# Initial-state-dependent quantum speed limit for dissipative state
preparation: Framework and optimization

Junjie Liu<EMAIL_ADDRESS>Department of Physics, International Center of
Quantum and Molecular Structures, Shanghai University, Shanghai, 200444, China
Hanlin Nie<EMAIL_ADDRESS>Centre for Quantum Technologies, National
University of Singapore, Block S15, 3 Science Drive 2, 117543, Singapore

(August 28, 2024)

###### Abstract

Dissipation has traditionally been considered a hindrance to quantum
information processing, but recent studies have shown that it can be harnessed
to generate desired quantum states. To be useful for practical applications,
the ability to speed up the dissipative evolution is crucial. In this study,
we focus on a Markovian dissipative state preparation scheme where the
prepared state is one of the energy eigenstates. We derive an initial-state-
dependent quantum speed limit (QSL) that offers a more refined measure of the
actual evolution time compared to the commonly used initial-state-independent
relaxation time. This allows for a passive optimization of dissipative
evolution across different initial states. By minimizing the dissipated heat
during the preparation process, conditioned on the minimization of evolution
time using the QSL, we find that the preferred initial state has a specific
permutation of diagonal elements with respect to an ordered energy eigenbasis
of increasing eigenvalues. In this configuration, the population on the
prepared state is the largest, and the remaining diagonal elements are sorted
in an order resembling that of a passive state in the same ordered energy
eigenbasis. We demonstrate the effectiveness of our strategy in a dissipative
Rydberg atom system for preparing the Bell state. Our work provides new
insights into the optimization of dissipative state preparation processes and
could have significant implications for practical quantum technologies.

##  I Introduction

The ability to prepare specific quantum states is a fundamental requirement in
many areas of quantum information, such as quantum error correction, fault
tolerant quantum computation Nielsen and Chuang (2002); Bravyi and Kitaev
(2005); Fowler  _et al._ (2012); Reed  _et al._ (2012); Nigg  _et al._ (2014);
Kelly  _et al._ (2015); Takeda  _et al._ (2022), and quantum simulation Bloch
_et al._ (2008); Lewenstein  _et al._ (2007); Häffner  _et al._ (2008); Blatt
and Roos (2012); Xiang  _et al._ (2013); Monroe and Kim (2013); Devoret and
Schoelkopf (2013). In these applications, a constant supply of ancillary
qubits in known low-entropy states is necessary to achieve reliable and
efficient processing. Furthermore, quantum computation models that achieve
exponential speed-up over classical ones require initialization of quantum
systems into pure states Knill and Laflamme (1998). In the case of one-way
quantum computing Raussendorf and Briegel (2001); Walther  _et al._ (2005),
for example, one should start with a multipartite system prepared in a cluster
state.

Conventional methods for quantum state preparation involve the use of quantum
gates that operate in a unitary manner Long and Sun (2001); Plesch and Brukner
(2011); Sun  _et al._ ; Zhang  _et al._ (2021, 2022). While this approach is
compatible with current unitary quantum computation and simulation designs,
its implementation is hindered by the accumulation of quantum unitary
operation errors during the process Long and Sun (2001); Plesch and Brukner
(2011); Sun  _et al._ , as well as the cost of an exponential number of
quantum gates or ancillary qubits Zhang  _et al._ (2021, 2022). Dissipative
state preparation (DSP) Harrington  _et al._ (2022) represents a departure
from traditional state preparation methods. Instead of viewing dissipation as
a source of noise to be minimized, DSP exploits it as a resource. By
harnessing the dissipation from surrounding environments, DSP enables the
generation of desired quantum states as steady states of engineered
dissipative evolution processes. Advances in theoretical Kraus  _et al._
(2008); Diehl  _et al._ (2008); Verstraete  _et al._ (2009); Leghtas  _et al._
(2013); Kastoryano  _et al._ (2011, 2013); Reiter  _et al._ (2013); Schuetz
_et al._ (2013); Cormick  _et al._ (2013); Mirrahimi  _et al._ (2014); Reiter
_et al._ (2016); Popkov and Presilla (2016); Puri  _et al._ (2017); Tuorila
_et al._ (2017); Li and Shao (2018); Zippilli and Vitali (2021); Zapletal  _et
al._ (2022) and experimental Barreiro  _et al._ (2011); Krauter  _et al._
(2011); Lin  _et al._ (2013); Shankar  _et al._ (2013); Kienzler  _et al._
(2015); Leghtas  _et al._ (2015); Morigi  _et al._ (2015); Kimchi-Schwartz
_et al._ (2016); Liu  _et al._ (2016); Ma  _et al._ (2021); Cole  _et al._
(2022) research have demonstrated the utility of DSP, making it a promising
scheme for preparing quantum states. In addition to the growing body of
research supporting the utility of DSP, empirical evidence indicates that this
approach may provide significant advantages over other state preparation
methods in certain physical systems. For instance, the fidelity scaling
achievable using DSP in the preparation of a maximally entangled state of two
atoms in an optical cavity surpasses that of any coherent unitary protocol
currently available Kastoryano  _et al._ (2011).

While DSP has shown great potential for practical state preparation, it is
essential to consider the speed of convergence for a given implementation.
State-of-the-art DSP builds upon Markovian evolution dynamics described by the
Lindblad master equation Kraus  _et al._ (2008); Diehl  _et al._ (2008);
Verstraete  _et al._ (2009), but its slow speed due to small damping rates is
a significant limitation Cormick  _et al._ (2013); Breuer and Petruccione
(2002). Faster evolution processes are needed to prevent the accumulation of
implementation errors, as well as to suppress heat dissipation and entropy
production Auffèves (2022). Several active control protocols Li  _et al._
(2016); Chen  _et al._ (2017); Zheng  _et al._ (2021); Alexander  _et al._
(2020); Wang  _et al._ (2021); Bao and Hou  have been developed to steer the
evolution through well-designed control fields. However, their experimental
implementation remains a significant challenge to date. Recently, a passive
speed-up strategy utilizing the Mpemba effect Carollo  _et al._ (2021);
Kochsiek  _et al._ (2022) has been proposed as an alternative. This approach
avoids using complex control fields and instead tunes initial states such that
the slowest decaying mode cannot be excited at the outset.

In the pursuit of minimizing the preparation time of DSP, it is crucial to
have a suitable measure of the evolution time of the underlying Markovian
process for benchmarking. In many existing studies Kraus  _et al._ (2008);
Verstraete  _et al._ (2009); Reiter  _et al._ (2013); Carollo  _et al._
(2021); Kochsiek  _et al._ (2022); Zapletal  _et al._ (2022), the relaxation
time, which is defined as the inverse of the Liouvillian spectral gap, is
often used as the measure. However, despite its simplicity and rigor, the
relaxation time defines a worst-case scenario and does not offer any physical
intuition on how to speed up the evolution process. Additionally, it should be
noted that the actual evolution time varies between different initial states
Wu  _et al._ (2014); Carollo  _et al._ (2021); Zhou  _et al._ (2021); Kochsiek
_et al._ (2022). Since the relaxation time is solely determined by the
spectrum of the Liouvillian super-operator, it fails to capture the initial-
state dependence of the actual evolution time. Given these limitations, a more
refined measure for the evolution time is required for DSP.

In this regard, it is worth noting that for unitary state preparation, a
quantum speed limit (QSL) has been derived Bukov  _et al._ (2019) that
provides a lower bound for the preparation time and can be used to develop
optimal driving protocols. However, a corresponding QSL for DSP has not yet
been established. In this work, we derive an initial-state dependent QSL for
measuring the preparation time of DSP by utilizing the relative purity
Mendonça  _et al._ (2008); del Campo  _et al._ (2013); Zhang  _et al._ (2014);
Meng  _et al._ (2015); Ektesabi  _et al._ (2017); Wu and Yu (2018); Liang  _et
al._ (2019); Pires and de Oliveira (2021) as an information-theoretic
quantifier for distinguishing two quantum states. Importantly, the relative
purity also serves as an experimentally friendly measure Giordani  _et al._
(2020). Notably, the obtained QSL depends solely on the overlap between the
initial and final prepared states, enabling a passive optimization over
initial states. Our optimization scheme thus distinguishes itself from
existing active ones Li  _et al._ (2016); Chen  _et al._ (2017); Zheng  _et
al._ (2021); Alexander  _et al._ (2020); Wang  _et al._ (2021); Bao and Hou
in that it requires no active controls during the evolution. In addition, it
differs from existing passive ones Carollo  _et al._ (2021); Kochsiek  _et
al._ (2022) as it does not require knowledge of the eigen-spectrum of the
Liouvillian super-operator.

To obtain a more comprehensive understanding, we explore the minimization of
the quantum speed limit (QSL) while simultaneously accounting for the
dissipated heat, which represents the system’s energy change during the
preparation process Van Vu and Saito (2022). This dissipated heat is a
critical quantity from an energetic standpoint Auffèves (2022). Recognizing
that the final prepared state is one of the system’s energy eigenstates Kraus
_et al._ (2008) and that only the diagonal elements of the initial states in
the energy basis are relevant when evaluating both the QSL and dissipated
heat, our investigation concentrates on diagonal initial states in the energy
basis for performing optimization. By minimizing dissipated heat while
optimizing the QSL, we identify the optimal permutation of a given set of
diagonal elements with respect to an ordered energy basis of increasing
eigenvalues. The preferred initial state with the optimal permutation exhibits
the largest population on the prepared state. Furthermore, the remaining
diagonal elements are sorted in an order that is reminiscent of a passive
state Allahverdyan  _et al._ (2004); Uzdin and Rahav (2021) in relation to the
same ordered energy basis [cf. Eq. (17)].

We illustrate the effectiveness of our optimization framework by applying it
to an open Rydberg atom system used to prepare the Bell state Li and Shao
(2018). Interestingly, the example shows that in the DSP scheme the prepared
state can be an excited energy eigenstate, in direct contrast to usual bit
reset and information erasure studies (see, e.g., Refs. Mohammady  _et al._
(2016); Klaers (2019); Miller  _et al._ (2020); Zhen  _et al._ (2021); Van Vu
and Saito (2022)) where the final ground state is typically assumed. As a
result, the dissipated heat can become negative in DSP with the preferred
initial state, indicating that the non-thermal environment injects energy into
the system to assist DSP, and moreover, the conventional Landauer principle
Landi and Paternostro (2021) is not applicable to such DSP processes since the
system entropy change is instead positive.

The paper is organized as follows. In Sec. II, we briefly review the
underlying idea of DSP for completeness and state the questions of interest.
We then introduce our initial-state dependent QSL based on the relative purity
in Sec. III and provide optimization details in Sec. IV. In Sec. V, we present
an example and demonstrate the effectiveness of our framework. Finally, we
conclude our study in Sec. VI. Two appendices provide complementary derivation
details.

##  II Background and question

State-of-the-art DSP harnesses Markovian dissipative evolution processes to
prepare useful quantum states Kraus  _et al._ (2008); Diehl  _et al._ (2008);
Verstraete  _et al._ (2009). Specifically, DSP utilizes the Lindblad master
equation governing the evolution of a reduced system density matrix $\rho_{t}$

$\dot{\rho}_{t}\leavevmode\nobreak\ =\leavevmode\nobreak\
-i[H_{S},\rho_{t}]+\sum_{\mu=1}\gamma_{\mu}\mathcal{D}[L_{\mu}]\rho_{t}.$ (1)

Here, $\dot{A}\equiv\frac{d}{dt}A$ is used throughout the study, $H_{S}$ is
the system Hamiltonian, $\gamma_{\mu}\geqslant 0$ is the damping coefficient
of channel $\mu$, $\mathcal{D}[L_{\mu}]\rho=L_{\mu}\rho
L_{\mu}^{\dagger}-\frac{1}{2}\\{L_{\mu}^{\dagger}L_{\mu},\rho\\}$ is the
Lindblad superoperator with $L_{\mu}$ the Lindblad jump operator for channel
$\mu$ and $\\{A,B\\}=AB+BA$ the anti-commutator between $A$ and $B$. The final
stationary state would be a pure one $|\Phi\rangle$ when the following
conditions are fulfilled Kraus  _et al._ (2008); Buca and Prosen (2012)

$\displaystyle H_{S}|\Phi\rangle\leavevmode\nobreak\ =\leavevmode\nobreak\
E_{n^{\ast}}|\Phi\rangle,$ $\displaystyle
L_{\mu}|\Phi\rangle\leavevmode\nobreak\ =\leavevmode\nobreak\
0,\leavevmode\nobreak\ \forall\mu.$ (2)

With the above conditions, one can easily check that the density matrix
$\rho_{f}=|\Phi\rangle\langle\Phi|$ is indeed the fixed point of the dynamics
with $\frac{d}{dt}\rho_{f}=0$. We denote $|\Phi\rangle=|E_{n^{\ast}}\rangle$
for later convenience. We remark that the Lindblad operators involved here
clearly do not fulfill the detailed balance condition Horowitz and Parrondo
(2013) and $|E_{n^{\ast}}\rangle$ can be an excited energy eigenstate of the
system.

Considering a generic initial state in an ordered energy eigenbasis
$\\{|E_{n}\rangle\\}$ of $H_{S}$ with eigenvalues sorted in an increasing
order, $E_{1}\leqslant E_{2}\leqslant\cdots\leqslant E_{N}$,

$\rho_{0}\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum_{nm}\rho_{0}^{nm}|E_{n}\rangle\langle E_{m}|.$ (3)

For our purpose, we decompose it as $\rho_{0}=\rho_{0}^{d}+\rho_{0}^{c}$ with
diagonal part $\rho_{0}^{d}$ and coherence part $\rho_{0}^{c}$ respectively
reading

$\left\\{\begin{array}[]{c}\rho_{0}^{d}\leavevmode\nobreak\
=\leavevmode\nobreak\ \sum\limits_{n}\lambda_{n}|E_{n}\rangle\langle
E_{n}|,\\\ \rho_{0}^{c}\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum\limits_{n\neq m}\rho_{0}^{nm}|E_{n}\rangle\langle
E_{m}|.\end{array}\right.$ (4)

Here, we denoted $\lambda_{n}\equiv\rho_{0}^{nn}$. Noting that in the Lindblad
master equation the evolution of populations and coherences in the energy
eigenbasis are decoupled Breuer and Petruccione (2002); Lidar , then for a
generic initial state, the DSP consists of two independent evolution processes

$\left\\{\begin{array}[]{c}\rho_{0}^{d}\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ \rho_{f},\\\ \rho_{0}^{c}\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ 0.\end{array}\right.$ (5)

Here we are interested in purely the state preparation process and hence limit
ourselves to initial states that are diagonal in the energy eigenbasis,
namely, $\rho_{0}=\rho_{0}^{d}$ hereafter.

Evolving the system state to a final pure state also results in changes in the
averaged system energy in terms of the dissipated heat $Q$ Van Vu and Saito
(2022),

$Q\leavevmode\nobreak\ =\leavevmode\nobreak\
-\int_{0}^{T_{f}}\mathrm{Tr}[H_{S}\dot{\rho}_{t}]dt\leavevmode\nobreak\
=\leavevmode\nobreak\ \mathrm{Tr}[H_{S}\rho_{0}]-E_{n^{\ast}}.$ (6)

Here, we have denoted $T_{f}$ as the final time with $\rho_{T_{f}}=\rho_{f}$.
We note that the change in system von Neumann entropy $\Delta
S=-\mathrm{Tr}[\rho_{0}\ln\rho_{0}]+\mathrm{Tr}[\rho_{f}\ln\rho_{f}]=-\mathrm{Tr}[\rho_{0}\ln\rho_{0}]=-\sum_{n}\lambda_{n}\ln\lambda_{n}\geq
0$ is fixed once initial populations $\\{\lambda_{n}\\}$ are fixed. Taking
into account that the dissipated heat depends solely on the diagonal elements
of $\rho_{0}$, we now pose the following intriguing question: For a given set
of populations $\\{\lambda_{n}\\}$, where $\lambda_{n}\geqslant 0$,
$\lambda_{n}\neq\lambda_{m\neq n}$, and $\sum_{n}\lambda_{n}=1$, what is the
optimal permutation of $\\{\lambda_{n}\\}$ with respect to the ordered basis
$\\{|E_{n}\rangle\\}$ such that the resulting initial state
$\rho_{0}=\sum\limits_{n}\lambda_{n}|E_{n}\rangle\langle E_{n}|$ can minimize
the evolution time of the process $\rho_{0}\Rightarrow\rho_{f}$ while
simultaneously maintaining the dissipated heat at the lowest possible level?

##  III Quantum speed limit

To quantify the evolution time from an initial state $\rho_{0}$ to the final
pure state $\rho_{f}$, we resort to the QSL which provides a lower bound on
the evolution time. By noting that $\rho_{f}$ is pure, we adopt the Uhlmann
fidelity, or equivalently in this scenario, the relative purity
$\mathrm{Tr}[\rho_{t}\rho_{f}]$ Mendonça  _et al._ (2008); del Campo  _et al._
(2013); Zhang  _et al._ (2014); Meng  _et al._ (2015); Ektesabi  _et al._
(2017); Wu and Yu (2018); Liang  _et al._ (2019); Pires and de Oliveira (2021)
to measure the distance between $\rho_{f}$ and $\rho_{t}$,

$\mathcal{F}(\rho_{t},\rho_{f})\leavevmode\nobreak\ =\leavevmode\nobreak\
[\mathrm{Tr}(\sqrt{\rho_{t}}\rho_{f}\sqrt{\rho_{t}})^{1/2}]^{2}\leavevmode\nobreak\
=\leavevmode\nobreak\ \mathrm{Tr}[\rho_{t}\rho_{f}].$ (7)

We highlight that here we take the final pure state as the reference, in sharp
contrast to existing studies del Campo  _et al._ (2013); Zhang  _et al._
(2014); Meng  _et al._ (2015); Wu and Yu (2018); Kobayashi and Yamamoto (2020)
on QSL which instead chose the initial state as the reference.

We parameterize the fidelity with an angle $\Theta_{t}$ changing from a given
initial value $\Theta_{0}\in\Big{(}0,\frac{\pi}{2}\Big{]}$ to a final value
$\Theta_{f}=0$

$\mathcal{F}(\rho_{t},\rho_{f})\leavevmode\nobreak\ =\leavevmode\nobreak\
\cos\Theta_{t}.$ (8)

We then consider the equation of motion for $\Theta_{t}$,

$\displaystyle\dot{\Theta}_{t}$ $\displaystyle=$
$\displaystyle\frac{-1}{\sqrt{1-\mathrm{Tr}(\rho_{t}\rho_{f})^{2}}}\mathrm{Tr}\left(\frac{d\rho_{t}}{dt}\rho_{f}\right)$
(9) $\displaystyle=$
$\displaystyle\frac{1}{\sin\Theta_{t}}\mathrm{Tr}\left[\left(i[\rho_{f},H_{S}]-\sum_{\mu}\gamma_{\mu}\mathcal{D}^{\dagger}[L_{\mu}]\rho_{f}\right)\rho_{t}\right]$
$\displaystyle=$
$\displaystyle\frac{-1}{\sin\Theta_{t}}\mathrm{Tr}\left[\left(\sum_{\mu}\gamma_{\mu}L_{\mu}^{\dagger}\rho_{f}L_{\mu}\right)\rho_{t}\right].$

In arriving at the second line, we have defined
$\mathcal{D}^{\dagger}[L_{\mu}]\rho=L_{\mu}^{\dagger}\rho
L_{\mu}-\frac{1}{2}\\{L_{\mu}^{\dagger}L_{\mu},\rho\\}$. In getting the last
line, we have used the properties of $|\Phi\rangle$ such that
$[\rho_{f},H_{S}]=0$ and $L_{\mu}\rho_{f}=0=\rho_{f}L_{\mu}^{\dagger}$
($L_{\mu}^{\dagger}\rho_{f}\neq 0$ generally).

To proceed, we note that the above equation can be rewritten as

$\dot{\Theta}_{t}\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{-1}{\sin\Theta_{t}}\mathrm{Tr}\left[\left(\sum_{\mu}\gamma_{\mu}L_{\mu}^{\dagger}\rho_{f}L_{\mu}\right)(\rho_{t}-\rho_{f})\right]$
(10)

by noting $\mathrm{Tr}[L_{\mu}^{\dagger}\rho_{f}L_{\mu}\rho_{f}]=0$. Using the
Cauchy-Schwarz inequality for operators
$|\mathrm{Tr}(A^{\dagger}B)|\leqslant||A||_{F}||B||_{F}$ with
$||A||_{F}=\sqrt{\mathrm{Tr}(A^{\dagger}A)}$ the Frobenius or Hilbert-Schimidt
norm, we find from the above equation that

$\displaystyle\left|\dot{\Theta}_{t}\right|$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{\sin\Theta_{t}}\left|\left|\sum_{\mu}\gamma_{\mu}L_{\mu}^{\dagger}\rho_{f}L_{\mu}\right|\right|_{F}||\rho_{f}-\rho_{t}||_{F}$
(11) $\displaystyle\leqslant$
$\displaystyle\frac{\sqrt{2}}{\sin\Theta_{t}}\left|\left|\sum_{\mu}\gamma_{\mu}L_{\mu}^{\dagger}\rho_{f}L_{\mu}\right|\right|_{F}\sqrt{1-\cos\Theta_{t}}.$

Here, we have noted that
$||\rho_{f}-\rho_{t}||_{F}=\sqrt{\mathrm{Tr}(\rho_{t}^{2}-2\rho_{t}\rho_{f}+\rho_{f}^{2})}\leqslant\sqrt{2-2\mathrm{Tr}(\rho_{t}\rho_{f})}=\sqrt{2-2\cos\Theta_{t}}$
due to the properties $\mathrm{Tr}(\rho_{t}^{2})\leqslant 1$ and
$\mathrm{Tr}[\rho_{f}^{2}]=1$. Integrating both sides of the above inequality
from $0$ to the final time $T_{f}$, we find

$\left|\int_{\Theta_{0}}^{\Theta_{f}}d\Theta_{t}\frac{\sin\Theta_{t}}{\sqrt{1-\cos\Theta_{t}}}\right|\leavevmode\nobreak\
\leqslant\leavevmode\nobreak\ \sqrt{2}\mathcal{A}T_{f},$ (12)

yielding the following initial-state dependent QSL for $T_{f}$

$T_{f}\leavevmode\nobreak\ \geqslant\leavevmode\nobreak\
T_{\mathrm{QSL}}\leavevmode\nobreak\ \equiv\leavevmode\nobreak\
\frac{\sqrt{2-2\cos\Theta_{0}}}{\mathcal{A}}.$ (13)

Here, we have defined a non-negative coefficient
$\mathcal{A}\equiv||\sum_{\mu}\gamma_{\mu}L_{\mu}^{\dagger}\rho_{f}L_{\mu}||_{F}$
which is independent of the initial state and remains fixed once $L_{\mu}$ and
$|\Phi\rangle$ are specified. We note that by following the procedures in Ref.
del Campo  _et al._ (2013) one can obtain a slightly different form of QSL
which is, however, looser than $T_{\mathrm{QSL}}$ defined above; see appendix
A for derivation details. It is worth mentioning that (i) $T_{\mathrm{QSL}}$
is independent of dynamics such that one can evaluate it without solving the
evolution equation Eq. (1), and (ii) Only the numerator of $T_{\mathrm{QSL}}$
depends on the initial state which brings simplicity when performing
optimizations. We further note that $T_{\mathrm{QSL}}$ is a monotonic
decreasing function of $0\leqslant\cos\Theta_{0}<1$ and attains its maximum
when $\cos\Theta=0$, that is, when $\rho_{0}$ is orthogonal to $\rho_{f}$.

##  IV Optimization over initial states

We now turn to the question mentioned at the end of Sec. II. For a given set
of populations $\\{\lambda_{n}\\}$ with $\lambda_{n}\geq 0$,
$\lambda_{n}\neq\lambda_{m\neq n}$ and $\sum_{n}\lambda_{n}=1$, one can
generate a set of diagonal initial states $\rho_{0}$ each characterized by a
specific permutation of $\\{\lambda_{n}\\}$ with respect to the ordered energy
basis $\\{|E_{n}\rangle\\}$. We identify the optimal permutation among them by
optimizing two incompatible objectives: the evolution time characterized by
the QSL $T_{\mathrm{QSL}}$ [cf. Eq. (13)] and the dissipated heat $Q$ [cf. Eq.
(6)]; In Appendix B, we illustrate this incompatibility by showing that
minimizing $Q$ alone yields a passive initial state Allahverdyan  _et al._
(2004); Uzdin and Rahav (2021) which in general does not minimize
$T_{\mathrm{QSL}}$ simultaneously as we will see.

We note that minimizing $T_{\mathrm{QSL}}$ amounts to maximizing the relative
purity $\mathrm{Tr}[\rho_{0}\rho_{f}]$ between initial and final states, hence
our optimization is of a Pareto-type Van Vu and Saito (2022) and one can
assign a multi-objective functional
$\mathcal{W}=gQ-(1-g)\mathcal{F}(\rho_{0},\rho_{f})$ with $g\in[0,1)$ a
weighting factor. In the present study, we are particularly interested in the
case with $g>0$ and $(1-g)/g\gg 1$ such that while both the heat and relative
purity are optimized the latter takes precedence over the former. In general,
solving the optimization problem with respect to the functional $\mathcal{W}$
subject to an arbitrary initial state should resort to numerical treatments.
However, with a given set of populations $\\{\lambda_{n}\\}$, we find that
$\mathcal{W}=g\sum_{n}\lambda_{n}E_{n}-gE_{n^{\ast}}-(1-g)\lambda_{n^{\ast}}$
is a linear function of $\\{\lambda_{n}\\}$; $n^{\ast}$ marks the index
corresponding to the final prepared state (see Eq. (II)). Hence, we can tackle
the optimization as follows: We first maximize the relative purity by
selecting the population among $\\{\lambda_{n}\\}$ on the state
$|E_{n^{\ast}}\rangle$. We next determine the arrangement of the remaining
populations on the remaining energy eigenstates by minimizing the dissipated
heat.

Figure 1: Relative purity $\mathrm{Tr}[\rho_{t}\rho_{f}]$ as a function of
evolution time obtained by solving Eq. (18) with three initial conditions
$\rho_{0}^{A,B,C}=\sum_{n}\lambda_{n}^{A,B,C}|E_{n}\rangle\langle E_{n}|$. We
indicate the arrangements of permutations $\\{\lambda_{n}^{A,B,C}\\}$ (see
detailed definitions in Eq. (V)) with respect to the ordered energy basis
$\\{|E_{n}\rangle\\}$ on the top. Permutation $A$ is the optimal one according
to Eq. (17). Inset: we show the corresponding values of $T_{\mathrm{QSL}}$
[cf. Eq. (13)] and $Q$ [cf. Eq. (6)] for all possible permutations of the same
set of populations as in Eq. (V)), with permutations $A,\leavevmode\nobreak\
B$ and $C$ highlighted. In the calculations, we set $\Omega=2\pi$ MHz as the
unit.

To maximize the relative purity, we consider the following function

$\mathrm{Tr}(U\rho_{0}U^{\dagger}\rho_{f})$ (14)

with $U$ a permutation operation. Noting that both $\rho_{0,f}$ are diagonal
in the energy eigenbasis and $\rho_{f}$ is pure, we find from the general
results in Stoustrup  _et al._ (1995) that

$\mathop{\mathrm{max}}[\mathrm{Tr}(U\rho_{0}U^{\dagger}\rho_{f})]\leavevmode\nobreak\
=\leavevmode\nobreak\ \lambda_{\mathrm{max}}.$ (15)

Here, $\lambda_{\mathrm{max}}$ is the maximum element of the set
$\\{\lambda_{n}\\}$. Hence, we infer that an initial state that overlaps with
the final prepared state $|E_{n^{\ast}}\rangle$ maximally yields the minimal
$T_{\mathrm{QSL}}$, which is both compelling and natural in retrospect. To
determine the arrangement of other populations with $\lambda_{\mathrm{max}}$
excluded, we then turn to the dissipated heat $Q$,

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\mathrm{Tr}[H_{S}\rho_{0}]-E_{n^{\ast}}$ (16) $\displaystyle=$
$\displaystyle\sum_{n\neq
n^{\ast}}E_{n}\lambda_{n}+(\lambda_{\mathrm{max}}-1)E_{n^{\ast}}.$

For a given set $\\{\lambda_{n}\\}$, the second term
$(\lambda_{\mathrm{max}}-1)E_{n^{\ast}}$ on the right-hand-side of the above
equation is fixed, therefore we only need to optimize the first term. Noting
that the energy eigenvalues $\\{E_{n}\\}$ is sorted in an increasing order,
according to the the Majorization theory (see, e.g., Ref. Mohammady  _et al._
(2016)), the first term attains its minimum when the probabilities
$\lambda_{n}$ are sorted in a decreasing order,
$\lambda_{1}\geqslant\lambda_{2}\cdots\geqslant\lambda_{N}$. Putting together,
we can conclude that an initial state with the optimal permutation of
populations should have the following structure

$\rho_{0}^{\mathrm{opt}}\leavevmode\nobreak\ =\leavevmode\nobreak\ \sum_{n\neq
n^{\ast}}\lambda_{n}|E_{n}\rangle\langle
E_{n}|+\lambda_{\mathrm{max}}|E_{n^{\ast}}\rangle\langle E_{n^{\ast}}|,$ (17)

where the first term on the right-hand-side resembles the structure of a
passive state Allahverdyan  _et al._ (2004); Uzdin and Rahav (2021) in the
sense of the same orderings
$\lambda_{1}\geqslant\lambda_{2}\cdots\geqslant\lambda_{N}$ and
$E_{1}\leqslant E_{2}\cdots\leqslant E_{N}$; Noting $n^{\ast}$ is excluded. We
remark that by writing down the above form we implicitly assume that the final
prepared state is an excited state of $H_{S}$, namely, $n^{\ast}\neq 1$. We
note that if the desired final state $|E_{n^{\ast}}\rangle$ happens to be the
ground state of $H_{S}$, then the optimal initial state is strictly a passive
one.

##  V Example: Coupled Rydberg atom system

To validate the above theoretical expectations, we consider a setup consisting
of two $\Lambda$-type three-level Rydberg atoms, each contains two ground
states $|0\rangle$ and $|1\rangle$, and one Rydberg state $|r\rangle$. It was
recently showed Li and Shao (2018) that the setup can be utilized to realize
the DSP of the Bell state $|\Phi\rangle=(|00\rangle-|11\rangle)/\sqrt{2}$ with
$|00(11)\rangle$ being understood as $|0(1)\rangle\otimes|0(1)\rangle$ . The
corresponding Lindblad master equation reads Li and Shao (2018)

$\dot{\rho}_{t}\leavevmode\nobreak\ =\leavevmode\nobreak\
-i[H_{S},\rho_{t}]+\frac{\gamma}{2}\sum_{\mu=1}^{4}\left[L_{\mu}\rho_{t}L_{\mu}^{\dagger}-\frac{1}{2}\\{L_{\mu}^{\dagger}L_{\mu},\rho_{t}\\}\right].$
(18)

Here, the system Hamiltonian takes the following form

$\displaystyle H_{S}$ $\displaystyle=$
$\displaystyle\Omega_{2}(|10\rangle\langle r0|+|01\rangle\langle
0r|)+\omega\Big{[}(|11\rangle+|00\rangle)$ (19) $\displaystyle\otimes(\langle
01|+\langle 10|)\Big{]}+\mathrm{H.c.}$

with $\mathrm{H.c.}$ denoting Hermitian conjugate, and we have four decaying
channels with corresponding Lindblad jump operators

$\displaystyle L_{1}$ $\displaystyle=$ $\displaystyle|01\rangle\langle
0r|,\leavevmode\nobreak\ \leavevmode\nobreak\ L_{2}\leavevmode\nobreak\
=\leavevmode\nobreak\ |00\rangle\langle 0r|,$ $\displaystyle L_{3}$
$\displaystyle=$ $\displaystyle|10\rangle\langle r0|,\leavevmode\nobreak\
\leavevmode\nobreak\ L_{4}\leavevmode\nobreak\ =\leavevmode\nobreak\
|00\rangle\langle r0|.$ (20)

$\gamma$ is a damping rate. One can easily check that the Bell state satisfies
$H_{S}|\Phi\rangle=0$ and $L_{1,2,3,4}|\Phi\rangle=0$ as required by the DSP
scheme. With the above Lindblad operators, we find
$\mathcal{A}=\sqrt{2}\gamma/4$ in Eq. (13). In the following analysis, we
adopt experimental values for the parameters
$(\Omega_{2},\omega,\gamma)=2\pi\times(0.02,0.01,0.03)$ MHz Grankin  _et al._
(2014).

Under the adopted parameter set, we obtain via exact diagonalization the
following ordered six energy eigenstates of $H_{S}$ expressed in the local
basis of
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle,|0r\rangle,|r0\rangle\\}$,

$\displaystyle|E_{1}\rangle=\frac{1}{\sqrt{8}}(|00\rangle+|11\rangle+|0r\rangle+|r0\rangle)-\frac{1}{2}(|01\rangle+|10\rangle),$
$\displaystyle|E_{2}\rangle=\frac{1}{2}(|01\rangle+|r0\rangle)-\frac{1}{2}(|10\rangle+|0r\rangle),$
$\displaystyle|E_{3}\rangle=-\frac{1}{2}(|00\rangle+|11\rangle)+\frac{1}{2}(|0r\rangle+|r0\rangle),$
$\displaystyle|E_{4}\rangle=\frac{1}{\sqrt{2}}(|00\rangle-|11\rangle)\leavevmode\nobreak\
=\leavevmode\nobreak\ |\Phi\rangle,$
$\displaystyle|E_{5}\rangle=\frac{1}{2}(|10\rangle+|r0\rangle)-\frac{1}{2}(|01\rangle+|0r\rangle),$
$\displaystyle|E_{6}\rangle=\frac{1}{\sqrt{8}}(|00\rangle+|11\rangle+|0r\rangle+|r0\rangle)+\frac{1}{2}(|01\rangle+|10\rangle).$
(21)

We remark that the prepared Bell state corresponds to the fourth excited state
of $H_{S}$.

Since we are just interested in the permutations of a given set
$\\{\lambda_{n}\\}$, we can take an arbitrary yet physical set of
$\\{\lambda_{n}\\}$ for numerical demonstration. As an illustration, we first
compare the following three permutations containing the same elements with
unity sum with respect to the ordered energy basis $\\{|E_{n}\rangle\\}$:

$\displaystyle\\{\lambda_{n}^{A}\\}$ $\displaystyle=$
$\displaystyle(0.2,0.15,0.1,0.4,0.08,0.07),$
$\displaystyle\\{\lambda_{n}^{B}\\}$ $\displaystyle=$
$\displaystyle(0.07,0.08,0.1,0.15,0.2,0.4),$
$\displaystyle\\{\lambda_{n}^{C}\\}$ $\displaystyle=$
$\displaystyle(0.4,0.2,0.15,0.1,0.08,0.07).$ (22)

In this case, permutation $A$ emerges as the optimal choice among all
permutations, as it exhibits the maximum population of $0.4$ on state
$|E_{4}\rangle$, with other elements arranged in a decreasing order according
to Eq. (17). On the other hand, permutation $C$ represents a passive state,
while permutation $B$ is chosen due to its natural ordering, which serves as
the inverse of its counterpart, $C$. We emphasize that there is nothing
special about the chosen elements of $\\{\lambda_{n}\\}$ in Eq. (V). We depict
the evolution dynamics of the relative purity $\mathrm{Tr}[\rho_{t}\rho_{f}]$
in Fig. 1 obtained by solving the Lindblad master equation Eq. (18) subject to
initial states $\rho_{0}=\sum_{n}\lambda_{n}|E_{n}\rangle\langle E_{n}|$
generated by the permutations in Eq. (V). We highlight the arrangements of Eq.
(V) on the top of Fig. 1. In the inset we further list the values of the QSL
$T_{\mathrm{QSL}}$ [cf. Eq. (13)] and dissipated heat $Q$ [cf. Eq. (6)] for
all possible permutations of the same elements as in Eq. (V).

As demonstrated in Fig. 1 and its inset, it becomes clear that the optimal
permutation $A$ achieves the minimal QSL while also exhibiting the smallest
dissipated heat among the set of permutations that yield the same QSL. This
result illustrates the methodology behind our Pareto-type optimization
strategy. In comparison, the permutation $\\{\lambda_{n}^{B}\\}$ fails to
minimize both the QSL $T^{\mathrm{QSL}}$ and the dissipated heat $Q$. While
the permutation $\\{\lambda_{n}^{C}\\}$ minimizes $Q$ as expected, it leads to
the slowest evolution dynamics among the three scenarios in Eq. (V). From the
results, we remark that the dissipated heat $Q$ can be negative as the final
prepared state is an excited eigenstate of the system, implying that the
environment injects energy into the system to assist the DSP process. This
seems at odds with the usual Landauer bound Landi and Paternostro (2021) which
states instead a positive $Q$ as $\Delta S$ is positive here. However, we
should point out that the engineered bath used in DSP is usually non-thermal
(see, e.g., Ref. Leghtas  _et al._ (2015)) and one cannot assign a meaningful
thermodynamic temperature to it. Without a temperature, it is difficult to
relate heat to entropy production, hence the usual Landauer bound is not
applicable. In this regard, a refined Landauer bound Timpanaro  _et al._
(2020) may help.

Figure 2: Relative purity $\mathrm{Tr}[\rho_{t}\rho_{f}]$ as a function of
evolution time obtained by solving Eq. (18) with three initial conditions
$\rho_{0}^{A,B,C}=\sum_{n}\lambda_{n,th}^{A,B,C}|E_{n}\rangle\langle E_{n}|$.
Here, $\\{\lambda_{n,th}^{A,B,C}\\}$ are permutations of thermal populations
$\\{e^{-\beta E_{n}}/\sum_{n}{e^{-\beta E_{n}}}\\}$ according to the same
ordering rule in Eq. (V). Permutation $A$ is again the optimal one. Inset: we
show the corresponding values of $T_{\mathrm{QSL}}$ [cf. Eq. (13)] and $Q$
[cf. Eq. (6)] for all possible permutations of the same set of thermal
populations $\\{e^{-\beta E_{n}}/\sum_{n}{e^{-\beta E_{n}}}\\}$, with
permutations $A,\leavevmode\nobreak\ B$ and $C$ highlighted. For a better
view, we shift the data set containing permutation $C$ upwards by an amount of
0.05 $\gamma^{-1}$. In the calculation, we set $\Omega=2\pi$ MHz as the unit
and take an inverse temperature $\beta=20$.

Although we take an arbitrary set of populations in the above, we remark that
the distinctions between three orderings $\\{\lambda_{n}^{A,B,C}\\}$ are
independent of detailed values of $\\{\lambda_{n}\\}$. To see this, we instead
consider populations $\\{\lambda_{n,th}\\}=\\{e^{-\beta
E_{n}}/\sum_{n}{e^{-\beta E_{n}}}\\}$ generated from a thermal initial state
$e^{-\beta H_{S}}/\mathrm{Tr}[e^{-\beta H_{S}}]$ with $\beta$ an inverse
temperature; One can couple the system to a thermal bath initially to obtain
such a thermal initial state. In Fig. 2, we depict such a set of results
obtained from three permutations $\\{\lambda_{n,th}^{A,B,C}\\}$ according to
the same ordering rule in Eq. (V) by fixing the temperature value. Compared
with Fig. 1, it is evident that one still observes the same behaviors of the
dynamics of $\mathrm{Tr}[\rho_{t}\rho_{f}]$ as well as dissipated heat $Q$.
Particularly, from the inset of Fig. 2, we see that the permutation $A$ again
minimizes the QSL together with a smallest $Q$ among the set with the same
QSL. We note that for the chosen parameter set $E_{3}\simeq-1.05\times
10^{-18}$ and $E_{4}=0$ are very close in energy in our example, hence the
permutations $\\{\lambda_{n,th}^{B}\\}$ and $\\{\lambda_{n,th}^{C}\\}$ sorted
in respectively an increasing and a decreasing order in magnitude result in
almost indistinguishable dynamical evolution of the relative purity
$\mathrm{Tr}[\rho_{t}\rho_{f}]$ as can be seen from Fig. 2. In the inset, we
shift the data set containing the permutation $C$ upwards for a better view.
Nevertheless, the dissipated heat $Q$ results from permutations $B$ and $C$
are different.

##  VI Conclusion and outlook

In this study, we derived an initial-state dependent quantum speed limit (QSL)
for measuring the preparation time of dissipative state preparation (DSP)
processes. Based on the obtained QSL, we performed a Pareto-type initial state
optimization under a fixed set of populations in the system energy eigenbasis.
By conditionally minimizing the dissipated heat based on an optimized QSL, we
determined the optimal permutation of populations with respect to an ordered
energy eigenbasis. Our findings revealed that the initial state corresponding
to this optimal permutation exhibits the maximum overlap with the final
prepared states among all possible states generated from permutations of
populations. Furthermore, the remaining population elements are sorted in an
order that resembles a passive state. We demonstrated the effectiveness of our
framework in a model system consisting of two three-level Rydberg atoms used
to dissipatively prepare the Bell state.

In the context of preparing desired states, it is commonly assumed that the
process essentially involves cooling the system to the ground state of an
appropriate many-body Hamiltonian. Hence the dissipated heat out of the system
is always positive, just as existing qubit reset and information erasure
studies Mohammady  _et al._ (2016); Klaers (2019); Miller  _et al._ (2020);
Zhen  _et al._ (2021); Van Vu and Saito (2022) showed. However, our numerical
analysis of a dissipative Rydberg atom system for preparing the excited Bell
state unveils a distinct thermodynamic behavior within the DSP scheme, namely,
dissipated heat can be negative under optimal scenarios. This result suggests
that the environment injects energy into the system to facilitate DSP
processes, enabling rapid preparation of an excited state.

Furthermore, conventional expressions for total entropy production are not
applicable in DSP Abe (2003); Santos  _et al._ (2017), highlighting the need
for a self-consistent quantum thermodynamic framework for DSP. The development
of such a framework remains an area for future research and promises to be a
valuable contribution to the field of dissipative quantum information
processing.

###### Acknowledgements.

J. Liu acknowledges supports from Shanghai Pujiang Program (Grant No.
22PJ1403900), the National Natural Science Foundation of China (Grant No.
12205179), and start-up funding of Shanghai University. H. Nie is supported by
CQT PhD programme.

##  Appendix A Another derivation for quantum speed limit

To obtain a QSL, one can directly start from the equation of motion for the
fidelity del Campo  _et al._ (2013),

$\frac{d}{dt}\mathrm{Tr}[\rho_{t}\rho_{f}]\leavevmode\nobreak\
=\leavevmode\nobreak\
\mathrm{Tr}[\rho_{f}(\mathcal{L}\rho_{t})]\leavevmode\nobreak\
=\leavevmode\nobreak\ \mathrm{Tr}[(\mathcal{L}^{\dagger}\rho_{f})\rho_{t}]$
(A1)

Here, we introduced Liouvillian superoperator $\mathcal{L}$ and its adjoint

$\mathcal{L}^{\dagger}\rho\leavevmode\nobreak\ =\leavevmode\nobreak\
i[H_{S},\rho]+\sum_{\mu}\gamma_{\mu}\left[L_{\mu}^{\dagger}\rho
L_{\mu}-\frac{1}{2}\\{L_{\mu}^{\dagger}L_{\mu},\rho\\}\right].$ (A2)

Using properties of $\rho_{f}$, we find
$\mathcal{L}^{\dagger}\rho_{f}=\sum_{\mu}\gamma_{\mu}L_{\mu}^{\dagger}\rho_{f}L_{\mu}$.
Using the Cauchy-Schwarz inequality, we have

$\left|\frac{d}{dt}\mathrm{Tr}[\rho_{t}\rho_{f}]\right|\leavevmode\nobreak\
\leqslant\leavevmode\nobreak\ \mathcal{A}||\rho_{t}||_{F}\leavevmode\nobreak\
\leqslant\leavevmode\nobreak\ \mathcal{A}.$ (A3)

Recalled that
$\mathcal{A}\equiv||\sum_{\mu}\gamma_{\mu}L_{\mu}^{\dagger}\rho_{f}L_{\mu}||_{F}$.
Adopting the parametrization and integrating both sides of the above
inequality, we receive

$T\leavevmode\nobreak\ \geqslant\leavevmode\nobreak\
T_{\mathrm{QSL},2}\leavevmode\nobreak\ \equiv\leavevmode\nobreak\
\frac{\int_{\Theta_{0}}^{\Theta_{f}}d|\cos\Theta_{t}|}{\mathcal{A}}\leavevmode\nobreak\
=\leavevmode\nobreak\ \frac{1-\cos\Theta_{0}}{\mathcal{A}}.$ (A4)

Since $\cos\Theta_{0}\in[0,1]$ with $\Theta_{0}\in[0,\pi/2]$, one can easily
proof that $1-\cos\Theta_{0}\leqslant\sqrt{2(1-\cos\Theta_{0})}$, therefore
the bound $T_{\mathrm{QSL}}$ obtained in the main text is tighter than
$T_{\mathrm{QSL},2}$ got here, namely,

$T\leavevmode\nobreak\ \geqslant\leavevmode\nobreak\
T_{\mathrm{QSL}}\leavevmode\nobreak\ \geqslant\leavevmode\nobreak\
T_{\mathrm{QSL},2}.$ (A5)

##  Appendix B Passive initial states minimize the heat exchange alone

Since the final state is fixed, we have

$Q\leavevmode\nobreak\ \geqslant\leavevmode\nobreak\
\mathop{\mathrm{min}}\limits_{\rho_{0}}\left(\mathrm{Tr}[H_{S}\rho_{0}]\right)-E_{n^{\ast}}.$
(B1)

To obtain the minimum of $\mathrm{Tr}[H_{S}\rho_{0}]$, we resort to the
Majorization theory Mohammady  _et al._ (2016): Noting that the energy basis
we considered is already ordered with ordered eigenvalues $E_{1}\leqslant
E_{2}\leqslant\cdots\leqslant E_{N}$, the minimum of
$\mathrm{Tr}[H_{S}\rho_{0}]$ is attained when we consider a passive initial
state Allahverdyan  _et al._ (2004); Uzdin and Rahav (2021)

$\sigma_{0}\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum_{n=1}^{N}\tilde{\lambda}_{n}|E_{n}\rangle\langle E_{n}|$ (B2)

with $\\{\tilde{\lambda}_{n}\\}$ a permutation of $\\{\lambda_{n}\\}$ such
that probabilities are sorted in a decreasing order,
$\tilde{\lambda}_{1}\geqslant\tilde{\lambda}_{2}\geqslant\cdots\geqslant\tilde{\lambda}_{N}$.
By doing so, we find

$\displaystyle\mathop{\mathrm{min}}\limits_{\rho_{0}}\left(\mathrm{Tr}[H_{S}\rho_{0}]\right)\leavevmode\nobreak\
=\leavevmode\nobreak\ \mathrm{Tr}[H_{S}\sigma_{0}]\leavevmode\nobreak\
=\leavevmode\nobreak\ \sum_{n=1}^{N}\tilde{\lambda}_{n}E_{n},$
$\displaystyle\Longrightarrow Q\leavevmode\nobreak\
\geqslant\leavevmode\nobreak\
\sum_{n=1}^{N}\tilde{\lambda}_{n}E_{n}-E_{n^{\ast}}.$ (B3)

We remark that since the DQSP is a non-unitary process it is possible for a
passive state to exchange heat with the environment. Now let us turn to the
fidelity

$\mathrm{Tr}[\sigma_{0}\rho_{f}]\leavevmode\nobreak\ =\leavevmode\nobreak\
\tilde{\lambda}_{n^{\ast}},$ (B4)

which is in general smaller than $\tilde{\lambda}_{1}$ and hence $\sigma_{0}$
is not optimal for minimizing the QSL $T_{\mathrm{QSL}}$ among other
permutations of $\\{\lambda_{n}\\}$.

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11institutetext:  1 School of Science and Engineering, The Chinese University
of Hong Kong(Shenzhen)
2 Shenzhen Research Institute of Big Data
3 School of Biomedical Engineering & Suzhou Institute for Advanced Research,
University of Science and Technology of China, Suzhou, China
4 Institute of Computing Technology, Chinese Academy of Sciences, Beijing,
China
5 Institute of Urology, The Third Affiliated Hospital of Shenzhen
University(Luohu Hospital Group), Shenzhen, China.
11email<EMAIL_ADDRESS>
# Shallow Attention Network for Polyp Segmentation

Jun Wei 1122†† Yiwen Hu 112255†† Ruimao Zhang 1122 Zhen Li 1122🖂🖂
S.Kevin Zhou 113344 Shuguang Cui 1122

###### Abstract

Accurate polyp segmentation is of great importance for colorectal cancer
diagnosis. However, even with a powerful deep neural network, there still
exists three big challenges that impede the development of polyp segmentation.
(i) Samples collected under different conditions show inconsistent colors,
causing the feature distribution gap and overfitting issue; (ii) Due to
repeated feature downsampling, small polyps are easily degraded; (iii)
Foreground and background pixels are imbalanced, leading to a biased training.
To address the above issues, we propose the Shallow Attention Network (SANet)
for polyp segmentation. Specifically, to eliminate the effects of color, we
design the color exchange operation to decouple the image contents and colors,
and force the model to focus more on the target shape and structure.
Furthermore, to enhance the segmentation quality of small polyps, we propose
the shallow attention module to filter out the background noise of shallow
features. Thanks to the high resolution of shallow features, small polyps can
be preserved correctly. In addition, to ease the severe pixel imbalance for
small polyps, we propose a probability correction strategy (PCS) during the
inference phase. Note that even though PCS is not involved in the training
phase, it can still work well on a biased model and consistently improve the
segmentation performance. Quantitative and qualitative experimental results on
five challenging benchmarks confirm that our proposed SANet outperforms
previous state-of-the-art methods by a large margin and achieves a speed about
72FPS.

###### Keywords:

Polyp segmentation Colonoscopy Colorectal cancer

##  1 Introduction

Colorectal Cancer (CRC) has become a serious threat for human health, causing
the fourth highest cancer death rate worldwide [7]. Polyps in the intestinal
mucosa are considered the harbinger of CRC, which are easily transformed into
malignant lesions [9]. Therefore, early diagnosis and treatment of polyps are
of great significance. Fortunately, with the assistance of computer
technology, a lot of automatic polyp segmentation models [1, 4, 5, 6, 13, 15,
17, 22, 23, 25] have been developed and achieved remarkable progress.

However, polyp segmentation has always been a challenging task due to the
inconsistent color distribution and small size of the targets. As shown in
Fig. 1(a), polyp samples collected under the same conditions usually
correspond to the same color, while those collected under different conditions
appear in different colors. Thus, a strong correlation between color and polyp
segmentation is implicitly contained in the dataset, which is harmful to the
model training and will cause the model to overfit the color. Besides, most of
polyp areas are very small. As shown in Fig. 1(b), the vast majority of polyps
have an area which is less than 0.1 of the total image area. This brings two
difficulties for existing segmentation models. First, small polyps are prone
to get lost and hard to restore because of the repeated feature downsampling.
Second, for images with small polyps, there exists a large imbalance between
foreground and background pixels, leading to a biased model and a poor
performance.

Figure 1: (a) Polyp samples with different colors. (b) The histogram of polyp
size. The horizontal axis shows the proportion of the polyp area to the image
area. The vertical axis shows the proportion of polyp samples of a specific
size to the total samples. One training set and five testing sets are shown in
different colors.

To break the correlation between color and polyp segmentation, we propose the
color exchange (CE) operation. Specifically, for each input image, we randomly
select another one and transfer its color to the input image. By multiple CE
operations, each input image could appear in different colors, thus the
connection between color and polyp segmentation is reduced and the model will
not overfit to the fake causality. Besides, small polyp segmentation relies
more on shallow features. Because they have higher resolutions and contain
richer details, compared with deep ones [19]. Unfortunately, shallow features
are too noisy to be used directly. Thus, we propose the shallow attention
module (SAM), which could help to remove the background noise, using clearer
deep features. Furthermore, to ease the imbalance between foreground and
background, we propose the probability correction strategy (PCS) during
inference, which can adjust the biased prediction to the correct one with
simple post-processing.

In summary, our contributions are four-folds: (1) We propose the color exhange
operation to decouple the contents and colors, which reduces the overfitting
issue; (2) We design the novel SANet to focus more on shallow features with
richer details, improving the small polyp segmentation; (3) We introduce the
probability correction strategy (PCS) for inference to balance the biased
predictions, which can consistently improve model performance with negligible
computation cost; (4) Extensive experiments demonstrate that the proposed
model achieves state-of-the-art performace on five widely used public
benchmarks.

##  2 Related Work

Figure 2: An overview of the proposed model.

Polyp segmentation has went through two periods of development. Earlier models
mainly rely on hand-crafted features (e.g., color and texture) [13, 24, 17],
which can hardly capture the global context information and are not robust to
complex scenarios. Recently, fully convolutional network (FCN [12]) has been
applied to polyp segmentation and made great progress. For example, U-Net [15]
is a famous structure for medical image segmentation, which consists of a
contracting path to capture context and an expanding path to restore the
precise detail. SegNet [20] adopts a similar structure to U-Net, but utilizes
the max pooling indices to restore features in the upsampling operation.
U-Net++ [25] and ResUNet++ [11] further improve original U-Net by dense
connection and better pretrained backbone, achieving promising segmentation
performance.

Though the body part of polyps could be well handled, boundaries are ignored.
To enhance the boundary segmentation, Psi-Net [14] proposes to combine both
body and boundary features in the segmentation model. Furthermore, SFA [6]
explicitly applies an area-boundary constraint to supervise the learning of
both polyp regions and boundaries. PraNet [5] proposes the reverse attention
to firstly locate the polyp areas and then refine object boundaries
implicitly. Though the above models have made great progress, polyp
segmentation still faces big challenges because of the limited data. Thus, we
propose the SANet to further improve the polyp segmentation, as shown in the
Fig. 1.

##  3 Method

Fig. 2 depicts the concrete architecture of the proposed SANet, where Res2Net
[8] is adopted as the encoder backbone. According to the feature scale,
Res2Net could be divided into five blocks, which have different receptive
fields. Considering Wu et al. [21] have shown that low-level features bring
too much computational cost with limited performance gains, thus we use the
features of last three blocks $\\{f_{i}|i\in(3,4,5)\\}$ for the following
experiments.

###  3.1 Color Exchange

Input: Img1, Img2

Output: Out1, Out2

1 transform Img1, Img2 from RGB space to LAB space, then get Lab1, Lab2

2 calculate the channel mean and channel std of Lab1 and Lab2

3 Lab1 = (Lab1-mean1)/std1*std2+mean2

4 Lab2 = (Lab2-mean2)/std2*std1+mean1

transform Lab1, Lab2 from LAB space to RGB space, then get Out1, Out2

Algorithm 1 Color Exchange

As show in Fig. 1(a), polyp samples collected under different conditions show
very different color distributions. However, these colors are far less
important than shapes and structures for polyp segmentation, which actually
leads to the overfitting issue due to the limited dataset. To avoid this
effect, we propose the color exchange (CE) operation to explicitly decouple
the image content and color. Specifically, for each input image, we randomly
pick another one from the dataset and transfer its color to the input image.
Alg. 1 shows the specific steps. We calculate the mean and standard deviation
of the colors in the LAB space and exchange these statistics between images.
As show in Fig. 2, after color exchange, we could get the new input image with
exactly the same content but a different color. Exchanging with different
auxiliary images, the same input image could show a variety of colors but
correspond to the same ground truth, so the model will focus more on the image
contents and will not be affected by the color distribution, which could
largely alleviate the influence of color distribution on model training. It is
worth noting that color exchange is performed only during training. For
inference, we directly use the original images, thus it will not bring any
time overhead during testing.

###  3.2 Shallow Attention Module

As shown in Fig. 1(b), small polyps make up the majority of both training and
testing datasets, facing the serious information loss during the repeated
downsampling in CNNs. To avoid this limitation, shallow features $f_{s}$
deserve more attention due to their high resolutions. Namely, $f_{s}$ have
clear object boundaries, which are important for the accurate polyp
segmentation. However, due to the limitation of receptive field, these
features are submerged by background noise and hard to be used directly. In
contrast, deep features $f_{d}$ are coarse in boundaries but have clean
background. Therefore, we propose the shallow attention module (SAM) to filter
out background noise of $f_{s}$ with the assistance of $f_{d}$, as shown in
Fig. 2. Specifically, SAM involves in both $f_{s}$ and $f_{d}$. Different from
self-attention, SAM makes use of the complementarity between different
features. In SAM, $f_{d}$ will be firstly upsampled into the same size with
$f_{s}$ and then regarded as the attention maps for $f_{s}$. Finally, these
maps will be multiplied with $f_{s}$ to help suppress the background noise.
The whole process is shown in following equations:

$\displaystyle Att$ $\displaystyle=\sigma(Up(f_{d})),$ (1) $\displaystyle
f_{s}$ $\displaystyle=Att\otimes f_{s},$ (2)

where $Up(\cdot)$ represents the upsampling operation. $\sigma(\cdot)$ is the
ReLU function and $\otimes$ is the element-wise multiplication. After SAM,
shallow features $f_{s}$ will become much cleaner and provide important cues
for the segmentation of small polyps. Furthermore, SAM plays an important role
in balancing features of different blocks. Instead of aggregating features
from all levels with the same weight, SAM could dynamically assign weight to
different features according to their contributions.

###  3.3 Probability Correction Strategy

Probability correction strategy (PCS) is an effective way to improve final
predictions, especially for small polyps where foreground and background
pixels are extremely unbalanced. This imbalance brings in difficulties for the
model training and leads to a biased model. Namely, negative samples
(background pixels) are dominant in the training process, which leads to the
tendency of the model to give lower confidence to positive samples (foreground
pixels). To enhance the predictions of the positive samples, we propose to
explicitly correct the predicted probability through logit reweighting.
Specifically, for each image, we extract the predicted features before the
Sigmoid function as the target (i.e., logit) to be corrected. We count the
proportion ($rate^{p}$) of positive samples (i.e., $logit>0$) in the image,
which is relatively small. At the same time, due to the model bias, the logit
of the positive sample also is small. Thus, we propose to normalize the logit
of positive samples with its proportion, as shown in Eq. 3.

$logit^{p_{norm}}_{ij}=logit^{p}_{ij}/rate^{p}$ (3)

where $i,j$ represent the coordinate of the sample. Similarly, the logits of
negative (i.e., $logit<0$) samples could also be normalized, as shown in Eq.
4.

$logit^{n_{norm}}_{ij}=logit^{n}_{ij}/rate^{n}$ (4)

Figure 3: Visual comparison between the predictions with PCS and without PCS.

After normalization, the effect of sample number has been attenuated and the
segmentation results will be more accurate. It is worth nothing that PCS is
only applied in the inference phase with very little computation cost. Fig. 3
visualizes some predictions. Obviously, predictions with PCS have clearer
boundaries.

###  3.4 Loss Function

We use binary cross entropy (BCE) loss and Dice loss for supervision, as shown
in Eq. 5, where $P,G$ are the prediction and ground truth, respectively.
$\lambda_{1},\lambda_{2}$ are the weighting coefficients, which are set to 1
for simplification.

$Loss={\lambda_{1}BCE(P,G)+\lambda_{2}Dice(P,G)}$ (5)

##  4 Experiments

###  4.1 Datasets and Training Settings

To evaluate the performance of the proposed SANet, five polyp segmentation
datasets are adopted, including Kvasir [10], CVC-ClinicDB [2], CVC-ColonDB
[3], EndoScene [18] and ETIS [16]. To keep the fairness of the experiments, we
follow [5] advice and take exactly the same training and testing dataset
division. Besides, six state-of-the-art methods are used for comparison,
namely U-Net [15], U-Net++ [25], ResUNet [23], ResUNet++ [11], SFA [6] and
PraNet [5]. We use Pytorch to implement our model. All input images are
uniformly resized to 352×352. For data augmentation, we adopt the random flip,
random rotation and multi-scale training. The whole network is trained in an
end-to-end way, using stochastic gradient descent (SGD). Initial learning rate
and batch size are set to 0.04 and 64, respectively. We train the entire model
for 128 epoches.

###  4.2 Quantitative Comparison

Table 1: Performance comparison with different polyp segmentation models. The highest and second highest scores are highlighted in red and blue colors, respectively. | Kvasir | ClinicDB | ColonDB | EndoScene | ETIS
---|---|---|---|---|---
Methods | mDice | mIoU | mDice | mIoU | mDice | mIoU | mDice | mIoU | mDice | mIoU
U-Net | 0.818 | 0.746 | 0.823 | 0.750 | 0.512 | 0.444 | 0.710 | 0.627 | 0.398 | 0.335
U-Net++ | 0.821 | 0.743 | 0.794 | 0.729 | 0.483 | 0.410 | 0.707 | 0.624 | 0.401 | 0.344
ResUNet | 0.791 | - | 0.779 | - | - | - | - | - | - | -
ResUNet++ | 0.813 | 0.793 | 0.796 | 0.796 | - | - | - | - | - | -
SFA | 0.723 | 0.611 | 0.700 | 0.607 | 0.469 | 0.347 | 0.467 | 0.329 | 0.297 | 0.217
PraNet | 0.898 | 0.840 | 0.899 | 0.849 | 0.712 | 0.640 | 0.871 | 0.797 | 0.628 | 0.567
SANet(Ours) | 0.904 | 0.847 | 0.916 | 0.859 | 0.753 | 0.670 | 0.888 | 0.815 | 0.750 | 0.654

To prove the effectiveness of the proposed SANet, six state-of-the-art models
are used for comparison, as shown in Tab. 1. SANet achieves the best scores
across five datasets on both mIoU and mDice, demonstrating the superior
performance of the proposed model. In addition, Fig. 4 shows the dice values
of above models under different thresholds. From these curves, we could
observe that SANet consistently outperforms other models, which proves its
good capability for polyp segmentation. Furthermore, SANet achieves a speed
about 72FPS on RTX 2080Ti GPU, faster than the 64FPS of previous PraNet [5].

Figure 4: Dice curves under different thresholds on three polyp datasets.

###  4.3 Visual Comparison

Fig. 5 visualizes some predictions of different models. Compared with other
counterparts, our method could not only clearly highlight the polyp regions
but also suppress the background noise. Even for the challenging scenarios,
our model could handle well and generate accurate segmentation mask.

###  4.4 Ablation Study

To investigate the importance of each component in SANet, both ColonDB and
Kvasir datasets are used for controlled experiments. As shown in Tab. 2, all
the modules or strategies are necessary for the final predictions. Combining
all the proposed methods, our model achieves the new state-of-the-art
performance.

Table 2: Ablation study for SANet on the ColonDB and Kvasir datasets. Settings | ColonDB | Kvasir
---|---|---
mDice | mIoU | mDice | mIoU
backbone | 0.676 | 0.608 | 0.853 | 0.780
backbone+SAM | 0.728 | 0.645 | 0.882 | 0.821
backbone+SAM+CE | 0.745 | 0.662 | 0.896 | 0.838
backbone+SAM+CE+PCS | 0.753 | 0.670 | 0.904 | 0.847
Figure 5: Visual comparison between the proposed method and four state-of-the-
art ones.

##  5 Conclusion

Because of the limited dataset, polyp segmentation model is easy to corrupt
due to overfitting. In this paper, we attempt to alleviate this problem from
two aspects. For the false color causality, we propose to decouple the image
color and content by color exchange. For the difficult small polyp
segmentation, we design the shallow attention to reduce the data noise. All of
these can reduce the interference of irrelevant factors on the model. In the
future, we will combine more prior knowledge to design more robust features to
remove the interference of independent factors.

##  6 Acknowledgement

The work was supported in part by Key Area R&D Program of Guangdong Province
with grant No.2018B030338001, by the National Key R&D Program of China with
grant No.2018YFB1800800, by Shenzhen Outstanding Talents Training Fund, by
Guangdong Research Project No.2017ZT07X152, by NSFC-Youth 61902335, by
Guangdong Regional Joint Fund-Key Projects 2019B1515120039, by The National
Natural Science Foundation Fund of China (61931024), by helix0n biotechnology
company Fund and CCF-Tencent Open Fund.

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# CMS RPC data taking during the LHC Run-2 and activities during Long Shutdown
2

Kevin Mota Amarilo  on behalf of CMS Collaboration  Rio de Janeiro State
University, Institute of Physics, R. São Francisco Xavier, 524, Rio de Janeiro
- RJ, 20550-013, Brasil<EMAIL_ADDRESS>
###### Abstract

The CMS experiment collected around 150 fb-1 of proton-proton collision data
at $\sqrt{s}$ = 13 TeV during the Run-2 data taking period of LHC. The CMS RPC
system provided redundant information for robust muon triggering,
reconstruction and identification. To ensure stable data taking, the CMS RPC
collaboration has performed detector operation, calibration and performance
studies. After the end of Run-2, it was started the second LHC long shutdown
period (LS2), an important opportunity for maintenance and preparation for the
next data taking period (Run-3) and the installation of services in
preparation for the Phase-II upgrade. The activities included maintenance of
power, gas and online systems. In this presentation, the overall performance
of the CMS RPC system during the Run-2 period is summarized as well as all the
activities done in preparation for future data taking periods.

##  1 Introduction

One of the main actors of the CMS (Compact Muon Solenoid) experiment [1] is
its Muon System [2], which uses gaseous detector technologies to accomplish
muon triggering, identification, transverse momentum and charge measurement.

Through the end of Run 2 in 2018, the components of the CMS Muon system were:
drift tubes (DT) at barrel region ($|\eta|<1.2$), cathode strip chambers (CSC)
at endcap region ($1.2<|\eta|<2.4$) and Resistive Plate Chambers (RPC), in
barrel and endcap regions ($|\eta|<1.9$). During LS2 a new type of detector
was installed, the gas electron multiplier (GEM), as part of the Muon System
Phase-2 Upgrades to complement measurements in the higher $\eta$ region for
Phase-II upgrade [3], which will also include new RPC chambers, the so called
improved-RPC (iRPC).

The CMS RPC system is composed of double gap phenolic resin (called bakelite)
chambers, operating in avalanche mode with gas mixture 95.2%
$C_{2}H_{2}F_{4}$, to enhance the ionization caused by incident particles,
4.5% $iC_{4}H_{10}$ as a quencher gas to reduce streamer formation and 0.3%
$SF_{6}$ to control secondary ionization. The RPCs are designed and calibrated
to have a time resolution of around 2 ns and a number of adjacent strips fired
in a single muon hit (Cluster size) between 2 and 3.

##  2 CMS RPC Run-2 data taking

Between 2015 to 2018 during Run-2, CMS recorded proton-proton collisions with
$\sqrt{s}$ = 13 TeV from LHC, with total integrated luminosity of 150.26 fb-1,
out of which only 0.15 % was lost due to RPC problems. The total accumulated
charge in Run-1 and Run-2 of the RPC system was 2.3 mC/cm2 in barrel and 7.5
mC/cm2 in endcap. Studies at the CERN Gamma Irradiation Facility ++ (GIF++)
have showed no drop in efficiency of RPCs up to an integrated charge of 153
mC/cm2 [8]

###  2.1 RPC efficiency and cluster size stability

Hit efficiency and cluster size (CLS) are important parameters for RPC
performance. An RPC hit’s geometrical coordinates are calculated as the
geometrical center of the cluster formed by the fired adjacent strips. The CLS
of RPC hits should be kept less than 4 strips to avoid fake muon triggers,
therefore, proper calibration is very important to keep good performance.

The calibration is done by analyzing the efficiency and CLS dependence on the
effective high voltage (HV), which is the high voltage applied corrected by
the environmental pressure and temperature variations. HV working point scans
were performed once or twice a year in dedicated collision runs, effective
voltage is applied in values between 8600 and 9800 V. The collected data is
analyzed and the proper working points are selected. More information on the
calibration can be found in [4].

Figure 1: RPC average efficiency vs integrated luminosity during Run-2 for
barrel stations. Red vertical lines show the planned technical stops (TS) and
the grey ones — Year-End-Technical stops (YETS). Figure 2: RPC average cluster
size vs integrated luminosity during Run-2 for barrel stations. Red vertical
lines show the planned technical stops (TS) and the grey ones — Year-End-
Technical stops (YETS).

The Figures 1 and 2 shows the Run-2 efficiency and CLS history. Each point
corresponds to the average efficiency or CLS per barrel station. Each change
in the trends is related to the deployment of a new working point. With
exception to a drop in efficiency and CLS between in August 2018 caused by a
configuration setting issue, it is possible to see that the RPC system had a
stable performance with efficiency greater than 95 % and CLS between 2 and 3,
within the CMS Muon Trigger requirements.

###  2.2 RPC contribution to the CMS Muon Trigger

The main contribution of the RPC system is to the CMS Level-1 (L1) Muon
Trigger [5], which is divided into 3 $\eta$-regions: the barrel muon track
finder (BMTF) for $|\eta|<0.83$, the overlap muon track finder (OMTF) for
$0.83<|\eta|<1.24$ and the endcap muon track finder (EMTF) for $|\eta|>1.24$.
Each one of the track finders can access information of the detectors
concurrently, building tracks and assigning pT and exploiting the redundancies
of the muon system.

In the BMTF, RPC timing information is used to improve DT trigger primitives
and bunch crossing assignment. In the OMTF, RPC-only segments are build from
the 8 chambers available, resulting in a substantial gain in in efficiency.
Figure 3 demonstrates that using RPC information in the OMTF increases
efficiency by 15 %. Finally, in the EMTF, RPC hit position is used in case of
CSC trigger primitive absence. More information on this subject can be found
in [6].

Figure 3: Trigger efficiency versus pT for the OMTF derived from algorithm
emulation applied on real data. Using (Red) and not using (Blue) RPC
information. Ref. [6].

###  2.3 RPC ohmic current monitoring

The ohmic current is defined as the current with no beam, up to around 7000 V,
where the gas amplification contribution is negligible and the currents follow
the Ohm’s law.

Figure 4 shows the history of ohmic current measured in four RPC stations. An
increase in the currents was observed in all stations, correlated to the
background, the RE+4 and RE-4 background rate is about 40 Hz/cm2 and W+0 and
RE-1 is less than 10 Hz/cm2. In November of 2018, at the start of the Heavy
Ion period where the luminosity is very low and the background is effectively
zero, the currents started to decrease. In RE-4 the decrease was faster with
the increase of the gas flux.

The increase was correlated with the production of flourine ions (F-) in the
gas gap due to the electrical discharge inside the gap. The flourine can
combine with the water in the gas and form HF, that can damage the gap. This
explains the velocity of decrease of currents correlation with gas flux, as
the gas can flush out the F-ions before the combination with water. More
information on the fluorine formation inside RPC gas gap can be found in [7].

Figure 4: RPC ohmic current monitoring history for the stations W+0, RE+1,
RE+4 and RE-4.

##  3 LS2 Activities

During LS2, CMS is undergoing an intensive upgrade and maintenance program
during the second two-year-long shutdown period (LS2). To ensure an excellent
performance of the detector in the subsequent physics program, the RPC group
pursued in the present shutdown a thorough detector consolidation program to
repair most of the hardware problems. In preparation for future installation
of new iRPC chambers, cooling and cable services for the new detectors were
installed. This includes thousands of kilometres of high voltage and low
voltage cables, stainless steel gas pipes between predistribution gas racks in
the Service cavern (USC) and gas distribution racks in the experimental cavern
(UXC), copper pipes between distribution racks and chambers, gas impedance
boxes, support equipment, and optical fibers for the iRPCs.

In order to keep the optimal performance of the system, an extensive HV and
low voltage (LV) maintenance campaign was performed. The goal of HV
maintenance was to identify the problematic parts of the HV power system and
to fix it in the best possible way, recuperating the performance of the
chambers. A total of 65 HV channels were repaired. The LV maintenance aim was
to ensure a proper operation and configuration of the detector electronics and
ensure a good functionality of the LV power boards and communication buses. A
total of 12 LV problems were fixed.

A very important activity was the extraction of the chambers from the two RE4
stations to allow the CSC ME4/1 chamber extraction for electronics
refurbishment. The chambers were brought to the surface, and accommodated to a
new laboratory with controlled environmental conditions and new gas lines to
provide the standard RPC gas mixture. All needed reparation and re-validation
was done in the laboratory before the re-installation.

The activity with most priority during LS2, was gas system consolidation. The
aim was to minimize the environmental impact of the RPC system, as the
standard gas mixture is mainly composed of fluorine composed gases (F-gases)
with high global warming potential (GWP). The actions taking place during LS2
are:

  * •

Gas Leak identification and repair, where 49 out of 99 leaky chambers in
barrel where repaired.

  * •

Recuperation of the Exhaust, which was not working during Run-2 for the
installation of the first $C_{2}H_{2}F_{4}$ recuperation system with
efficiency of 80 %, which have been developed by CERN EP-DT Gas team.

  * •

By end of 2021, CERN EP-DT Group is going to install automatic pressure
regulation valves on the redistribution gas racks in USC to minimize pressure
variations in the chambers, which can be a possible source of new leaks.

  * •

Turn off the remaining leaky chambers which was not possible to repair (about
3.5 % of RPC system), to keep the amount of fresh gas added to the system at a
minimum.

##  4 RPC commissioning with cosmic rays data during LS2

In Figure 5, the efficiency of the barrel chambers is compared using cosmic
rays runs of 2018 (End of Run-2) and in 2021 (after all the repairs of LS2).
The average efficiency of the chambers with efficiency greater than 70 % is
compared and is about the same (gain of 0.6 %). The fraction of chambers with
efficiency greater than 70 % increased in around 6 % as a result of all the
repairs, which is very important for the muon system redundancy.

Figure 5: RPC barrel efficiency comparison between cosmic runs of 2018 (Red)
and 2021 (Blue).

##  5 Conclusion

The CMS RPC system showed a stable performance after operation in Run-1 and
Run-2 ($\sim$185 fb-1) with average efficiency greater than 95 % and average
CLS between 2 and 3 strips/hit. We observed reversible ohmic current increase
in high background regions correlated to the F- formation inside the gaps.
During LS2 RPC did a massive maintenance campaign to repair gas leaks, HV and
LV in order to keep the optimal performance and is planning to keep the
greenhouse gases emissions at a minimum by keeping all the remaining leaky
chambers off and using the new $C_{2}H_{2}F_{4}$ recuperation system. The
performance of the RPC System is improved with respect to the end of Run-2 and
the system is ready and commissioned for Run-3 data taking.

## References

## References

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