the presence of other IR cutoffs have also been
addressed in Refs. KHDEmorad ; KHDE2 . In general, since the non-extensive
features of gravity, and also the properties of DE, are not completely known,
there is no restriction on the values of $\kappa$ at present step, and more
accurate observations as well as other parts of physics may help us impose
some constraints on its values. Therefore, it is expectable to see different
intervals for $\kappa$ in meeting the observational requirement, depending on
the primary presumptions such as the IR cutoff morad2020 ; KHDEmorad ; KHDE2 .
On the other hand, the Brans-Dicke (BD) gravity is an alternative to general
relativity in which the gravitational coupling $G$, is not a constant, and is
replaced with the inverse of a scalar field $(\phi)$ 4not . It is true that
although BD theory can explain accelerated expansion without resorting to DE
models, the obtained values for BD parameter ($\omega$) is lower than the
observational limit jcap ; pav . Therefore, there has been much attempt to
solve this problem by introducing different DE models in BD theory Gong ;
Setare ; Banerjee2 ; Banerjee ; Xu ; Jamil ; Khodam . Hence, according to
dynamic behavior of the HDE, it is much appropriate to study it in the BD
dynamic framework. In this work, we would like to study the results of using
the KHDE model with the Hubble radius as the cut-off in the BD cosmology. In
the next section, we extract the KHDE density in BD gravity and investigate
its cosmological parameters for the non-interacting case. In section III, we
consider interaction KHDE model and investigate its cosmological evolution.
Section IV is devoted to stability of model and our conclusions are drawn in
section V.

##  II  Non-interacting Kaniadakis holographic dark energy in the Brans-Dicke
cosmology

We consider a homogeneous and isotropic FRW universe described by the line
element

${\rm d}s^{2}=-{\rm d}t^{2}+a^{2}(t)\left(\frac{{\rm
d}r^{2}}{1-kr^{2}}+r^{2}{\rm d}\Omega^{2}\right),$ (2)

where $k=0,1,-1$ represent a flat, closed and open maximally symmetric space,
respectively. The Brans-Dicke field equations can be written as Banerjee

$\frac{3}{4\omega}\phi^{2}\Big{(}H^{2}+\frac{k}{a^{2}}\Big{)}-\frac{\dot{\phi}^{2}}{2}+\frac{3}{2\omega}H\dot{\phi}\phi=\rho_{M}+\rho_{\Lambda},$
(3)
$\frac{-\phi^{2}}{4\omega}\Big{(}\frac{2\ddot{a}}{a}+H^{2}+\frac{k}{a^{2}}\Big{)}-\frac{1}{\omega}H\dot{\phi}\phi-\frac{1}{2\omega}\ddot{\phi}\phi-\frac{\dot{\phi}^{2}}{2}\Big{(}1+\frac{1}{\omega}\Big{)}=p_{\Lambda},$
(4)
$\ddot{\phi}+3H\dot{\phi}-\frac{3}{2\omega}\Big{(}\frac{\ddot{a}}{a}+H^{2}+\frac{k}{a^{2}}\Big{)}\phi=0.$
(5)

where $\phi^{2}=\omega/(2\pi G_{\rm eff})$, $G_{\rm eff}$ is the effective
gravitational constant 4not , $H=\dot{a}/a$ is the Hubble parameter and
$\rho_{m}$, $\rho_{\Lambda}$ and $p_{\Lambda}$ are the pressureless dark
matter density, dark energy density and pressure of DE, respectively.
Following Banerjee2 , we assume that the BD field $\phi$ is proportional to
scalar field as $(\phi\propto a^{n})$, then we get

$\dot{\phi}=nH\phi,$ (6)

and hence

$\ddot{\phi}=n^{2}H^{2}\phi+n\dot{H}\phi,$ (7)

where a dot denotes derivative with respect to time. According to the
Kaniadakis generalized entropy, that it is independent of gravitational
theory, and holographic principle to introduce a KHDE model in BD gravity, one
arrives at the following relation for the Kanadakis holographic energy
density(KHDE) in which the apparent horizon in the flat Universe is considered
as the IR cut-off $(L=H^{-1})$

$\rho_{D}=\frac{3c^{2}\phi^{2}H^{4}}{4\omega\kappa}\sinh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right).$ (8)

Here, $c^{2}$ is an unknown constant HDE3 . It is clear that in the limit
$\kappa\rightarrow 0$ and $G_{\rm eff}\rightarrow G$, the energy density of
HDE in standard cosmology is restored. Also, for the limiting case $G_{\rm
eff}\rightarrow G$, the above equation yields the KHDE density in the standard
Einstein gravity KHDEmorad . Defining the critical density as
$\rho_{cr}=\frac{3\phi^{2}H^{2}}{4\omega}$, the dimensionless density
parameters are presented by

$\displaystyle\Omega_{m}=\frac{\rho_{m}}{\rho_{cr}}$ $\displaystyle=$
$\displaystyle\frac{4\omega\rho_{m}}{3\phi^{2}H^{2}},$
$\displaystyle\Omega_{D}=\frac{\rho_{D}}{\rho_{cr}}$ $\displaystyle=$
$\displaystyle\frac{4\omega\rho_{D}}{3\phi^{2}H^{2}},$
$\displaystyle\Omega_{\phi}=\frac{\rho_{\phi}}{\rho_{cr}}$ $\displaystyle=$
$\displaystyle 2n\Big{(}\frac{n\omega}{3}-1\Big{)}.$ (9)

Using the above definition, we can write Eq. (3) as follows

$\Omega_{m}+\Omega_{D}=1+\Omega_{\phi}.$ (10)

Using the second line of definitions (II) along with Eq. (8), the
dimensionless density parameter of KHDE is obtained as

$\Omega_{D}=\frac{c^{2}H^{2}}{\kappa}\sinh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right).$ (11)

Here, we also assume that two dark sectors of the universe do not interact
with each other, i.e., there is no energy exchange between these cosmic
sectors, then the energy conservation equations are given as follow

$\dot{\rho}_{D}+3H(1+\omega_{D})\rho_{D}=0,$ (12)

and

$\dot{\rho}_{m}+3H\rho_{m}=0,$ (13)

where $\omega_{D}=\frac{p_{D}}{\rho_{D}}$ denotes the equation of state (EoS)
parameter of DE. Taking the time derivative of Eq. (8), we have

$\displaystyle\dot{\rho}_{D}$ $\displaystyle=$ $\displaystyle
2H\rho_{D}\left(n+\frac{\dot{2H}}{H^{2}}\right)$ (14) $\displaystyle+$
$\displaystyle\frac{3\pi
c^{2}\phi^{2}H^{3}}{2\omega}\cosh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right)\left(n-\frac{\dot{H}}{H^{2}}\right).$

Taking the time derivative of Eq. (3) and combining the result with Eqs. (6),
(7), (13) and (14), one can easily get

$\displaystyle\frac{\dot{H}}{H^{2}}$ $\displaystyle=$
$\displaystyle\Big{[}-\frac{9\Omega_{m}^{0}H_{0}^{2}\phi_{0}^{2}}{4\omega\phi^{2}H^{2}}+(2\omega
n^{2}-6n-3)\frac{n}{2\omega}$ $\displaystyle\\!\\!\\!+$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{3nc^{2}\pi^{2}\phi^{2}}{\omega^{2}}\cosh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right)+\frac{3nc^{2}H^{2}}{2\omega\kappa}\sinh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right)\Big{]}$ $\displaystyle\\!\\!\\!\times$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Bigg{[}\frac{3c^{2}\pi^{2}\phi^{2}}{\omega^{2}}\cosh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right)-\frac{3c^{2}H^{2}}{\omega\kappa}\sinh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right)\Bigg{]}^{-1},$

where $\Omega_{m}^{0}$, $H_{0}$ and $\phi_{0}$ are present values of matter
density parameter, hubble parameter and BD field, respectively. Combining Eqs.
(12) and (14), one can obtain the EoS parameter for the KHDE model in BD
gravity as

$\displaystyle\omega_{D}=-1-\frac{2n}{3}-\frac{4\kappa\pi^{2}\phi^{2}}{3\omega
H^{2}}\coth\left(\frac{2\kappa\pi^{2}\phi^{2}n}{\omega H^{2}}\right)$ (16)
$\displaystyle-\frac{4}{3}\Big{[}1-\frac{4\kappa\pi^{2}\phi^{2}}{3\omega
H^{2}}\coth\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right)\Big{]}\frac{\dot{H}}{H^{2}}.$

One can also find out that, in the limiting case $\kappa\rightarrow 0$, this
equation is reduced to the EoS parameter for the original HDE model in BD
gravity Ghaffari1 . It is also clear that the EoS parameter for KHDE in
standard cosmology is recovered at the appropriate limit $n\rightarrow 0$ and
$(\omega\rightarrow 0)$ KHDEmorad .

Also, using the deceleration parameter relation $q=-1-\frac{\dot{H}}{H^{2}}$
and Eq. (II), one gets

$\displaystyle q=-1-\Big{[}-\frac{9\Omega_{m}^{0}H_{0}^{2}}{4\omega
H^{2}}+(2\omega n^{2}-6n-3)\frac{n}{2\omega}$
$\displaystyle+\frac{3nc^{2}\pi^{2}\phi^{2}}{\omega^{2}}\cosh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right)\\!+\\!\frac{3nc^{2}H^{2}}{2\omega\kappa}\sinh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right)\Big{]}$
$\displaystyle\times\Bigg{[}\frac{3c^{2}\pi^{2}\phi^{2}}{\omega^{2}}\cosh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right)\\!-\\!\frac{3c^{2}H^{2}}{\omega\kappa}\sinh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right)\Bigg{]}^{-1}.$

One can also examine the fate of the universe filled with DM and KHDE
components. To this end, we should consider the effective EoS parameter

$\omega_{\rm eff}=\frac{P}{\rho}=\frac{p_{DE}}{\rho_{M}+\rho_{D}}.$ (18)

The behavior of the $\omega_{\rm eff}$ is plotted in Figs. 2 and 4.

###  II.1 Cosmological Evolution

In what follows, we investigate the cosmological behavior of a universe filled
with KHDE and DM. As it is apparent form Eqs. (11), (16) and (II), the
cosmological parameters such as dimensionless density, EoS and deceleration
parameters completely depend on the behavior of Hubble parameter function
$H(z)$ and its differential equation with number of constant parameters. As
differential equation (II) cannot be solved analytically, we proceed with
solving it numerically for fixed values $H_{0}=67.9$ HubbleOb ,
$\Omega_{m}=0.3$ OmegamOb , $n=0.005$, $\omega=10$ at the present time
$(z=0)$.
In the upper panel of Fig. (1) we have sketched the behavior of dimensionless
density parameter $\Omega_{D}$ in terms of redshift. From this figure we
clearly see that at the early times of the universe ($z\rightarrow\infty$), we
have $\Omega_{D}\rightarrow 0.2$ in full agreement with Eq. (10), while at the
late-time ($z\rightarrow-1$) the DE dominates ($\Omega_{D}\rightarrow 1$). It
is clear that for decreasing values of $c$ and $\kappa$ parameters
$\Omega_{D}$ become smaller at the late time, which means that more energy is
transferred to the matter component.
In the middle panel of Fig. (1) we have plotted the evolution of the
deceleration parameter $q(z)$ versus redshift $z$ where we have considered the
initial condition as mentioned above and various values of $c,\kappa$ and
$\phi_{0}$ in order to study the effects of the model parameters. As we can
see, the KHDE model with the Hubble cutoff in BD gravity can describe the
current accelerated expansion of the universe, even in the absence of an
interaction between the two dark sectors of cosmos unlike the standard HDE in
the BD gravity Xu . The transition from deceleration to acceleration is
realized around $0.25<z<0.8$ which is quite in agreement with cosmological
observations Daly ; Komatsu ; Salvatelli . Also, we observe, the behavior of
the deceleration parameter depends on the values of $c,\kappa$ and $\phi_{0}$.
For decreasing model parameters $c$ and $\kappa$, the transition point moves
to older universe and also the value of deceleration parameter at the present
time $q(z=0)=q_{0}$ decreases.
Finally, in the lower panel, we depict $\omega_{D}(z)$ for various values of
$\kappa,\phi_{0},c$ parameters. As is clear, there is a transition from the
quintessence regime to the phantom regime at the certain values of redshift.
This value of the redshift depends on the $c$ and $\kappa$ parameters of KHDE
model, so that, with increasing values of $c$ and $\kappa$ this transition
will occur in the near present time (smaller redshift) and also the EoS
parameter at the present time $(\omega_{D}(z=0))$ will be smaller.

We have plotted the evolution of $\omega_{\rm eff}(z)$ for KHDE in BD gravity
in Fig. 2. As it is obvious in this figure, at the early time we have the
$\omega_{\rm eff}\rightarrow 0$, which shows a pressureless DM dominated
universe, while at the late time we have $\omega_{\rm eff}\rightarrow-1$,
which means ending of the universe in a Big-Rip singularity.

Figure 1: The evolution of the $\omega_{D}(z)$, $q(z)$ and $\omega_{D}(z)$
parameters for Non-interacting KHDE in BD gravity, in which fix parameters
$H_{0}=67.9,\Omega_{m}=0.3,n=0.005,\omega=10$ and various values of
$\kappa,\phi_{0},c$ are adopted. Figure 2: The evolution of the $\omega_{\rm
eff}(z)$ for non-interacting KHDE in BD gravity, in which fix parameters
$H_{0}=67.9,\Omega_{m}=0.3,n=0.005,\omega=10$ and various values of
$\kappa,\phi_{0},c$ are adopted.

##  III Interacting KHDE model

In this section, we focus on the interaction between KHDE model and DM. For
the FRW universe filled with DE and DM interacting with each other, the
conservation equations are given by

$\dot{\rho}_{D}+3H(1+\omega_{D})\rho_{D}=-Q,$ (19)
$\dot{\rho}_{m}+3H\rho_{m}=Q,$ (20)

where $Q$ represents the interaction term, and we assume that it has the form
$Q=H(\alpha\rho_{m}+\beta\rho_{D})$ pavonQ , in which $\alpha$ and $\beta$ are
coupling constants. Taking the time derivative of Eq. (3) along with using
Eqs. (6), (7), (14) and (20), we get

$\displaystyle\frac{\dot{H}}{H^{2}}=\Bigg{[}-\frac{3\Omega_{m}^{0}H_{0}^{2}H_{0}^{2}}{4\phi^{2}H^{2}}+(2\omega
n^{2}-6n-3)\frac{n}{6}+\frac{\alpha\Omega_{m}}{4}$ (21) $\displaystyle+$
$\displaystyle\\!\\!\\!\frac{nc^{2}\pi^{2}\phi^{2}}{\omega}\cosh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right)+\frac{c^{2}H^{2}(n+\frac{\beta}{2})}{2\kappa}\sinh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right)\Bigg{]}$ $\displaystyle\times$
$\displaystyle\\!\\!\\!\Bigg{[}\frac{c^{2}\pi^{2}\phi^{2}}{\omega}\cosh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right)-\frac{c^{2}H^{2}}{\kappa}\sinh\left(\frac{2\kappa\pi^{2}\phi^{2}}{\omega
H^{2}}\right)$ $\displaystyle+$ $\displaystyle\\!\\!\\!n-\frac{\omega
n^{2}}{3}+\frac{1}{2}\Bigg{]}^{-1}.$

In the absence of interaction term $(\alpha=\beta=0)$, Eq. (21) is reduced to
its respective relation in the previous section.

Combining Eqs. (19) and (14), one can obtain the following expression for EoS
parameter

$\displaystyle\omega_{D}$ $\displaystyle=$
$\displaystyle-1+\frac{1}{3}(\alpha-\beta-2n)$ $\displaystyle-$
$\displaystyle\frac{4\kappa\pi^{2}\phi^{2}}{3\omega
H^{2}}\coth\left(\frac{2\kappa\pi^{2}\phi^{2}n}{\omega
H^{2}}\right)\left(n-\frac{\dot{H}}{H^{2}}\right)$ $\displaystyle+$
$\displaystyle\frac{\kappa\alpha(2\omega
n^{2}-2n-1)}{9c^{2}H^{2}\sinh\left(\frac{2\kappa\pi^{2}\phi^{2}n}{\omega
H^{2}}\right)}.$

###  III.1 Cosmological Evolution

Now, we study the cosmic evolution of the KHDE in the presence of the
interaction between two dark components. For this purpose we again solve Eq.
(21) numerically and then use it to plot the evolution of the cosmological
parameters. It should be noted that in this section, fixed values of
$H_{0}=67.9,\Omega_{m}=0.3,\kappa=0.8,\phi_{0}=0.4,n=0.06$ and $\omega=10$ are
considered. We have plotted the behavior of dimensionless density parameter
$\Omega_{D}$, the deceleration parameter $q$ and the EoS parameter with
respect $z$ in Fig. 3

Using Eqs. II and (21), we can obtain the evolution of $\Omega_{D}(z)$ for
interacting KHDE which has been plotted in the upper diagram in Fig. (3). From
this figure, it is clear that $\Omega_{D}\rightarrow 0.2$ at the early times
of the universe ($z\rightarrow\infty$), according to Eq. (10) and the DE
dominates ($\Omega_{D}\rightarrow 1$) at late-time ($z\rightarrow-1$).

Behavior of the deceleration parameter $q(z)$ is illustrated in the middle
diagram of Fig. 3. As it is observed, our universe undergoes a transition from
deceleration to acceleration phase at the redshift value around $0.3<z<0.6$.
Such a transition for $\alpha>0$ and $\beta<0$ happens at a higher redshift
(earlier universe).

From the latest diagram one can easily see that the EoS parameter can remain
in the quintessence regime or can also enter the phantom regime, which depends
on the coupling constants $\alpha$ and $\beta$.

Figure 3: The evolution of the $\Omega_{D}(z)$, $q$ for interacting KHDE for
different para. We have taken the
$H_{0}=67.9,\Omega_{m}=0.3,\kappa=0.8,\phi_{0}=0.4,n=0.06$ and $\omega=10$ as
the initial condition. Figure 4: The evolution of the effective EoS parameter
$\omega_{\rm eff}$ versus redshift parameter $z$ for interacting THDE.

##  IV Stability

In this section we would like to study the classical stability of the KHDE
model against small perturbations in the background. In the perturbation
theory, the sign of squared of the sound speed, $v_{s}^{2}$, determines the
stability of the background evolution. For $v_{s}^{2}>0$ the model is stable
against the perturbations, since the given perturbation propagates in the
environment, while for $v_{s}^{2}<0$ the amplitude of perturbations grows
within the environment and consequently the model is unstable. The squared
sound speed $v_{s}^{2}$ is given by

$v_{s}^{2}=\frac{dp}{d\rho_{D}}=\frac{\dot{p}}{\dot{\rho}_{D}}.$ (23)

By differentiating $p_{D}$ with respect to time together with inserting the
result into Eq. (23), and using Eq. (14), we can get

$v_{s}^{2}=\omega_{D}+\frac{\omega_{D}^{\prime}}{2\delta
n+2(2-\delta)\frac{\dot{H}}{H^{2}}},$ (24)

for the squared sound speed.

###  IV.1 Non-interacting case

Taking the time derivative of Eq. (16) and using Eqs. (11), (14), (II) and
(24), we can obtain $v_{s}^{2}$ for the non-interacting KHDE with the Hubble
cutoff in BD cosmology. Since this expression is too long, we shall not
present it here, and only plot it in Fig. 5. From this figure we see that the
non-interacting KHDE model in BD gravity is unstable $(v_{s}^{2}<0)$.

Figure 5: The evolution of $v_{s}^{2}$ against redshift parameter $z$ for non-
interacting KHDE for he initial condition
$H_{0}=67.9,\Omega_{m}=0.3,n=0.05,\omega=10$ and different values of
$\kappa,\phi_{0},c$.

###  IV.2 Interacting case

We now calculate the squared of the sound speed for the interacting KHDE
model. By taking the time derivative of Eq. (III), and combining the result
with Eqs. (8), (14), (21) and (24), we can obtain $v_{s}^{2}$. Again, since
this expression is too long, we do not present it here. We have plotted
$v_{s}^{2}$ of the interacting KHDE in the BD gravity in Figs. 6. As it is
clear, in the presence of interaction term, the squared of the sound speed is
positive in a period of time meaning that the interacting KHDE can be stable
for $\alpha>0$ (whether $\beta$ is positive or negative).

Figure 6: The evolution of $v_{s}^{2}(z)$ for interacting KHDE in BD gravity.
The initial condition $H_{0}=67.9,\Omega_{m}=0.3,n=0.05,\omega=10$ and
different values of $\kappa,\phi_{0},c$ are adopted.

##  V Concluding remarks

In this paper we studied the results of using the Kaniadakis entropy to
construct a holographic dark energy model (called Kaniadakis holographic DE)
in the framework of BD theory. We considered the Hubble horizon as the IR
cutoff and studied the behavior of KHDE model for both the interacting and
non-interacting cases. To this end we obtained the equation governing the
evolution of Hubble parameter and used its numerical solution in order to
study the evolution of the corresponding cosmological parameters.
From the behavior of deceleration parameter $q$, we found that, unlike the
standard HDE in the BD gravity, the KHDE with the Hubble cutoff in BD gravity
can provide a setting for current accelerated expansion of the universe, even
in the absence of an interaction between the two dark sectors of cosmos. In
addition, the behavior of EoS parameter, for both interacting and non-
interacting cases, shows that the KHDE model can cross the phantom divide
depending on the values of free model parameters. Moreover, the EoS parameter
in the presence of interaction shows an interesting behavior, i.e., it can
represent a quintessence regime ( $\omega_{D}>-1$), a phantom one
$(\omega_{D}<-1)$ or cosmological constant $(\omega_{D}=-1)$. In order to
study the fate of the universe filled with DM and KHDE, we have plotted the
behavior of the effective EoS parameter $\omega_{\rm eff}(z)$ for various
interacting and non-interacting parameters in Figs. 2 and 4. We observed that,
at high redshifts, the effective matter content of the universe is in the dust
form and evolves to a DE form at late times. This study indicated that as
$\omega_{\rm eff}\rightarrow-1$ for $z\rightarrow-1$, then the occurrence of a
Big-Rip singularity at a certain time in the future is possible.
Finally, the analysis of the squared of sound speed $v_{s}^{2}$ revealed that
non-interacting KHDE in BD gravity is unstable against small perturbations in
the background. This implies that considering an interaction between two dark
sectors seems more reasonable. We therefore found that interacting KHDE can
provide stability in the period of time.

## References

  * (1) A. G. Cohen, D. B. Kaplan, A. E. Nelson, Phys. Rev. Lett. 82, 4971 (1999).
  * (2) M. Li, Phys. Lett. B 603, 1 (2004).
  * (3) S. Wang, Y. Wang, M. Li, Phys. Rep. 696, 1 (2017).
  * (4) B. Guberina, R. Horvat, H. Nikolić, 01, 012 (2007).
  * (5) S. Ghaffari, M. H. Dehghani, A. Sheykhi, Phys. Rev. D 89, 123009 (2014).
  * (6) P. Horava, D. Minic, Phys. Rev. Lett. 85, 1610 (2000).
  * (7) S. Thomas, Phys. Rev. Lett. 89, 081301 (2002).
  * (8) S. D. H. Hsu, Phys. Lett. B 594, 13 (2004).
  * (9) Y. S. Myung, Phys. Lett. B 652, 223 (2007).
  * (10) C. Tsallis, L. J. L. Cirto, Eur. Phys. J. C 73, 2487 (2013).
  * (11) H. Moradpour, Int. J. Theor. Phys. 55, 4176 (2016).
  * (12) H. Moradpour, A. Bonilla, E.M.C. Abreu, J.A. Neto, Phys. Rev. D 96(12), 123504 (2017).
  * (13) N. Komatsu, Eur. Phys. J. C 77, 229 (2017).
  * (14) M. Abdollahi Zadeh, A. Sheykhi and H. Moradpour, Mod. Phys. Lett A 34, 1950086 (2019).
  * (15) A. Sayahian Jahromi et al, Phys. Lett. B 780, 21 (2018).
  * (16) H. Moradpour et al. Eur. Phys. J. C 78, 829 (2018).
  * (17) M. Tavayef, A. Sheykhi, K. Bamba and H. Moradpour, Phys. Lett. B. 781, 195 (2018).
  * (18) Moradpour, H., Ziaie, A. H. Zangeneh, M. K. Eur. Phys. J. C 80, 732 (2020).
  * (19) M. Abdollahi Zadeh et al. Eur. Phys. J. C, 78 11 (2018) 940.
  * (20) H. Moradpour, C. Corda, A.H. Ziaie, S. Ghaffari, EPL 127(6), 60006 (2019)
  * (21) H. Moradpour, C. Corda, and A. H. Ziaie, 2021 EPL 134 20003.
  * (22) Shababi, H., Ourabah, K. Eur. Phys. J. Plus 135, 697 (2020).
  * (23) J. D. Barrow, Phys. Letts. B 808, 135643 (2020).
  * (24) S. A. Hayward, Class. Quantum Gravity 15, 3147 (1998);
S.A. Hayward, S. Mukohyana, M.C. Ashworth, Phys. Lett. A 256, 347 (1999);
D. Bak, S.J. Rey, Class. Quantum Gravity 17, 83 (2000);
R.G. Cai, S.P. Kim, J. High Energy Phys. 0502, 050 (2005);
M. Akbar, R.G. Cai, Phys. Rev. D 75, 084003 (2007);
R.G. Cai, L.M. Cao, Y.P. Hu, Class. Quantum Gravity 26, 155018 (2009).

  * (25) Umesh Kumar Sharma, Vipin Chandra Dubey, A. H. Ziaie, H. Moradpour, arXiv:2106.08139.
  * (26) Niki Drepanoua, Andreas Lymperisb, E. N. Saridakisc, K. Yesmakhanovae, arxiv:2109.09181v1.
  * (27) S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972).
  * (28) C. Mathiazhagan, V. B. Johri, Class. Quantum Gravity 1, L29 (1984);
D. La, P. J. Steinhardt, Phys. Rev. Lett. 62, 376 (1989);
S. Das and N. Banerjee, Phys. Rev. D 78, 043512 (2008).

  * (29) N. Banerjee, D. Pavon, Phys. Rev. D 63, 043504 (2001).
  * (30) V. Acquaviva, L. Verde, JCAP. 0712, 001 (2007).
  * (31) Y. Gong, Phys. Rev. D 61 (2000) 043505;
Y. Gong, Phys. Rev. D 70 (2004) 064029, [hep-th/0404030];
B. Nayak, L. P. Singh, arXiv:0803.2930;
H. Kim, H. W. Lee, Y. S. Myung Phys. Lett. B 628, 11 (2005).

  * (32) M. R. Setare, Phys. Lett. B 644 99(2007) [hep-th/0610190].
  * (33) N. Banerjee, D. Pavon, Phys. Lett. B 647 477 (2007);
  * (34) N. Banerjee, D. Pavon, Class. Quant. Grav. 18, 593 (2001)
  * (35) L. Xu, J. Lu and W. Li, Eur. Phys. J. C 60, 135 (2009);
  * (36) M. Jamil et al. Int. J. Theor. Phys, 51, 604 (2012).
  * (37) S. Vagnozzi, E. Di Valentino, S. Gariazzo, A. Melchiorri, O. Mena, and J. Silk, to the BOSS: the galaxy power spectrum take on spatial curvature and cosmic concordance,” 10 2020.
  * (38) N. Aghanim et al. [Planck], Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys.641, A6 (2020) [erratum: Astron. Astrophys. 652, C4 (2021)] [arXiv:1807.06209].
  * (39) A. Khodam-Mohammadi, E. Karimkhani, A. Sheykhi, Int. J. Mod. Phys. D 23, 1450081 (2014).
  * (40) S. Ghaffari, H. Moradpour, I. P. Lobo, J. P. Morais Graça, Valdir B. Bezerra, Eur. Phys. J. C 78, 706 (2018).
  * (41) M. Quartin et al., J. Cosmol. Astropart. Phys. 05, 007 (2008); C. G. Boehmer, G. Caldera-Cabral, R. Lazkoz, R. Maartens, Phys. Rev. D 78, 023505 (2008); G. Caldera-Cabral, R. Maartens, L.A. Urena-Lopez, Phys. Rev. D 79, 063518 (2009); D. Pavon, W. Zimdahl, Phys. Lett. B 628, 206 (2005); D. Pavon, B. Wang, Gen. Relativ Gravity 41, 1 (2009).
  * (42) H. Wei, Nucl. Phys. B 845, 381 (2011).
  * (43) L. P. Chimento, Phys. Rev. D 81, 043525 (2010)
  * (44) L. P. Chimento, M. Forte, G. M. Kremer, Gen. Rel. Grav. 41, 1125 (2009)
  * (45) R. A. Daly et al., Astrophys. J. 677, 1 (2008)
  * (46) E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192, 18 (2011).
  * (47) V. Salvatelli, A. Marchini, L. L. Honorez and O. Mena, Phys. Rev. D 88, 023531 (2013).


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\field{month}{4}
\field{title}{Classification of Topologically Protected Gates for Local Stabilizer Codes}
\field{volume}{110}
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\field{pages}{170503}
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\verb 10.1103/PhysRevLett.110.170503
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{American Physical Society}%
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\field{journaltitle}{Phys. Rev. A}
\field{month}{1}
\field{title}{Fault-tolerant logical gates in quantum error-correcting codes}
\field{volume}{91}
\field{year}{2015}
\field{pages}{012305}
\range{pages}{1}
\verb{doi}
\verb 10.1103/PhysRevA.91.012305
\endverb
\endentry
\entry{Kubica_2015}{article}{}
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{IOP Publishing}%
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\field{journaltitle}{New Journal of Physics}
\field{month}{8}
\field{number}{8}
\field{title}{Unfolding the color code}
\field{volume}{17}
\field{year}{2015}
\field{pages}{083026}
\range{pages}{1}
\verb{doi}
\verb 10.1088/1367-2630/17/8/083026
\endverb
\endentry
\entry{yamasakiTimeEfficientConstantSpaceOverheadFaultTolerant2024}{article}{}
\name{author}{2}{}{%
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\field{abstract}{Scalable realization of quantum computing to attain substantial speedups over classical computing requires fault tolerance. Conventionally, protocols for fault-tolerant quantum computation (FTQC) demand excessive space overhead of physical qubits per logical qubit. A more recent protocol to achieve constant-space-overhead FTQC using quantum low-density parity-check (LDPC) codes thus attracts considerable attention but suffers from another drawback: it incurs polynomially long time overhead. To address these problems, we here introduce an alternative approach using a concatenation of multiple small-size quantum codes for the constant-space-overhead FTQC rather than a single large-size quantum LDPC code. We develop techniques for concatenating different quantum Hamming codes with growing sizes. As a result, we construct a low-overhead protocol to achieve constant space overhead and only quasi-polylogarithmic time overhead simultaneously. Our protocol accomplishes FTQC even if a decoder has non-constant runtime, unlike the existing constant-space-overhead protocol. These results establish a foundation for FTQC realizing a large class of quantum speedups within feasibly bounded space overhead yet negligibly short time overhead. This achievement opens a promising avenue for the low-overhead FTQC based on code concatenation.}
\field{eprinttype}{arxiv}
\field{issn}{1745-2473, 1745-2481}
\field{journaltitle}{Nature Physics}
\field{langid}{english}
\field{month}{1}
\field{title}{Time-{{Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation}}}
\field{year}{2024}
\verb{doi}
\verb 10.1038/s41567-023-02325-8
\endverb
\verb{file}
\verb /Users/shinho/Zotero/storage/N2EBN3CA/Yamasaki and Koashi - 2024 - Time-Efficient Constant-Space-Overhead Fault-Toler.pdf
\endverb
\keyw{Quantum Physics}
\endentry
\entry{gottesmanConstantOverhead}{article}{}
\name{author}{1}{}{%
\list{location}{1}{%
{Paramus, NJ}%
\list{publisher}{1}{%
{Rinton Press, Incorporated}%
\strng{namehash}{7547b3f3d9b6b3aebccf1577d598b7de}
\strng{fullhash}{7547b3f3d9b6b3aebccf1577d598b7de}
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\field{labeltitlesource}{title}
\field{abstract}{What is the minimum number of extra qubits needed to perform a large fault-tolerant quantum circuit? Working in a common model of fault-tolerance, I show that in the asymptotic limit of large circuits, the ratio of physical qubits to logical qubits can be a constant. The construction makes use of quantum low-density parity check codes, and the asymptotic overhead of the protocol is equal to that of the family of quantum error-correcting codes underlying the fault-tolerant protocol.}
\field{issn}{1533-7146}
\field{journaltitle}{Quantum Info. Comput.}
\field{month}{11}
\field{number}{15–16}
\field{title}{Fault-Tolerant Quantum Computation with Constant Overhead}
\field{volume}{14}
\field{year}{2014}
\field{pages}{1338\bibrangedash 1372}
\range{pages}{35}
\verb{doi}
\verb 10.5555/2685179.2685184
\endverb
\keyw{fault tolerance,quantum error correction,LDPC codes}
\endentry
\entry{fawziConstantOverheadQuantum2018}{inproceedings}{}
\name{author}{3}{}{%
\strng{namehash}{76b7aa87f384f03260902db5de028966}
\strng{fullhash}{cc85e6f3addb8441f71999e9a92e5f31}
\strng{bibnamehash}{cc85e6f3addb8441f71999e9a92e5f31}
\strng{authorbibnamehash}{cc85e6f3addb8441f71999e9a92e5f31}
\strng{authornamehash}{76b7aa87f384f03260902db5de028966}
\strng{authorfullhash}{cc85e6f3addb8441f71999e9a92e5f31}
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\field{sortinithash}{4f6aaa89bab872aa0999fec09ff8e98a}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{abstract}{We prove that quantum expander codes can be combined with quantum fault-tolerance techniques to achieve constant overhead: the ratio between the total number of physical qubits required for a quantum computation with faulty hardware and the number of logical qubits involved in the ideal computation is asymptotically constant, and can even be taken arbitrarily close to 1 in the limit of small physical error rate. This improves on the polylogarithmic overhead promised by the standard threshold theorem.}
\field{booktitle}{2018 {{IEEE}} 59th {{Annual Symposium}} on {{Foundations}} of {{Computer Science}} ({{FOCS}})}
\field{eprinttype}{arxiv}
\field{langid}{english}
\field{month}{10}
\field{title}{Constant Overhead Quantum Fault-Tolerance with Quantum Expander Codes}
\field{year}{2018}
\field{pages}{743\bibrangedash 754}
\range{pages}{12}
\verb{doi}
\verb 10.1109/FOCS.2018.00076
\endverb
\keyw{Computer Science - Information Theory,Quantum Physics}
\endentry
\entry{Edmonds1965}{article}{}
\name{author}{1}{}{%
\strng{namehash}{4512d981a2fac684e867257fcb2c5419}
\strng{fullhash}{4512d981a2fac684e867257fcb2c5419}
\strng{bibnamehash}{4512d981a2fac684e867257fcb2c5419}
\strng{authorbibnamehash}{4512d981a2fac684e867257fcb2c5419}
\strng{authornamehash}{4512d981a2fac684e867257fcb2c5419}
\strng{authorfullhash}{4512d981a2fac684e867257fcb2c5419}
\field{sortinit}{1}
\field{sortinithash}{4f6aaa89bab872aa0999fec09ff8e98a}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{journaltitle}{J. of Res. the Nat. Bureau of Standards}
\field{title}{Maximum Matching and a Polyhedron with $0,1$ Vertices}
\field{volume}{69~B}
\field{year}{1965}
\field{pages}{125\bibrangedash 130}
\range{pages}{6}
\endentry
\entry{denniskitaevlandalpreskill}{article}{}
\name{author}{4}{}{%
\strng{namehash}{894d9e7842dead2ca68c4a717b5ea22a}
\strng{fullhash}{282ea8a0ad5aa187ef091b089dad9ab1}
\strng{bibnamehash}{282ea8a0ad5aa187ef091b089dad9ab1}
\strng{authorbibnamehash}{282ea8a0ad5aa187ef091b089dad9ab1}
\strng{authornamehash}{894d9e7842dead2ca68c4a717b5ea22a}
\strng{authorfullhash}{282ea8a0ad5aa187ef091b089dad9ab1}
\field{sortinit}{1}
\field{sortinithash}{4f6aaa89bab872aa0999fec09ff8e98a}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{abstract}{{We analyze surface codes, the topological quantum error-correcting codes introduced by Kitaev. In these codes, qubits are arranged in a two-dimensional array on a surface of nontrivial topology, and encoded quantum operations are associated with nontrivial homology cycles of the surface. We formulate protocols for error recovery, and study the efficacy of these protocols. An order-disorder phase transition occurs in this system at a nonzero critical value of the error rate; if the error rate is below the critical value (the accuracy threshold), encoded information can be protected arbitrarily well in the limit of a large code block. This phase transition can be accurately modeled by a three-dimensional Z2 lattice gauge theory with quenched disorder. We estimate the accuracy threshold, assuming that all quantum gates are local, that qubits can be measured rapidly, and that polynomial-size classical computations can be executed instantaneously. We also devise a robust recovery procedure that does not require measurement or fast classical processing; however, for this procedure the quantum gates are local only if the qubits are arranged in four or more spatial dimensions. We discuss procedures for encoding, measurement, and performing fault-tolerant universal quantum computation with surface codes, and argue that these codes provide a promising framework for quantum computing architectures.}}
\field{issn}{0022-2488}
\field{journaltitle}{Journal of Mathematical Physics}
\field{month}{09}
\field{number}{9}
\field{title}{{Topological quantum memory}}
\field{volume}{43}
\field{year}{2002}
\field{pages}{4452\bibrangedash 4505}
\range{pages}{54}
\verb{doi}
\verb 10.1063/1.1499754
\endverb
\endentry
\entry{kubicaSingleshotQuantumError2022}{article}{}
\name{author}{2}{}{%
\strng{namehash}{3b43225373fa49a0fcf1c2e9eb12c01b}
\strng{fullhash}{3b43225373fa49a0fcf1c2e9eb12c01b}
\strng{bibnamehash}{3b43225373fa49a0fcf1c2e9eb12c01b}
\strng{authorbibnamehash}{3b43225373fa49a0fcf1c2e9eb12c01b}
\strng{authornamehash}{3b43225373fa49a0fcf1c2e9eb12c01b}
\strng{authorfullhash}{3b43225373fa49a0fcf1c2e9eb12c01b}
\field{sortinit}{1}
\field{sortinithash}{4f6aaa89bab872aa0999fec09ff8e98a}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{abstract}{We introduce a new topological quantum code, the three-dimensional subsystem toric code (3D STC), which is a generalization of the stabilizer toric code. The 3D STC can be realized by measuring geometrically-local parity checks of weight at most three on the cubic lattice with open boundary conditions. We prove that single-shot quantum error correction (QEC) is possible with the 3D STC, i.e., one round of local parity-check measurements suffices to perform reliable QEC even in the presence of measurement errors. We also propose an efficient single-shot QEC strategy for the 3D STC and investigate its performance. In particular, we numerically estimate the resulting storage threshold against independent bit-flip, phase-flip and measurement errors to be \$p\_{\textbackslash}text\{STC\} {\textbackslash}approx 1.045{\textbackslash}\%\$. Such a high threshold together with local parity-check measurements of small weight make the 3D STC particularly appealing for realizing fault-tolerant quantum computing.}
\field{eprinttype}{arxiv}
\field{issn}{2041-1723}
\field{journaltitle}{Nature Communications}
\field{langid}{english}
\field{month}{10}
\field{number}{1}
\field{title}{Single-Shot Quantum Error Correction with the Three-Dimensional Subsystem Toric Code}
\field{volume}{13}
\field{year}{2022}
\field{pages}{6272}
\range{pages}{1}
\verb{doi}
\verb 10.1038/s41467-022-33923-4
\endverb
\verb{file}
\verb /Users/shinho/Zotero/storage/2V7Q3SZC/Kubica and Vasmer - 2022 - Single-shot quantum error correction with the thre.pdf
\endverb
\keyw{Condensed Matter - Strongly Correlated Electrons,Quantum Physics}
\endentry
\entry{portnoy2023local}{misc}{}
\name{author}{1}{}{%
\strng{namehash}{5d30c491d355d626574c41901b7c21fd}
\strng{fullhash}{5d30c491d355d626574c41901b7c21fd}
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\strng{authorbibnamehash}{5d30c491d355d626574c41901b7c21fd}
\strng{authornamehash}{5d30c491d355d626574c41901b7c21fd}
\strng{authorfullhash}{5d30c491d355d626574c41901b7c21fd}
\field{sortinit}{4}
\field{sortinithash}{9381316451d1b9788675a07e972a12a7}
\field{labelnamesource}{author}
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\field{eprintclass}{quant-ph}
\field{eprinttype}{arXiv}
\field{title}{Local Quantum Codes from Subdivided Manifolds}
\field{year}{2023}
\verb{eprint}
\verb 2303.06755
\endverb
\endentry
\entry{lin2023geometrically}{misc}{}
\name{author}{3}{}{%
giveni={T\bibinithyphendelim C\bibinitperiod}}}%
giveni={M\bibinithyphendelim H\bibinitperiod}}}%
\strng{namehash}{0ecc2e0222fc69f247fe2f38f799436e}
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\field{eprintclass}{quant-ph}
\field{eprinttype}{arXiv}
\field{title}{Geometrically Local Quantum and Classical Codes from Subdivision}
\field{year}{2023}
\verb{eprint}
\verb 2309.16104
\endverb
\endentry
\entry{baspinwilliamson23}{misc}{}
\name{author}{2}{}{%
given={Dominic\bibnamedelima J.},
giveni={D\bibinitperiod\bibinitdelim J\bibinitperiod}}}%
\strng{namehash}{8a7281add46dbddeccd3d3a7bea5287d}
\strng{fullhash}{8a7281add46dbddeccd3d3a7bea5287d}
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\strng{authornamehash}{8a7281add46dbddeccd3d3a7bea5287d}
\strng{authorfullhash}{8a7281add46dbddeccd3d3a7bea5287d}
\field{sortinit}{4}
\field{sortinithash}{9381316451d1b9788675a07e972a12a7}
\field{labelnamesource}{author}
\field{labeltitlesource}{title}
\field{eprintclass}{quant-ph}
\field{eprinttype}{arXiv}
\field{title}{Layer Codes}
\field{year}{2023}
\verb{eprint}
\verb 2309.16503
\endverb
\endentry
\entry{pattisonHierarchicalMemoriesSimulating2023}{misc}{}
\name{author}{3}{}{%
given={Christopher\bibnamedelima A.},
giveni={C\bibinitperiod\bibinitdelim A\bibinitperiod}}}%
\list{publisher}{1}{%
\strng{namehash}{5da9c57ace77e86dc633a889a5eede70}
\strng{fullhash}{e8decd4736984b21924fce46ed6dba0a}
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\field{abstract}{Constant-rate low-density parity-check (LDPC) codes are promising candidates for constructing efficient fault-tolerant quantum memories. However, if physical gates are subject to geometric-locality constraints, it becomes challenging to realize these codes. In this paper, we construct a new family of [[N,K,D]] codes, referred to as hierarchical codes, that encode a number of logical qubits K = Omega(N/{\textbackslash}log(N)\^2). The N-th element of this code family is obtained by concatenating a constant-rate quantum LDPC code with a surface code; nearest-neighbor gates in two dimensions are sufficient to implement the corresponding syndrome-extraction circuit and achieve a threshold. Below threshold the logical failure rate vanishes superpolynomially as a function of the distance D(N). We present a bilayer architecture for implementing the syndrome-extraction circuit, and estimate the logical failure rate for this architecture. Under conservative assumptions, we find that the hierarchical code outperforms the basic encoding where all logical qubits are encoded in the surface code.}
\field{eprintclass}{quant-ph}
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\field{month}{3}
\field{number}{arXiv:2303.04798}
\field{shorttitle}{Hierarchical Memories}
\field{title}{Hierarchical Memories: {{Simulating}} Quantum {{LDPC}} Codes with Local Gates}
\field{year}{2023}
\verb{eprint}
\verb 2303.04798
\endverb
\verb{file}
\verb /Users/shinho/Zotero/storage/A9AQ3KQC/Pattison et al_2023_Hierarchical memories.pdf;/Users/shinho/Zotero/storage/5K4A8UGZ/2303.html
\endverb
\keyw{Quantum Physics}
\endentry
\entry{delfosseBoundsStabilizerMeasurement2021}{misc}{}
\name{author}{3}{}{%
given={Michael\bibnamedelima E.},
giveni={M\bibinitperiod\bibinitdelim E\bibinitperiod}}}%
given={Maxime\bibnamedelima A.},
giveni={M\bibinitperiod\bibinitdelim A\bibinitperiod}}}%
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\field{abstract}{In this work we establish lower bounds on the size of Clifford circuits that measure a family of commuting Pauli operators. Our bounds depend on the interplay between a pair of graphs: the Tanner graph of the set of measured Pauli operators, and the connectivity graph which represents the qubit connections required to implement the circuit. For local-expander quantum codes, which are promising for low-overhead quantum error correction, we prove that any syndrome extraction circuit implemented with local Clifford gates in a 2D square patch of \$N\$ qubits has depth at least \${\textbackslash}Omega(n/{\textbackslash}sqrt\{N\})\$ where \$n\$ is the code length. Then, we propose two families of quantum circuits saturating this bound. First, we construct 2D local syndrome extraction circuits for quantum LDPC codes with bounded depth using only \$O(n\^2)\$ ancilla qubits. Second, we design a family of 2D local syndrome extraction circuits for hypergraph product codes using \$O(n)\$ ancilla qubits with depth \$O({\textbackslash}sqrt\{n\})\$. Finally, we use circuit noise simulations to compare the performance of a family of hypergraph product codes using this last family of 2D syndrome extraction circuits with a syndrome extraction circuit implemented with fully connected qubits. While there is a threshold of about \$10\^\{-3\}\$ for a fully connected implementation, we observe no threshold for the 2D local implementation despite simulating error rates of as low as \$10\^\{-6\}\$. This suggests that quantum LDPC codes are impractical with 2D local quantum hardware. We believe that our proof technique is of independent interest and could find other applications. Our bounds on circuit sizes are derived from a lower bound on the amount of correlations between two subsets of qubits of the circuit and an upper bound on the amount of correlations introduced by each circuit gate, which together provide a lower bound on the circuit size.}
\field{eprintclass}{quant-ph}
\field{eprinttype}{arxiv}
\field{langid}{english}
\field{month}{9}
\field{number}{arXiv:2109.14599}
\field{title}{Bounds on Stabilizer Measurement Circuits and Obstructions to Local Implementations of Quantum {{LDPC}} Codes}
\field{year}{2021}
\verb{eprint}
\verb 2109.14599
\endverb
\verb{file}
\verb /Users/shinho/Zotero/storage/YHELKAHF/Delfosse et al. - 2021 - Bounds on stabilizer measurement circuits and obst.pdf
\endverb
\keyw{Computer Science - Information Theory,Quantum Physics}
\endentry
\entry{baspinLowerBoundOverhead2023}{misc}{}
\name{author}{3}{}{%
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\field{abstract}{We show that a quantum architecture with an error correction procedure limited to geometrically local operations incurs an overhead that grows with the system size, even if arbitrary error-free classical computation is allowed. In particular, we prove that in order to operate a quantum error correcting code in 2D at a logical error rate of {$\delta$}, a space overhead of {$\Omega$}( log(1/{$\delta$})) is needed for any constant depolarizing noise p {$>$} 0.}
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\field{month}{2}
\field{number}{arXiv:2302.04317}
\field{title}{A Lower Bound on the Overhead of Quantum Error Correction in Low Dimensions}
\field{year}{2023}
\verb{eprint}
\verb 2302.04317
\endverb
\verb{file}
\verb /Users/shinho/Zotero/storage/MHQVBKUG/Baspin et al. - 2023 - A lower bound on the overhead of quantum error cor.pdf
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\keyw{Quantum Physics}
\endentry
\entry{guSingleshotDecodingGood2023}{misc}{}
\name{author}{6}{}{%
given={Shin\bibnamedelima Ho},
giveni={S\bibinitperiod\bibinitdelim H\bibinitperiod}}}%
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\field{abstract}{Quantum Tanner codes constitute a family of quantum low-density parity-check (LDPC) codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Z{\textbackslash}'emor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols.}
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\verb 2306.12470
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\keyw{Computer Science - Information Theory,Quantum Physics}
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\entry{AlonChungGraham94}{article}{}
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given={R.\bibnamedelimi L.},
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\field{abstract}{A class of routing problems on connected graphs G is considered. Initially, each vertex v of G is occupied by a “pebble” that has a unique destination \$\pi ( v )\$ in G (so that \$\pi \$ is a permutation of the vertices of G). It is required that all the pebbles be routed to their respective destinations by performing a sequence of moves of the following type: A disjoint set of edges is selected, and the pebbles at each edge’s endpoints are interchanged. The problem of interest is to minimize the number of steps required for any possible permutation \$\pi \$.This paper investigates this routing problem for a variety of graphs G, including trees, complete graphs, hypercubes, Cartesian products of graphs, expander graphs, and Cayley graphs. In addition, this routing problem is related to certain network flow problems, and to several graph invariants including diameter, eigenvalues, and expansion coefficients.}
\field{journaltitle}{SIAM Journal on Discrete Mathematics}
\field{number}{3}
\field{title}{Routing Permutations on Graphs via Matchings}
\field{volume}{7}
\field{year}{1994}
\field{pages}{513\bibrangedash 530}
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\verb{doi}
\verb 10.1137/S0895480192236628
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\entry{Magicstate05}{article}{}
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{American Physical Society}%
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\field{journaltitle}{Phys. Rev. A}
\field{month}{2}
\field{title}{Universal quantum computation with ideal {Clifford} gates and noisy ancillas}
\field{volume}{71}
\field{year}{2005}
\field{pages}{022316}
\range{pages}{1}
\verb{doi}
\verb 10.1103/PhysRevA.71.022316
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\entry{choeLongrangeDataTransmission2022}{misc}{}
\name{author}{2}{}{%
given={Shin\bibnamedelima Ho},
giveni={S\bibinitperiod\bibinitdelim H\bibinitperiod}}}%
\list{publisher}{1}{%
\strng{namehash}{1c9180046313442603be8ff3d1dcb6a2}
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\field{abstract}{We propose a scheme for fault-tolerant long-range entanglement generation at the ends of a rectangular array of qubits of length \$R\$ and a square cross section of size \$d\textbackslash times d\$ with \$d=O(\textbackslash log R)\$. Up to an efficiently computable Pauli correction, the scheme generates a maximally entangled state of two qubits using a depth-\$6\$ circuit consisting of nearest-neighbor Clifford gates and local measurements only. Compared with existing fault-tolerance schemes for quantum communication, the protocol is distinguished by its low latency: starting from a product state, the entangled state is prepared in a time \$O(t\_\{\textbackslash textrm\{local\}\})\$ determined only by the local gate and measurement operation time \$t\_\{\textbackslash textrm\{local\}\}\$. Furthermore, the requirements on local repeater stations are minimal: Each repeater uses only \$\textbackslash Theta(\textbackslash log\^2 R)\$ qubits with a lifetime of order \$O(t\_\{\textbackslash textrm\{local\}\})\$. We prove a converse bound \$\textbackslash Omega(\textbackslash log R)\$ on the number of qubits per repeater among all low-latency schemes for fault-tolerant quantum communication over distance \$R\$. Furthermore, all operations within a repeater are local when the qubits are arranged in a square lattice. The noise-resilience of our scheme relies on the fault-tolerance properties of the underlying cluster state. We give a full error analysis, establishing a fault-tolerance threshold against general (circuit-level) local stochastic noise affecting preparation, entangling operations and measurements. This includes, in particular, errors correlated in time and space. Our conservative analytical estimates are surprisingly optimistic, suggesting that the scheme is suited for long-range entanglement generation both in and between near-term quantum computing devices.}
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\field{month}{9}
\field{number}{arXiv:2209.09774}
\field{title}{Long-Range Data Transmission in a Fault-Tolerant Quantum Bus Architecture}
\field{year}{2022}
\verb{eprint}
\verb 2209.09774
\endverb
\keyw{Quantum Physics}
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\entry{bravyiQuantumAdvantageNoisy2020}{article}{}
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\field{abstract}{Prior work has shown that there exists a relation problem which can be solved with certainty by a constant-depth quantum circuit composed of geometrically local gates in two dimensions, but cannot be solved with high probability by any classical constant depth circuit composed of bounded fan-in gates. Here we provide two extensions of this result. Firstly, we show that a separation in computational power persists even when the constant-depth quantum circuit is restricted to geometrically local gates in one dimension. The corresponding quantum algorithm is the simplest we know of which achieves a quantum advantage of this type. It may also be more practical for future implementations. Our second, main result, is that a separation persists even if the shallow quantum circuit is corrupted by noise. We construct a relation problem which can be solved with near certainty using a noisy constant-depth quantum circuit composed of geometrically local gates in three dimensions, provided the noise rate is below a certain constant threshold value. On the other hand, the problem cannot be solved with high probability by a noise-free classical circuit of constant depth. A key component of the proof is a quantum error-correcting code which admits constant-depth logical Clifford gates and single-shot logical state preparation. We show that the surface code meets these criteria. To this end, we provide a protocol for single-shot logical state preparation in the surface code which may be of independent interest.}
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\field{issn}{1745-2473, 1745-2481}
\field{journaltitle}{Nature Physics}
\field{month}{10}
\field{number}{10}
\field{title}{Quantum Advantage with Noisy Shallow Circuits in {{3D}}}
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\field{year}{2020}
\field{pages}{1040\bibrangedash 1045}
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\verb{doi}
\verb 10.1038/s41567-020-0948-z
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{American Physical Society}%
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\newpage
\appendix

\section{Local stochastic noise and simple  adaptive quantum circuits \label{sec:lemmasnoisycircuit}}
Local stochastic noise is
compatible with
Clifford circuits, as expressed by the following Lemma, (cf. e.g.,~\cite[Lemma 11]{bravyiQuantumAdvantageNoisy2020}  and~\cite{fawziConstantOverheadQuantum2018}).

\begin{lemma}[Properties of local stochastic noise] \label{lem:propertieslocalstochasticnoise}
Consider a system of~$n$ qubits. Then the following holds.
\begin{enumerate}[(i)]
\item
Suppose $E \sim \cN(p)$ and $E'$ is a random Pauli such that $\supp(E') \subseteq \supp(E)$ with probability $1$. Then $E' \sim \cN(p)$. \label{it:subset}
\item
Suppose $E \sim \cN(p)$ and $F \sim \cN(q)$ are random Paulis which may be dependent. Then $E \cdot F \sim \cN(q')$ where $q' = 2 \max \{\sqrt{p}, \sqrt{q}\}$. \label{it:productdependent}
\item
Suppose $E \sim \cN(p)$ is a random Pauli and let $U$ be a tensor product of single- and two-qubit Clifford unitaries. Then $UEU^{\dagger} \sim\cN(q')$ where $q' = \sqrt{2p}$.
\label{it:clifford}
\item
Let $A\subseteq [n]$ be a subset of qubits.
Suppose $E \sim \cN(p)$, $F \sim \cN(q)$ and $\supp(E) \subseteq A$ and $\supp(F) \subseteq A^c=:[n]\backslash A$ with probability $1$. Then $E \cdot F \sim \cN(q')$ where $q' = \max \{\sqrt{p},\sqrt{q}\}$. \label{it:productdisjoint}
\item
Let $\pi$ be a permutation on $[n]$, and let $\pi(E)$ be the $n$-qubit Pauli which acts on the $i$-th qubit as $\pi(i)$. Then $\pi(E) \sim \cN(p)$.
\label{it:permutation}
\end{enumerate}
\end{lemma}
\begin{proof}
See~\cite[Lemma 11]{bravyiQuantumAdvantageNoisy2020} for \eqref{it:subset}--\eqref{it:clifford}.
For \eqref{it:productdisjoint}, observe that  for any subset $I \subseteq [n]$, we have
\begin{align}
\Pr[I \subseteq \supp(E \cdot F)]
&= \Pr[I \cap A \subseteq \supp(E) \ \text{and} \ I \cap A^c \subseteq \supp(F)]\\
&\leq \min\{p^{\abs{I \cap A}}, q^{\abs{I \cap A^c}}\} \\
&\leq \min \{ \left(\max \{p,q\}\right)^{\abs{I \cap A}}, \left(\max \{p,q\}\right)^{\abs{I \cap A^c}} \} \\
&\leq \left(\max\{p,q\}\right)^{\frac{\abs{I \cap A} + \abs{I \cap A^c}}{2}} \\
&= \left(\max\{\sqrt{p},\sqrt{q}\}\right)^{\abs{I}} \, ,
\end{align}
where we used the definition of local stochastic noise and the fact that $\min \{a,b\} \leq \sqrt{ab}$ for any $a, b \geq 0$.
For \eqref{it:permutation}, observe that for any subset $I = \{i_1, \dots, i_k\} \subseteq [n]$, we have
$I \subseteq \supp(\pi(E))$ if and only if $\{\pi^{-1}(i_1), \dots, \pi^{-1}(i_k)\} =: \pi^{-1}(I) \subseteq \supp(E)$. Therefore,
\begin{align}
\Pr[I \subseteq \supp(\pi(E))] = \Pr[\pi^{-1}(I) \subseteq \supp(E)] \leq p^{\abs{\pi^{-1}(I)}} = p^{\abs{I}} \, ,
\end{align}
where the fact that $E$ is local stochastic noise is used.
\end{proof}

Lemma~\ref{lem:propertieslocalstochasticnoise} allows one to commute errors through adaptive circuits explicitly  if the adaptivity is expressed by linear functions.  We formulate this as follows:
\begin{lemma}[Noisy adaptive quantum circuits] \label{lem:noisyadaptivecircuits}
Let $n, n_1, n_2 \in \mathbb{N}$ with $n=n_1 + n_2$ and
\begin{align}
A = (a_{ij}) \ , \  B = (b_{ij}) \in \mathbb{F}_2^{n_1 \times n_2}
\end{align}
be matrices.
Consider a set of qubits~$\cA = \cA_1 \cup \cA_2 = [n]$ where~$\cA_1 = [n_1]$ and~$\cA_2 = \{n_1+1, \dots, n\}$.
For $x\in \mathbb{F}_2^n$
let $X(x)=\prod_{j=1}^n X_j^{x_j}$
and $Z(x)=\prod_{j=1}^n Z_j^{x_j}$
be products of Pauli-$X$ and Pauli-$Z$ operators with support defined by~$x$. For each~$x_1 \in \mathbb{F}_2^{n_1}$, let us denote by~$X(x_1)$ and~$Z(x_1)$ Pauli operators on~$\cA_1$ defined analogously.

Let $\cQ$ be an adaptive quantum circuit on the qubits~$\cA$ of the following:
\begin{enumerate}
\item
The circuit~$\cQ$ measures all qubits belonging to~$\cA_2$ in the computational basis, obtaining a measurement outcome~$z\in \mathbb{F}_2^{n_2}$.
\item Subsequently, an $n_1$-qubit Pauli gate~$C(z)\in \cP_{n_1}$
is applied to the qubits~$\cA_1$, where we assume that
\begin{align}
C(z) = X(Az) Z(Bz) \qquad \text{for all measurement outcomes} \quad z \in \mathbb{F}_2^{n_2} \ .
\end{align}
\end{enumerate}
For any Pauli operator~$E$ on~$\cA$, there exists a Pauli operator~$F=F(E)$ on~$\cA_1$ determined by~$E$ satisfying the following:
\begin{enumerate}[(i)]
\item \label{it:noisyEFequiv}
A noisy implementation~$\cQ_1=\cQ\circ \cU_E$ of the circuit~$\cQ$ associated with the error~$E$ acting on~$\cA$ before the (ideal) circuit~$\cQ$ is applied is equivalent to a noisy implementation~$\cQ_2=\cU_F\circ\cQ$ of the circuit~$\cQ$ associated with the error~$F$ acting on~$\cA_2$ after the (ideal) circuit~$\cQ$ is applied (see Fig.~\ref{fig:noisyadaptivecircuitsequiv}.)
\item \label{it:Fstrength}
Let $w \in \mathbb{N}$ with $n_1 \cdot w \leq n_2$, and
suppose that all non-zero row vectors of~$A$ and~$B$ have weight $w$ and all column vectors of~$A$ and~$B$ have weight at most 1, i.e.,
\begin{align}
\begin{matrix}
\sum_{j=1}^{n_2} a_{ij}&\in&\{0,w\} \ ,&\sum_{j=1}^{n_2} b_{ij}&\in&\{0,w\} &\qquad  \text{for all}& i \in[n_1] \\
\sum_{i=1}^{n_1} a_{ij}&\leq& 1 \ , &\sum_{i=1}^{n_1} b_{ij}&\leq&1 &\qquad \text{for all} &j \in[n_2] \ .
\end{matrix}
\end{align}
For any $p \in [0,1]$, if $E \sim \cN(p)$, then $F \sim \cN(p')$, where
\begin{align}\label{eq:Fstrength}
p' = 4 \cdot p^{\frac{1}{4w}} \ .
\end{align}
\end{enumerate}

\begin{figure}
\begin{quantikz}
\lstick[3]{$\mathcal{A}_1$}   & \gate[6]{E}   & \qw       &\gate[3]{Z(Bz)} & \gate[3]{X(Az)}      & \qw \\
&               & \qw       &                &                      & \qw \\
&               & \qw       &                &                      & \qw \\
\lstick[3]{$\mathcal{A}_2$}   &               & \meter{$z_1$}  &\cwbend{-1}     & \cwbend{-1}     &  \\
&               & \meter{$\vdots$}  &\cwbend{-2}     & \cwbend{-2}  & \\
&               & \meter{$z_{n_2}$}  &\cwbend{-3}     & \cwbend{-3}     &
\end{quantikz}
\begin{quantikz}
\lstick[3]{$\mathcal{A}_1$}   & \qw       &\gate[3]{Z(Bz)} & \gate[3]{X(Az)}      & \gate[3]{F} &\qw \\
& \qw       &                &                      & \qw &\qw \\
& \qw       &                &                      & \qw &\qw\\
\lstick[3]{$\mathcal{A}_2$}   & \meter{$z_1$}  &\cwbend{-1}     & \cwbend{-1}     & \\
& \meter{$\vdots$}  &\cwbend{-2}     & \cwbend{-2}  & \\
& \meter{$z_{n_2}$}  &\cwbend{-3}     & \cwbend{-3}     &
\end{quantikz}
\caption{Two different noisy implementations~$\cQ_1$ (left) and~$\cQ_2$ (right) of an adaptive quantum circuit~$\cQ$ are equivalent if two Paulis $E \in \cP_{n}$ and $F \in \cP_{n_2}$ satisfies Eq.~\eqref{eq:FintermsofE}.\label{fig:noisyadaptivecircuitsequiv}}
\end{figure}

\end{lemma}
\begin{proof}
We construct the Pauli operator~$F$ as follows.
Let us write
\begin{align}
E = X(e) Z(f) \ ,
\end{align}
\begin{align}
e = (e^{(1)}, e^{(2)}) , \ f = (f^{(1)}, f^{(2)}) \in \mathbb{F}_2^{n} \qquad \text{with} \qquad \ e^{(j)},  \ f^{(j)} \in \mathbb{F}_2^{n_j} \quad \text{for} \quad j \in \{1,2\} \ .
\end{align}
Define a Pauli operator~$F \in \cP_n$ on~$\cA$ as
\begin{align}\label{eq:FintermsofE}
F := X(e^{(1)} + Ae^{(2)}) Z(f^{(1)} + Be^{(2)}) \ .
\end{align}

We show that the Pauli~$F$ satisfies the property~\eqref{it:noisyEFequiv}. Observe that
\begin{align}
\left( X(Az) Z(Bz) \otimes \bra{z}_{\cA_2} \right) E
&= \left(F  X(A (z+e^{(2)})) Z(B(z+e^{(2)})) \right) \otimes \bra{z+e^{(2)}}_{\cA_2}  \ , \label{eq:localstochasticEtoF}
\end{align}
where a phase factor from the anti-commutativity of Pauli-$X$ and Pauli-$Z$ is ignored.
Let~$\cQ_j(\rho)$ denote the output state of the circuit~$\cQ_j$ with an input state~$\rho$ on $\cA$ for $j \in \{1,2\}$.
It follows from Eq.~\eqref{eq:localstochasticEtoF} that for any state~$\rho$ on the system $\cA$, we have
\begin{align}
\cQ_1(\rho) &= \sum_{z \in \mathbb{F}_2^m} \left( X(Az) Z(Bz) \otimes \bra{z} \right) E \rho E^* \left( (X(A z)Z(Bz))^* \otimes \ket{z} \right) \\
&= \sum_{z \in \mathbb{F}_2^m} \left(F X(A(z+e^{(2)}))Z(z+e^{(2)}) \otimes \bra{z+e^{(2)}} \right) \rho \left(h.\ c. \right) \\
&= \cQ_2(\rho) \ ,
\end{align}
and thus~$\cQ_1$ and~$\cQ_2$ are equivalent.

Next, we show that the Pauli~$F$ satisfies the property~\eqref{it:Fstrength}.
We first claim that
\begin{align}
X(Ae^{(2)}) \sim \cN(4 \cdot p^{1/w}) \qquad \text{and} \qquad Z(Be^{(e)}) \sim \cN(4 \cdot p^{1/w}) \ . \label{eq:XAelocalstochastic}
\end{align}
Here we only give a proof of the claim~\eqref{eq:XAelocalstochastic} for $X(Ae^{(2)})$ since the proof for $Z(Be^{(2)})$ mirrors this.
Let~$I = \{i_1, \dots, i_\ell\} \subseteq [n_1]$ be the set of indices such that the $i$-th row vector of $A$ is non-zero, and let us denote by $j^{(i)}_1 < \dots < j^{(i)}_w$ for each $i \in I$ the indices in $[n_2]$ consisting of the support of the $i$-th row vector of $A$, i.e.,
\begin{align}
a_{i j^{(i)}_k} = 1 \qquad \text{for all} \qquad k \in [w], \ i \in I \ ,
\end{align}
and define
\begin{align}
g^{(k)} = \left(e^{(2)}_{j_k^{(i_1)}}, \dots, e^{(2)}_{j_k^{(i_\ell)}}\right) \in \mathbb{F}_2^{\ell} \qquad \text{for all} \qquad k \in [w] \ .
\end{align}
The assumption about the columns of the matrix~$A$ implies that for each $k$, the indices~$\{j_k^{(i)}\}_{i \in I} \subseteq [n_2]$ are pairwise disjoint. Therefore, there exists some permutation $\pi$ on $[n_2]$ such that $g^{(k)}$ can be obtained by truncating  the last $n_2 - \ell$ bits of the vector $\pi(e^{(2)}):=(e^{(2)}_{\pi(1)}, \dots, e^{(2)}_{\pi(n_2)})$. Therefore, by the properties~\eqref{it:subset} and~\eqref{it:permutation} of local stochastic noise in Lemma~\ref{lem:propertieslocalstochasticnoise}, we have that
\begin{align}
X(g^{(k)}) \sim \cN(p) \qquad \text{for all} \qquad k \in [w] \ .
\label{eq:Xeklocalstochastic}
\end{align}
By construction, we have $A e^{(2)} = g^{(1)} + \dots + g^{(w)}$, and thus
\begin{align}
X(Ae^{(2)}) = \prod_{k=1}^w X(g^{(k)}) \sim \cN\left(4 \cdot p^{2^{- \ceil{\log_2 w}}} \right) \ , \label{eq:XAelocalstochasticwithlog}
\end{align}
where we applied Lemma~\ref{lem:propertieslocalstochasticnoise}~\eqref{it:productdependent} $w-1$ times together with the fact~\eqref{eq:Xeklocalstochastic} to compute noise strength of the product of $\{X(g^{(k)})\}_{k=1}^w$ in the form of a complete binary tree. The claim~\eqref{eq:XAelocalstochastic} follows from Eq.~\eqref{eq:XAelocalstochasticwithlog} together with the inequality $4 \cdot p^{2^{- \ceil{\log_2 w}}} \leq 4 \cdot p^{1/w}$.

Finally, we observe that $X(e^{(1)})Z(f^{(1)}) \sim \cN(p)$ by the property~\eqref{it:subset} of Lemma~\ref{lem:propertieslocalstochasticnoise}. Combining this with~\eqref{eq:XAelocalstochastic}, we have
\begin{align}
F = X(Ae^{(2)}) Z(Be^{(2)}) X(e^{(1)}) Z(f^{(1)}) \sim \cN\left(4 \cdot p^{\frac{1}{4w}}\right) \ ,
\end{align}
where Lemma~\ref{lem:propertieslocalstochasticnoise}~\eqref{it:productdependent} is used twice and a phase factor is again ignored.
\end{proof}
An important special case of Lemma~\ref{lem:noisyadaptivecircuits} is the parallel execution of entanglement-swapping circuits.
\begin{lemma}[Entanglement swapping on noisy Bell states]\label{lem:noisyentanglementswapping}
Consider the following entanglement swapping protocol which takes $k\ell$ Bell pairs
\begin{align}
\Psi_{\mathrm{in}} = \bigotimes_{i=1}^{k} \bigotimes_{j=1}^\ell \Phi_{Q_i^{(j)}{Q'_i}^{(j)}}
\end{align}
as an input and generates $\ell$ Bell pairs~$\Psi_{\mathrm{out}} = \bigotimes_{j=1}^{\ell} \Phi_{Q_1^{(j)} {Q'_k}^{(j)}}$:
\begin{enumerate}
\item Apply a Bell measurement to the qubits~${Q'_i}^{(j)}, Q_{i+1}^{(j)}$ for each $i \in [k-1]$ and $j \in [\ell]$, getting outcomes $(a_i^{(j)}, b_i^{(j)})$. \label{it:entanglementswapbellmeas}
\item Apply $X^{a^{(j)}}Z^{b^{(j)}}$ to qubit~${Q'_k}^{(j)}$ for all $j \in [\ell]$ where $a^{(j)} = \bigoplus_{i=1}^{k-1} a_i^{(j)}$ and $b^{(j)} = \bigoplus_{i=1}^{k-1} b_i^{(j)}$.
\end{enumerate}
Let $E \sim \cN(p)$ be a~$2k\ell$-qubit Pauli noise which is local stochastic of strength~$p$.
If the entanglement swapping protocol takes noisy $k\ell$ Bell pairs $E \Psi_{\mathrm{in}}$, then the output state is a noisy $\ell$-fold Bell state $F\Psi_{\mathrm{out}}$
for some $2\ell$-qubit local stochastic noise $F \sim \cN(p')$, where $p' = 4 (\sqrt{2p})^{\frac{1}{4(k-1)}}$.
\end{lemma}
\begin{proof}
Note that Step~\ref{it:entanglementswapbellmeas} can be decomposed into one layer~$U$ consisting of two-qubit Clifford unitaries followed by single-qubit measurements in computational basis.
By commuting~$E$ past the layer~$U$, we see that
the protocol with state~$E \Psi_{\mathrm{in}}$ is equivalent to the following protocol with the input state~$(U E U^\dagger)U\bigotimes_{i=1}^{k} \bigotimes_{j=1}^\ell \Phi_{Q_i^{(j)}{Q'_i}^{(j)}}$:
\begin{enumerate}
\item Apply computational basis measurement to the qubits~${Q'_i}^{(j)}, Q_{i+1}^{(j)}$ for each $i \in [k-1]$ and $j \in [\ell]$, getting outcomes $(a_i^{(j)}, b_i^{(j)})$.
\item Apply $X^{a^{(j)}}Z^{b^{(j)}}$ to qubit~${Q'_k}^{(j)}$ for all $j \in [\ell]$ where $a^{(j)} = \bigoplus_{i=1}^{k-1} a_i^{(j)}$ and $b^{(j)} = \bigoplus_{i=1}^{k-1} b_i^{(j)}$.
\end{enumerate}
Note that this has the form of an adaptive circuit as discussed in Lemma~\ref{lem:noisyadaptivecircuits}.
By Lemma~\ref{lem:noisyadaptivecircuits}, the resulting state is a noisy~$\ell$-fold Bell state affected by a Pauli noise~$F$ which is determined by~$UEU^\dagger$.
By Lemma~\ref{lem:propertieslocalstochasticnoise}~\eqref{it:clifford}, we have
\begin{align}\label{eq:equivalentinputerrorstrength}
UEU^\dagger \sim \cN\left(\sqrt{2p}\right) \ .
\end{align}

Let us label the qubits by the sets~$\cA_1$ and~$\cA_2$ defined as
\begin{align}
\cA_1 &= \bigcup_{j=1}^\ell \{Q_1^{(j)}, {Q'_k}^{(j)}\} = [2\ell] \\
\cA_2 &= \bigcup_{i=1}^{k-1} \bigcup_{j=1}^\ell \{{Q'_i}^{(j)}, Q_{i+1}^{(j)}\} = \{2\ell+1, 2\ell+2, \dots,   2k\}  \ ,
\end{align}
and construct a string~$z = (z_1, \dots, z_{(2k-2) \ell}) \in \mathbb{F}_2^{(2k-2) \ell}$ by collecting all bits
\begin{align}
\bigcup_{i=1}^{k-1} \bigcup_{j=1}^{\ell}  \{(a_i^{(j)}, b_i^{(j)})\} \ ,
\end{align}
where the ordering is consistent with the labeling~$\cA_2$.
Then, it is clear by inspection that
there are matrices~$A, B \in \mathbb{F}_2^{2\ell \times (2k-2) \ell}$ satisfying
\begin{align}
\bigotimes_{j=1}^{\ell} X^{a^{(j)}}_{{Q'_k}^{(j)}} Z^{b^{(j)}}_{{Q'_k}^{(j)}} = X(Az)Z(Bz) \ .
\end{align}
Furthermore, all non-zero row vectors of $A$ and $B$ have weight~$k-1$ and all column vectors of $A$ and $B$ have weight at most 1. As a result, we can apply Lemma~\ref{lem:noisyadaptivecircuits}~\eqref{it:Fstrength}.
We obtain $p'$ by inserting~$\sqrt{2p}$, which was obtained by Eq.~\eqref{eq:equivalentinputerrorstrength}, and $k-1$ respectively into $p$ and $w$ of Eq.~\eqref{eq:Fstrength}.
\end{proof}
Another consequence of Lemma~\ref{lem:noisyadaptivecircuits} is the following statement, which characterizes the effect of local stochastic noise on  teleportation circuit applied in parallel.

\begin{lemma}[Noisy implementation of the quantum teleportation protocol] \label{lem:noisyquantumteleportation}
Consider the following quantum teleportation protocol which takes a $3\ell$-qubit state
\begin{align}
\Psi'_{\mathrm{in}} = \Psi_{Q_1 \dots Q_\ell} \bigotimes_{j=1}^{\ell} \Phi_{R_j R'_j}
\end{align}
as an input and outputs an~$\ell$-qubit state~$\Psi'_{\mathrm{out}}=\Psi_{R'_1 \dots R'_\ell}$:
\begin{enumerate}
\item Apply a Bell measurement to the qubits $Q_j, R_j$ for each $j \in [\ell]$, getting outcomes~$(a_j, b_j)$.
\item Apply $X^{a_j} Z^{b_j}$ to the qubit $R'_j$ for all~$j \in [\ell]$.
\end{enumerate}
Let $E \sim \cN(p)$ be a $3\ell$-qubit Pauli noise which is local stochastic noise of strength $p$. If the quantum teleportation protocol takes a noisy state~$E \Psi'_{\mathrm{in}}$ as an input, then the output state is a noisy state~$F \Psi'_{\mathrm{out}}$ for some $\ell$-qubit local stochastic noise $F \sim \cN(p')$, where $p' = 2^{\frac{17}{8}} p^{\frac{1}{8}}$.
\end{lemma}
\begin{proof}
The proof mirrors that of Lemma~\ref{lem:noisyentanglementswapping}, so we only give a sketch.
We choose~$\sqrt{2p}$ and $1$ respectively for the parameters $p$ and $w$ of Eq.~\eqref{eq:Fstrength} to obtain $p'$.
\end{proof}

\end{document}