# Large-scale GHZ states through topologically protected zero-energy mode in a
superconducting qutrit-resonator chain

Jin-Xuan Han1 Jin-Lei Wu1<EMAIL_ADDRESS>Yan Wang1 Yan Xia2 Yong-Yuan
Jiang1 Jie Song1,3,4,5<EMAIL_ADDRESS>1School of Physics, Harbin Institute
of Technology, Harbin 150001, China
2Department of Physics, Fuzhou University, Fuzhou 350002, China
3Key Laboratory of Micro-Nano Optoelectronic Information System, Ministry of
Industry and Information Technology, Harbin 150001, China
4Key Laboratory of Micro-Optics and Photonic Technology of Heilongjiang
Province, Harbin Institute of Technology, Harbin 150001, China
5Collaborative Innovation Center of Extreme Optics, Shanxi University,
Taiyuan, Shanxi 030006, People’s Republic of China

###### Abstract

We propose a superconducting qutrit-resonator chain model, and analytically
work out forms of its topological edge states. The existence of the zero-
energy mode enables to generate a state transfer between two ends of the
chain, accompanied with state flips of all intermediate qutrits, based on
which $N$-body Greenberger-Horne-Zeilinger (GHZ) states can be generated with
great robustness against disorders of coupling strengths. Three schemes of
generating large-scale GHZ states are designed, each of which possesses the
robustness against loss of qutrits or of resonators, meeting a certain
performance requirement of different experimental devices. With experimentally
feasible qutrit-resonator coupling strengths and available coherence times of
qutrits and resonators, it has a potential to generate large-scale GHZ states
among dozens of qutrits with a high fidelity. Further, we show the
experimental consideration of generating GHZ states based on the circuit QED
system, and discuss the prospect of realizing fast GHZ states.

##  I Introduction

Greenberger-Horne-Zeilinger (GHZ) states constitute an important class of
entangled many-body states. A general form of $n$-qubit GHZ states can be
expressed as Greenberger  _et al._ (1990); Mukherjee  _et al._ (2020); Reiter
_et al._ (2016); Omran  _et al._ (2019a); Zheng (2001)

$\frac{1}{\sqrt{2}}(|\alpha_{1}\alpha_{2}\alpha_{3}\cdots\alpha_{n}\rangle+e^{i\phi}|\beta_{1}\beta_{2}\beta_{3}\cdots\beta_{n}\rangle).$
(1)

where $\alpha_{j}+\beta_{j}=1~{}(\alpha_{j},\beta_{j}\in\\{0,1\\})$. These
states play a key role in quantum science and technologies, including open-
destination quantum teleportation Yang  _et al._ (2004a), concatenated error-
correcting codes Knill (2005), quantum simulation Song  _et al._ (2018), and
high-precision spectroscopy Leibfried  _et al._ (2004). In principle, the
benchmark for quantum information capability is the number of particles that
can be reliably entangled in a quantum processor. In experiment, multi-body
entanglement is achieved recently by capturing 20 trapped ions with around the
fidelity of $63.2\%$ Friis  _et al._ (2018), 12 photons with $59.8\%$ Zhong
_et al._ (2018), 18 photonic qubits exploiting three degrees of freedom of six
photons with $72.4\%$ Wang  _et al._ (2018), 12 superconducting qubits with
$55.6\%$ Gong  _et al._ (2019), 18 superconducting qubits with $53.0\%$ Song
_et al._ (2019) and 20 Rydberg atoms with $54.2\%$ Omran  _et al._ (2019b).
Among various physical systems, the ubiquitous noise and device imperfections
are, however, unavoidable and limit the range where multi-body entanglement
can be realized with a high fidelity.

Topological insulator Qi and Zhang (2011a); Chiu  _et al._ (2016a), a new kind
of novel state of matters, is characterized by the conducting edge states and
the insulating bulk states. The special conducting edge states are protected
by the energy gap of the topological system, leading the edge state to be
insensitive to local perturbations and disorders Kitaev (2001); Kane and Mele
(2005); Hasan and Kane (2010). To this end, the topological state of matter
has been studied in many systems, such as optical lattice systems Guo (2016);
Seo and Tian (2015), spin Fermi systems Ren  _et al._ (2015); Ho and Lin
(2016), supercooled atoms Jotzu  _et al._ (2014); Aidelsburger  _et al._
(2015) and synthetic materials Sun and Jia (2017). These novel properties
support many potential applications of topological insulators in quantum
information processing and computing. For instance, several proposals for
quantum state transfer (QST) have been presented Yao  _et al._ (2013); Dlaska
_et al._ (2017); Mei  _et al._ (2018); Longhi (2019); Qi  _et al._ (2020a);
Tan  _et al._ (2020); Zheng  _et al._ (2020); Qi  _et al._ (2020b) with
topologically protected channels. A topologically protected channel for QST
between remote quantum nodes mediated by the edge mode of a chiral spin liquid
was proposed and analyzed Yao  _et al._ (2013); Dlaska  _et al._ (2017). Mei
$et~{}al.$ presented an experimentally feasible mechanism for implementing
robust QST via the topological edge states by connecting superconducting Xmon
qubits into a one-dimensional chain Mei  _et al._ (2018). Also, topologically
protected entangled photonic states and its transport via edge states have
been reported Rechtsman  _et al._ (2016); Blanco Redondo  _et al._ (2018);
Wang  _et al._ (2019a); Hu  _et al._ (2020). In experiment, the topological
protection of spatially entangled biphoton states was demonstrated Wang  _et
al._ (2019a). The robustness in crucial features of the topological biphoton
correlation map in the presence of deliberately introduced disorder was found
in the silicon nanophotonic structure. Recently, a spatially entangled two-
particle NOON state is proposed by topological Thouless pumping in one-
dimensional disordered lattices Hu  _et al._ (2020).

In this paper, we propose a superconducting-circuit model to generate large-
scale GHZ states, where the model is a chain consisting of $N$ flux qutrits
connected by $(N-1)$ resonators, and the entangled states are protected by
topological zero-energy mode. We analytically derive the wave function of the
topological edge state with zero energy of the qutrit-resonator chain, through
which a state transfer between two ends of the chain, accompanied with the
state flips of all intermediate qutrits, can be implemented by designing
qutrit-resonator coupling strengths. The key advantage that topology offers in
such processes is the inherent protection of boundary edge states lying in the
band gap of the dispersion relation when the bulk is topologically nontrivial.
Based on such a peculiar state transfer protected topologically, we show three
schemes of generating large-scale GHZ states, providing feasible and visible
methods to generate the robust large-scale GHZ states by meeting the
performance requirements of different experimental devices in the
superconducting qutrit-resonator chain. Meanwhile, we take into account the
experimental consideration, such as the implementation of the model,
preparation of the initial state, and realization of tunable couplings in the
circuit-QED chain. In addition, we discuss the potential of realizing fast GHZ
states by speeding up the adiabatic proscess, which provides more
possibilities for obtaining high-fidelity multi-body entanglement.

Our work may facilitate the potential applications of topological matter in
quantum information processing, due to the following advantages and interests.
First, the topological qutrit-resonator chain can be used to create a large-
scale GHZ state theoretically with the size being far more than $N=20$ that is
the particle number of multi-body entanglement realized experimentally up to
now Friis  _et al._ (2018); Zhong  _et al._ (2018); Wang  _et al._ (2018);
Gong  _et al._ (2019); Song  _et al._ (2019); Omran  _et al._ (2019b). Second,
as the core foundation of implementing large-scale GHZ states, we derive
theoretically the wave function of an edge state with zero-energy mode, whose
form involves the state flips of all intermediate qutrits, different from that
in the frequently studied standard Su-Schrieffer-Heeger (SSH) model Mei  _et
al._ (2018); Qi  _et al._ (2020a); Tan  _et al._ (2020); Zheng  _et al._
(2020). Finally, there are three schemes proposed for generating large-scale
GHZ states, which provide potential choices, depending on different device
requirements, i.e., coherence times of qutrits and resonators.

##  II Physical model and wave function of an edge state

Figure 1: (a) A superconducting qutrit-resonator chain. The $n$-th unit cell
contains one flux qutrit and one single-mode resonator, labeled as $A_{n}$ and
$B_{n}$, respectively, and holds an intra-cell qutrit-resonator coupling
strength $J_{1}$. Between two adjacent cells, a qutrit $A_{n+1}$ is coupled to
the resonator $B_{n}$ with an inter-cell coupling strength $J_{2}$. The
resonator $B_{n}$ drives the transition $|L\rangle\leftrightarrow|e\rangle$
($|R\rangle\leftrightarrow|e\rangle$) of the two nearest-neighbor qutrits
$A_{n}$ and $A_{n+1}$ when $n$ is odd (even). (b) Circuit schematic of one
unit cell in the superconducting qutrit-resonator chain. The coupling strength
can be dynamically tuned by a coupler of SQUID. The energy level space of the
qutrit can be tuned by FBL. (c) Schematics of energy level transitions for
qutrits $A_{1}$, $A_{n}$ ($1<n<N$), and $A_{N}$. The energy level structure of
a flux qutrit holds two ground states ($|L\rangle$ and $|R\rangle$) and one
excited state ($|e\rangle$). We denote that coupling strengths $J_{L}=J_{1}$
and $J_{R}=J_{2}$ ($J_{L}=J_{2}$ and $J_{R}=J_{1}$) when $n$ is even (odd).

###  II.1 Physical model

The setup of the superconducting qutrit-resonator chain for generating large-
scale GHZ states is illustrated in Fig. 1(a). The chain contains $(2N-1)$
lattice sites, $N$ qutrits and $(N-1)$ resonators. Each unit cell in the chain
contains one flux qutrit $A_{n}$ and one resonator $B_{n}$, whose circuit
schematic is described by Fig. 1(b). The energy levels of qutrit and the
coupling strengths are adjustable via the magnetic flux provided by the flux-
bias line (FBL) Schmidt and Koch (2013) and superconducting quantum
interference device (SQUID). As described in Fig. 1(c), each flux qutirt holds
a three-level structure, involving two ground states $|L\rangle$ and
$|R\rangle$, and one excited state $|e\rangle$. The interaction in the chain
can be described by the following interaction-picture Hamiltonian ($\hbar$ =
1)

$H_{I}=\sum_{n=1}^{N-1}\big{(}J_{1}|e\rangle_{n}\langle
j_{n}|+J_{2}|e\rangle_{n+1}\langle j_{n}|\big{)}b_{n}+\rm{H.c.},$ (2)

where $j_{n}=L~{}(R)$ when $n$ is odd (even), and $b_{{n}}$ the annihilation
operator of the resonator $B_{n}$. $J_{1}$ and $J_{2}$ can be tuned through
adopting controlled voltage pulses generated by an arbitrary waveform
generator (AWG) to tune the flux threading the loop DiCarlo  _et al._ (2009).
Further, $H_{I}$ can be rewritten as

$H=\sum_{n=1}^{N-1}J_{1}b_{n}\sigma_{n}^{+}(\sigma_{n}^{x})^{n}+J_{2}b_{n}\sigma_{n+1}^{+}(\sigma_{n+1}^{x})^{n}+\rm{H.c.},$
(3)

where we define $\sigma_{n}^{+}=|e\rangle_{n}\langle R|$,
$\sigma_{n}^{x}=|L\rangle_{n}\langle R|+|R\rangle_{n}\langle L|$. The
existence of the $(\sigma_{n}^{x})^{n}$ renders two different transitions
$|L\rangle\leftrightarrow|e\rangle$ and $|R\rangle\leftrightarrow|e\rangle$
for qutrit $A_{n}$ depending on the odevity of $n$. For instance,
$\sigma_{n}^{+}(\sigma_{n}^{x})^{n}=|e\rangle_{n}\langle
R|~{}(|e\rangle_{n}\langle L|)$ with $n$ being even (odd).

Such a adjustable chain can be analogous to an SSH model Su  _et al._ (1980);
Asbóth  _et al._ (2016); Su  _et al._ (1979) which describes quanta (e.g.,
electrons, photons, or phonons) hopping on a chain (one-dimensional lattice),
with staggered hopping amplitudes. The chain of SSH model consists of $N$ unit
cells, each unit cell hosting two sites, one on sublattice $A$, and the other
on sublattice $B$. The Hamiltonian of the standard SSH model is of the form
$H_{SSH}=(\nu\sum_{m=1}^{N}|m,B\rangle\langle
m,A|+\mu\sum_{m=1}^{N-1}|m+1,A\rangle\langle m,B|)+\rm{H.c.}$. Here
$|m,A\rangle$ and $|m,B\rangle$, with $m\in{1,2,\cdots N}$, denote the states
of the chain where the hopping quantum is on sites $A$ and $B$, respectively,
in the unit cell $m$ Su  _et al._ (1980); Asbóth  _et al._ (2016). The
Hamiltonian (3) holds a form of the SSH model whose topological phase is
characterized by winding number Ryu  _et al._ (2010), except for an additional
operator $(\sigma_{n}^{x})^{n}$ that works for flipping qutrit states and is
the key to realizing GHZ states. Based on the bulk-edge correspondence Qi and
Zhang (2011b); Hasan and Kane (2010); Chiu  _et al._ (2016b), the SSH model
possesses zero-energy edge modes at open boundaries in the topologically
nontrivial phase, which are protected by the topological properties of the
system.

###  II.2 Wave function of an edge state

The appearance of topologically protected gapless edge states within the bulk
gap is a manifestation of the topological insulator. The number of such
gapless edge modes is specified by topological invariants. As for an SSH
model, the zero energy mode, one of the characteristics of the topological
nontrivial SSH phase, is regarded as the topological invariant in the edge
state and will protect the edge state topologically from local disorders.
Thus, it is critical to obtain the wave function of the edge state with zero
energy mode in our model.

The edge states of the chain with a single excitation are exponentially
localized at the boundaries. The wave function of an edge state can be
described by the following ansatz, analogous to the standard SSH model Su  _et
al._ (1980); Asbóth  _et al._ (2016); Su  _et al._ (1979)

$|\varphi_{e}\rangle=\sum_{n=1}^{N}\lambda^{n}\Big{[}\gamma\sigma_{n}^{+}(\sigma_{n}^{x})^{n-1}\bigotimes^{n-1}_{l=1}\sigma_{l}^{x}+\eta
b_{{n}}^{\dagger}\bigotimes^{n}_{l=1}\sigma_{l}^{x}\Big{]}|G\rangle,$ (4)

where

$|G\rangle=|\underbrace{RLR\cdots}_{N}\rangle_{A_{N}}\otimes|\underbrace{000\cdots}_{N-1}\rangle_{B_{N-1}}$

denotes a decoupled state of the qutrit-resonator chain with all resonators in
$|0\rangle$ (i.e., zero-photon Fock state), while qutrits $A_{1}$, $A_{2}$,
$A_{3}$, $\cdots$ in states $|R\rangle$, $|L\rangle$, $|R\rangle$, $\cdots$,
repsectively. $\lambda$ is the localized index, $\gamma$ and $\eta$ being the
probability amplitudes of the gap states. When $\lambda\textless 1$
($\lambda\textgreater 1$), the probability amplitude of the site $n$ decays
(increases) exponentially with distance $n$ which means the left (right) edge
state with the wave function normalized. When $\gamma=0$ ($\eta=0$), according
to Eq. (4) the resonators (qutrits) are occupied by the edge state whose
eigenenergy is $E=0$. In particular, in order to generate large-scale GHZ
states with binary quantum information carried by two states in qutrits, we
choose $\gamma=1$ and $\eta=0$ to render the qutrit in each unit cell to
occupy the left ($\lambda<1$) and right ($\lambda>1$) edge states with $E=0$.
Thus, the wave function of an edge state occupied by qutrits can be written as
$|\varphi^{\prime}_{e}\rangle=\sum_{n=1}^{N}\lambda^{n}\sigma_{n}^{+}(\sigma_{n}^{x})^{n-1}\bigotimes^{n-1}_{l=1}\sigma_{{l}}^{x}|G\rangle$.
Substituting $|\varphi_{e}^{\prime}\rangle$ into the eigenvalue equation
$E|\varphi_{e}^{\prime}\rangle=H|\varphi_{e}^{\prime}\rangle$, one can obtain

$\displaystyle
E\sum_{n=1}^{N}\lambda^{n}\sigma_{n}^{+}(\sigma_{n}^{x})^{n-1}\bigotimes^{n-1}_{l=1}\sigma_{{l}}^{x}|G\rangle$
$\displaystyle=\Big{(}J_{1}\lambda^{n}b_{{n}}^{\dagger}\bigotimes_{l=1}^{n}\sigma_{{l}}^{x}+J_{2}\lambda^{n+1}b_{n}^{\dagger}\bigotimes_{l=1}^{n}\sigma_{{l}}^{x}\Big{)}|G\rangle.$
(5)

Through these analyses, the edge state can be worked out as

$|\varphi_{E}\rangle=\sum_{n=1}^{N}\lambda^{n}\sigma^{+}_{{n}}(\sigma_{n}^{x})^{n-1}\bigotimes^{n-1}_{l=1}\sigma_{{l}}^{x}|G\rangle,$
(6)

with $\lambda=-{J_{1}}/{J_{2}}$.

Figure 2: (a) Energy spectra of the chain versus $J_{1}/J_{2}$. (b)
Distribution of the zero energy mode versus $J_{1}/J_{2}$. (c) Energy spectra
of the chain versus $g_{0}t$. (d) Distribution of the zero energy mode versus
$g_{0}t$. The size of the chain is $L=2N-1=49$.

To illustrate the topological property of the qutrit-resonator chain more
clearly, we take the size $L\equiv 2N-1=49$ of the chain as an example to plot
the energy spectrum of the system and the distribution of the topological edge
states, respectively, in Figs. 2(a) and (b) with varying $J_{1}/J_{2}$. We
find that the energy spectrum possesses a unique zero energy mode keeping
unchanged with varying $J_{1}/J_{2}$, which denotes a non-evolutive state. The
closest distance between the zero energy state and the bulk appears at the
point $J_{1}/J_{2}=1$. In Fig. 2(b), the zero energy state is localized near
the left extremity when $J_{1}/J_{2}<1$, while for $J_{1}/J_{2}>1$ it is
localized near the right extremity. It means that one can achieve the
topological state transfer assisted by the zero energy mode between the first
and the last qutrits via varying the parameter $J_{1}/J_{2}$, which is
consistent with conclusion obtained from Eq. (6). Therefore, the Hamiltonian
$H$ is a modulated model if varying the coupling strength $J_{1}$ and $J_{2}$.
Specifically, the shapes of the coupling strengths are engineered as Gauss
functions

$\begin{split}J_{1}&=g_{0}\exp{[-(t-3\tau)^{2}/\tau^{2}]},\\\
J_{2}&=g_{0}\exp{[-(t-2\tau)^{2}/\tau^{2}]},\end{split}$ (7)

where $\tau=T/7$ is set as the value of the pulse width, as well as the delay
of $J_{1}$ and $J_{2}$, with $T$ being the evolution time. The forms of
$J_{1}$ and $J_{2}$ in Eq. (7) satisfy the state transfer conditions
$J_{1}/J_{2}|_{t\rightarrow 0}=0$ and $J_{1}/J_{2}|_{t\rightarrow T}=+\infty$
well. The idea of engineering the coupling strengths as Gauss functions is to
achieve a temporal soft quantum control starting from and ending at a zero
amplitude, which enables on-resonant couplings among a desired set of target
levels, while efficiently avoiding unwanted off-resonant contributions coming
from others Haase  _et al._ (2018); Wu  _et al._ (2020).

Based on time-dependent Gaussian coupling strengths, as shown in Fig. 2(c),
there exists a zero energy mode among all eigenstates during the whole
evolutionary process. Simultaneously, the topological edge state at zero
energy is well separated from bulk states. In Fig. 2(d), we plot the state
distribution of the zero mode. The qutrit-resonator chain has not only a bulk
part but also boundaries (which we refer to ends or edges). The qutrits
$A_{1}$ and $A_{N}$ are regarded as two edges, while the other qutrits and
resonators are the bulk part. The distribution of the zero mode state decays
exponentially on the qutrits under the condition of $g_{0}t<1800$.
Particularly, the eigenstate of the zero energy is localized near the first
lattice site $A_{1}$ (left edge) and the distribution of left edge state with
zero mode is equal to unity when $g_{0}t<1800$. When $g_{0}t$ increases
continuously, the distribution of zero mode increases exponentially on the
qutrits. Similarly, the eigenstate is localized near the last lattice site
$A_{25}$ (right edge) and the distribution of right edge state with zero mode
is equal to unity when $g_{0}t>2200$.

##  III Large-scale GHZ states

###  III.1 Scheme A for generating large-scale GHZ states

Now we focus on the generation of large-scale GHZ states among the $N$ qutrits
in the superconducting qutrit-resonator chain. In Fig. 2(d), the edge state
concentrates towards the left (right) end when $g_{0}t<1800$ ($g_{0}t>2200$).
In particular, when $g_{0}t\rightarrow 0$ and $g_{0}t\rightarrow+\infty$, the
edge states become

$\displaystyle|l\rangle$ $\displaystyle=$ $\displaystyle|eLR\cdots
m\rangle_{A_{N}}\otimes|000\cdots 0\rangle_{B_{N-1}},$
$\displaystyle|r\rangle$ $\displaystyle=$ $\displaystyle|LRL\cdots
e\rangle_{A_{N}}\otimes|000\cdots 0\rangle_{B_{N-1}},$ (8)

where $m=R~{}(L)$ when $N$ is odd (even). For generating GHZ states, the
superconducting qutrit-resonator chain is assumed initially in the state

$\displaystyle|\phi_{0}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(|G\rangle+|l\rangle\right).$ (9)

Going through the evolution, the first decoupled state component in the Eq.
(9) does not evolve because no photon in the resonators can be absorbed to
excite the ground-state qutrits. From the Fig. 2(d), we learn that the second
term of Eq. (9) is essentially the left edge state with zero energy, which can
evolve into the right edge state along the topologically protected process.
Thus, the following evolution occurs

$|\phi_{0}\rangle~{}\mapsto~{}\frac{1}{\sqrt{2}}\left[|G\rangle-(-1)^{N}|r\rangle\right].$

Here we set that $|R\rangle$ is used to carry the logical state 1 and
$|L\rangle$ is used to carry the logical state 0 for the qutrit $A_{n}$
($1\leq n<N$), while $|e\rangle$ of the last qutrit $A_{N}$ is encoded as the
logic state 1(0) when $N$ is even (odd). Except for the last qutrit $A_{N}$,
the excitation state $|e\rangle$ is an auxiliary state without carrying any
quantum information. Accordingly, the final state of $N$ qutrits after the
evolution with omitting the zero-photon product state of resonators is

$\frac{1}{\sqrt{2}}(|101\cdots 1(0)\rangle_{A_{N}}-(-1)^{N}|010\cdots
0(1)\rangle_{A_{N}}),$ (10)

which is exactly an $N$-body GHZ state according to Eq. (1).

As an example, with the full Hamiltonian Eq. (3) we take $N=25$ to numerically
plot in Fig. 3 the population evolution of the initial state, the ideal state
$|\phi_{ideal}\rangle=\frac{1}{\sqrt{2}}(|RLR\cdots
R\rangle_{A_{25}}+|LRL\cdots e\rangle_{A_{25}})\otimes|000\cdots
0\rangle_{B_{24}}$, and two edge states $|l_{\rm A}\rangle=|eLR\cdots
R\rangle_{A_{25}}\otimes|000\cdots 0\rangle_{B_{24}}$ and $|r_{\rm
A}\rangle=|LRL\cdots e\rangle_{A_{25}}\otimes|000\cdots 0\rangle_{B_{24}}$.
Evidently, the population of $|\phi_{ideal}\rangle$ evolves from $0.25$ to
$1$, which indicates the successful creation of a $25$-body GHZ state. When
$g_{0}t\approx 1800$, the populations of $|\phi_{ideal}\rangle$ and $|r_{\rm
A}\rangle$ suddenly increase, which shows the same result as the distribution
of the zero mode in Fig. 2(d).

Figure 3: Time evolution of populations for $|\phi_{0}\rangle$,
$|\phi_{ideal}\rangle$, $|l_{\rm A}\rangle$ and $|r_{\rm A}\rangle$ with the
size of the superconducting qutrit-resonator chain $L=49$. We choose
$g_{0}T=3600$ as a total evolution time. Figure 4: (a) Time evolution of
$\log_{10}(1-F)$ for generating $N$-body GHZ states with $N$ from 10 to 60 at
intervals of 5. (b) The fitting function and numerical scatters between the
number of qutrits and evolution time with $99.9\%$ fidelity. (c) The fidelity
of GHZ state against the coupling strength with disorder $\delta$
corresponding to 11, 18 and 25 qutrits. (d) The fidelity of GHZ state versus
the varying $\delta/g_{0}$ and $g_{0}t/10^{3}$ for $11$-body GHZ state.

###  III.2 Shortest time for generating high-fidelity $N$-body GHZ states and
robustness against disorder of coupling strengths

In the protocol above, the $N$-body GHZ state is achieved through an
excitation displacement of the zero-energy mode along the chain. The evolution
time $T$ must be chosen as long as possible in order to obtain an adiabatic
excitation displacement from one extremity to the other of the qutrit-
resonator chain. The adiabaticity condition can be written as
$\dot{\theta}\ll\Delta E$, where $\theta=\arctan(J_{1}/J_{2})$ and $\Delta E$
is the energy gap between the zero-energy mode and the bulk modes Mei  _et
al._ (2018); D’Angelis  _et al._ (2020). In the above numerical simulation,
the evolution time is appropriate but not the shortest time of generating a
high-fidelity ($0.999$) ideal GHZ state.

In order to get the information of the shortest time of generating $N$-body
GHZ states, we numerically calculate the fidelity of ideal $N$-body GHZ
states, and plot in Fig. 4(a) the time evolution of $\log_{10}(1-F)$ with $N$
from 10 to 60 at intervals of 5. The numerical results exhibit that a higher
fidelity needs a longer evolution time with increasing $N$. For example, the
evolution time for $10$-body GHZ state is $g_{0}t=661$ with $99.9\%$ fidelity.
However, $60$-body GHZ state with $99.9\%$ fidelity requires $g_{0}t=2.5\times
10^{4}$. In practice, the value of $g_{0}$ can be chosen about $2\pi\times$
$50$ MHz Mundada  _et al._ (2019). Thus, the evolution time of realizing
$60$-body GHZ state is $T\approx 76~{}\mu$s. Up to now, the superconducting
resonator lifetimes can be achieved between $1$ and $10$ ms Reagor  _et al._
(2013, 2016); Axline  _et al._ (2016). And the decoherence time of the
superconducting magnetic flux qubtrit achieved on $1$ ms by designing a
$\pi$-phase difference across the Josephson junction in circuit has been
reported Pop  _et al._ (2014), which is far more than the evolution time $T$.
As shown in Fig. 4(b), by selecting different $N$ and corresponding evolution
times of generating GHZ states with the fidelity of $99.9\%$ as numerical
samples, $g_{0}t$ versus $N$ can be fitted by a quadratic function
$g_{0}t=6.9419N^{2}+2.455N-59.8933$. The evolution time should be larger
enough to satisfy the adiabatic condition for a longer size of the chain.

Because of the existence of some circuit imperfections in a superconducting
qutrit-resonator chain such as the mutual inductance and self-inductance of
the circuit, and the unstable magnetic flux threading to the SQUID loop, which
may cause variation in ideal couplings. In order to study the robustness of
the scheme, we add a random disorder into coupling strength
$J^{\prime}_{1(2)}=J_{1(2)}(1+\rm{rand}~{}[-\delta,\delta])$ for each qutrit-
resonator coupling, where $\rm{rand}~{}[-\delta,\delta]$ denotes a random
number within the range of $[-\delta,\delta]$. Figure 4(c) shows the
relationship between the fidelity of GHZ state and the disorder
$\delta\in[0,0.5]$ for 11, 18 and 25 qutrits. Note that the disorder is
randomly sampled 101 times, and then the fidelity is taken as the mean value
of the 101 results. Corresponding to a shorter size of the superconducting
qutrit-resonator chain, the fidelity decreases more slowly and the GHZ state
is more robust to the disorder of coupling strength. On account of the process
of generating the GHZ state via the topological zero mode protected by the
energy gap, the width of the gap in a superconducting qutrit-resonator chain
usually exhibits an exponential decay behavior with the size of chain
increasing Chiu  _et al._ (2016a). Furthermore, the disorder has little effect
on the GHZ state with respect to $\delta/g_{0}<[0,0.2]$. Even if
$\delta/g_{0}\in[0.2,0.5]$, the fidelity still keeps over $0.993$, which
reflects the topological protection property of the system. As shown in Fig.
4(d), we plot the fidelity versus the varying evolution time and the disorder
of coupling strength $\delta$ for 11 qutrits. We can learn that the damage to
fidelity caused by coupling defects $\delta/g_{0}\in[0.2,0.5]$ can be
compensated by longer evolution time. Under the condition of
$\\{g_{0}t=830,\delta=0.5\\}$, the fidelity of 11-body GHZ state can be
achieved with fidelity over $0.995$. The robustness of $N$-body GHZ state to
the disorder and perturbation provides much more convenience for the
experimental realization and the practical application of multi-particle
entanglement state.

###  III.3 Influence of the losses in the superconducting qutrit-resonator
chain

We now give a discussion for the effect of qutrit-resonator losses on the
fidelity of generating GHZ states. Two dominant channels are considered in the
loss mechanism: (i) The loss of qutrits with decay rates $\gamma_{n}$; (ii)
The loss of resonators with decay rates $\kappa_{n}$.

The effect of losses during the evolution time can be evaluated by using a
conditional Hamiltonian Pachos and Walther (2002); Huang  _et al._ (2018)

$\displaystyle H_{cond}$ $\displaystyle=$ $\displaystyle
H-\frac{i\kappa_{n}}{2}\sum_{n=1}^{N-1}b^{\dagger}_{n}b_{n}-\frac{i\gamma_{n}}{2}\sum_{n=1}^{N}|e\rangle_{n}\langle
e|,$ (11)

where $H$ is the lossless Hamiltonian for the system in Eq. (3). The second
and third terms represent the losses of resonators and qutrits, respectively.
For convenience, we assume that $\kappa_{n}=\kappa$ and $\gamma_{n}=\gamma$.
We take $N=11$, 18, and 25 as examples. The fidelity is formulated as
$F=\langle\Psi_{ideal}|\rho|\Psi_{ideal}\rangle$, where
$|\Psi_{ideal}\rangle=1\ \sqrt{2}(|G\rangle-(-1)^{N}|r\rangle)$ is the output
state of an ideal system, and $\rho$ is the density operator of the system
dominated by the conditional Hamiltonian Eq. (11).

We now numerically simulate the fidelity of ideal GHZ states by solving the
non-Hermitian Liouville equation $\dot{\rho}=-i(H_{cond}\rho-\rho
H^{\dagger}_{cond})$. Figure 5 shows the relationship between the fidelity $F$
and decay of qutrits or resonators. As the decay rate of qutrits or resonators
increases, the fidelity of the ideal state shows a trend of decline. Comparing
those lines, the size of chain increases, the fidelity decreases slightly. And
the higher fidelities indicate that the Scheme A for generating $N$-body GHZ
states is insensitive to the value of $\kappa$. The fidelity of the 11-, 18-
or 25-body GHZ state plummets to $25\%$ roughly with $\gamma/g_{0}=0.01$.
Evidently, the value of $\gamma$ has a greater influence on the fidelity.
Owing to the distribution of edge states $|l_{\rm A}\rangle$ and $|r_{\rm
A}\rangle$ closed to 0.5 at end of the evolution time in Fig. 2(d), the loss
of qutrits in the excited state $|e\rangle$ in edge states during the whole
evolution time causes a significant destructive effect on the fidelity of GHZ
states. As shown in Fig. 5, we plot the fidelity of GHZ states without
considering the losses in left and right edge states, i.e, setting
$\gamma_{1}=\gamma_{n}=0$. Under the condition of $\gamma_{n}/g_{0}=0.01$ for
$1<n<N$, the fidelities of 11-, 18- and 25-body GHZ states are $55.24\%$,
$44.72\%$ and $33.92\%$, respectively, which indicates a notable improvement
of the fidelities than before. Consequently, how to suppress the loss of
excited state $|e\rangle$ of qutrits, especially of $A_{1}$ and $A_{N}$, is
the key to enhancing the fidelity of GHZ states. In the next section, we
propose two alternative schemes that are of improved robustness against losses
of qutrits.

Figure 5: Fidelities for GHZ states versus decay rates of qutrits or
resonators for different $N$, where $\gamma$ ($\kappa$) in the legend
represents decoherence involving only the qutrit (resonator) decay. The
evolution time $T$ is chosen as $g_{0}T=6.9416N^{2}+2.455N-59.8933$ that
ensures a $99.9\%$ fidelity for generating a lossless $N$-body GHZ state.
Figure 6: (a) and (b): Schematics of energy level transitions for qutirts
$A_{1}$ and $A_{N}$ in Scheme B. (c) and (d): Schematics of energy level
transitions for qutirts $A_{1}$ and $A_{N}$ in Scheme C. The interaction
diagram of other cells is the same as Fig. 2 in both Scheme A and B. The
choice of coupling strengths and driving fields of last qutrits $A_{N}$
depends on the odevity of $N$.

##  IV Alternative schemes

In the process of generating GHZ states, the distribution of excitations in
two edge states is 0.5 at the start and end of the evolution time (see Fig.
3). The losses of qutrits in the excited states have a great destructive
effect on the final fidelities of GHZ states. In this section, we propose two
improvement protocols, labeled Scheme B and Scheme C, to focus on suppressing
the loss of qutrits in the excited state.

###  IV.1 Scheme B for suppressing excitation of qutrits $A_{1}$ and $A_{N}$

The energy level space of a qutrit can be readily adjusted via changing the
external flux applied to the SQUID loop Neeley  _et al._ (2008); Yang  _et
al._ (2010); You and Nori (2005) or/and provided by an FBL Schmidt and Koch
(2013). As shown in Figs. 6(a) and (b), an extra long-lived state $|P\rangle$
is introduced into the right potential well of the qutrit $A_{1}$, upper than
and separated enough from the ground state $|R\rangle$. The transition between
$|e\rangle\leftrightarrow|L\rangle$ in qutrit $A_{1}$ is coupled off-
resonantly to the resonator $B_{1}$ with coupling strength $J_{1}^{\prime}$
and detuning $\Delta_{1}$. For the qutrit $A_{N}$ with $N$ being odd (even),
the transition between $|e\rangle\leftrightarrow|R\rangle~{}(|L\rangle$) is
coupled off-resonantly to resonator $B_{N-1}$ with the coupling strength
$J_{2}^{\prime}$ and detuning $\Delta_{2}$. In addition, an auxiliary
classical field is introduced to drive the transition off-resonantly between
$|e\rangle\leftrightarrow|P\rangle$ for qutrit $A_{1}$ with Rabi frequency
$\Omega_{1}$ and detuning $\Delta_{1}$. The other classical field with Rabi
frequency $\Omega_{2}$ and detuning $\Delta_{2}$ drives the transition of
$A_{N}$ off-resonantly, $|e\rangle\leftrightarrow|L\rangle~{}(|R\rangle$),
with $N$ being odd (even). The interaction Hamiltonians for the first and last
qutrits coupled to their adjacent resonators can be written as

$H_{1}=J_{1}^{\prime}b_{1}|e\rangle_{1}\langle
L|e^{i\Delta_{1}t}+\Omega_{1}|e\rangle_{1}\langle
P|e^{i\Delta_{1}t}+\rm{H.c.},$ (12a)
$H_{N}=J_{2}^{\prime}b_{N-1}|e\rangle_{N}\langle
m|e^{i\Delta_{2}t}+\Omega_{2}|e\rangle_{N}\langle
n|e^{i\Delta_{2}t}+\rm{H.c.},$ (12b)

with $m=R~{}(L)$ and $n=L~{}(R)$ when $N$ is odd (even), while the interaction
Hamiltonian for other cells is identical with Eq. (3).

For suppressing losses of qutrits in the excited state $|e\rangle$, the value
of $\Delta_{1(2)}$ is supposed to be as large as possible to satisfy the
condition $\Delta_{1(2)}\gg\\{J_{1(2)}^{\prime},\Omega_{1(2)}\\}$. According
to the theory of second-order perturbation James and Jerke (2007), one can
eliminate adiabatically the excited states of qutrits $A_{1}$ and $A_{N}$.
Besides, off-resonant-interaction-induced ground-state Stark shifts can be
offset by introducing auxiliary off-resonant fields Zhao  _et al._ (2017),
phase compensations Vepsäläinen  _et al._ (2019), or detuning compensations
Han  _et al._ (2020). Then, the Hamiltonians in Eqs. (12a) and (12b) can be
reduced into, respectively

$H_{1\rm{eff}}=J_{1\rm{eff}}(|P\rangle_{1}\langle L|+|L\rangle_{1}\langle
P|),$ (13a) $H_{N\rm{eff}}=J_{2\rm{eff}}(|R\rangle_{N}\langle
L|+|L\rangle_{N}\langle R|),$ (13b)

with $J_{1(2)\rm{eff}}=J_{1(2)}^{\prime}\Omega_{1(2)}/\Delta_{1(2)}$, which
involve solely long-lived states. When the chain with the size $L=2N-1=49$ is
initially in the left edge state $|l_{\rm
B}\rangle=|\psi_{1}\rangle=|PLR\cdots LR\rangle_{A_{25}}\otimes|000\cdots
00\rangle_{B_{24}}$ under the interaction Hamiltonians (12a) and (12b), the
system evolves in the finite space $\\{|\psi_{n}\rangle\\}$

$\begin{split}&|\psi_{1}\rangle=|PLR\cdots LR\rangle_{A_{25}}\otimes|000\cdots
0\rangle_{B_{24}},\\\ &|\psi_{2}\rangle=|eLR\cdots
LR\rangle_{A_{25}}\otimes|000\cdots 0\rangle_{B_{24}},\\\
&|\psi_{3}\rangle=|LLR\cdots LR\rangle_{A_{25}}\otimes|100\cdots
0\rangle_{B_{24}},\\\ &|\psi_{4}\rangle=|LeR\cdots
LR\rangle_{A_{25}}\otimes|000\cdots 0\rangle_{B_{24}},\\\ &\quad\vdots\\\
&|\psi_{49}\rangle=|LRL\cdots RR\rangle_{A_{25}}\otimes|000\cdots
1\rangle_{B_{24}},\\\ &|\psi_{50}\rangle=|LRL\cdots
Re\rangle_{A_{25}}\otimes|000\cdots 0\rangle_{B_{24}},\\\
&|\psi_{51}\rangle=|LRL\cdots RL\rangle_{A_{25}}\otimes|000\cdots
0\rangle_{B_{24}}.\end{split}$ (14)

To further evaluate the topological properties of zero energy states in the
Scheme B, we plot the distribution of the zero mode on component states
$|\psi_{n}\rangle$, as shown in Fig. 7(a). The components $|\psi_{1}\rangle$
and $|\psi_{51}\rangle$, which play roles of left and right edge states,
$|l_{\rm B}\rangle$ and $|r_{\rm B}\rangle$, respectively, are populated with
the maximal distributions in the regions of $g_{0}t\in[0,1800)$ and
$g_{0}t\in(2200,4000]$. In comparison between Figs. 2(d) and 7(a), the
systematic degree of freedom in the evolution subspace changes from $2N-1$ to
$2N+1$ due to the introductions of classical drives on the two extremity
qutrits of the chain. In the Scheme B, an extra long-lived state $|P\rangle$
is introduced in the qutrit $A_{1}$, and the large detuning condition is
satisfied so as to eliminate adiabatically the excited state $|e\rangle$. In
other words, the state transformations of Scheme A and Scheme B are,
respectively

$\displaystyle|l_{\rm A}\rangle=|eLR\cdots R\rangle_{A_{25}}\otimes|000\cdots
0\rangle_{B_{24}}$ $\displaystyle\Downarrow$ $\displaystyle|r_{\rm
A}\rangle=|LRL\cdots e\rangle_{A_{25}}\otimes|000\cdots 0\rangle_{B_{24}},$

and

$\displaystyle|l_{\rm B}\rangle=|PLR\cdots LR\rangle_{A_{25}}\otimes|000\cdots
0\rangle_{B_{24}}$ $\displaystyle\Downarrow$ $\displaystyle|r_{\rm
B}\rangle=|LRL\cdots RL\rangle_{A_{25}}\otimes|000\cdots 0\rangle_{B_{24}}.$

Thus, the distribution and spacing of bright and dark fringes in Fig. 7(a) are
different from that in Fig. 2(d). The dark fringes in Fig. 7(a) appearing on
$|\psi_{2}\rangle$ and $|\psi_{50}\rangle$ indicate that the population
elimination of the excited state $|e\rangle$ in qutrits $A_{1}$ and $A_{25}$.

Figure 7: Distributions of the zero energy mode for (a) Scheme B and (b)
Scheme C, respectively, on component states $|\psi_{n}\rangle$ and
$|\phi_{n}\rangle$. The size of the superconducting qutrit-resonator chain is
$L=2N-1=49$.

For sake of generating large-scale GHZ states, the superconducting qutrit-
resonator chain is assumed initially in the state

$|\Phi^{\prime}_{0}\rangle=\frac{1}{\sqrt{2}}\Big{(}|G\rangle+|l^{\prime}_{\rm
B}\rangle\Big{)}.$ (15)

where $|l^{\prime}_{B}\rangle=|PLR\cdots m\rangle_{A_{N}}\otimes|000\cdots
0\rangle_{B_{N-1}}$ and $m=R~{}(L)$ when $N$ is odd (even). After the
evolution along the topologically protected zero-energy mode, similar to the
Scheme A, one can obtain the final state

$|\Phi^{\prime}_{ideal}\rangle=\frac{1}{\sqrt{2}}\Big{(}|G\rangle-(-1)^{N}|r^{\prime}_{B}\rangle\Big{)}.$
(16)

in which $|r^{\prime}_{B}\rangle=|LRL\cdots n\rangle_{A_{N}}\otimes|000\cdots
0\rangle_{B_{N-1}}$ with $n=L~{}(R)$ when $N$ is odd (even). Here $|R\rangle$
is used to carry the logical state 1 while $|L\rangle$ carries the logical
state 0 for all qutrits. Accordingly, the initial state without considering
the zero-photon resonators after the evolution becomes

$\frac{1}{\sqrt{2}}\left(|101\cdots 1(0)\rangle_{A_{N}}-(-1)^{N}|010\cdots
0(1)\rangle_{A_{N}}\right).$ (17)

which is exactly an $N$-body GHZ state.

In Fig. 8(a), with the same parameters as in Fig. 3 we numerically plot the
time evolution of populations for the ideal state
$|\Phi_{ideal}\rangle=\frac{1}{\sqrt{2}}(|RLR\cdots
R\rangle_{A_{25}}+|LRL\cdots L\rangle_{A_{25}})\otimes|000\cdots
0\rangle_{B_{24}}$, the initial state
$|\Phi_{0}\rangle=\frac{1}{\sqrt{2}}(|RLR\cdots R\rangle_{A_{25}}+|PLR\cdots
R\rangle_{A_{25}})\otimes|000\cdots 0\rangle_{B_{24}}$, and two edge states
$|l_{\rm B}\rangle$ and $|r_{\rm B}\rangle.$ Obviously, the population of
$|\Phi_{ideal}\rangle$ ($|\Phi_{0}\rangle$) reaches nearly 1 (0.25) and keeps
steady at the end of evolution time, which proves the feasibility of Scheme B.
The populations of two edge states have the identical trend with Fig. 3.

In order to verify the effectiveness of suppressing the losses of qutrits in
the excited state, we simulate the fidelity of the ideal GHZ state
$|\Phi_{ideal}\rangle$ by solving the non-Hermitian Liouville equation. In
Fig. 8(b), the fidelities for 11-, 18-, and 25-body GHZ states hold on
$98.3\%$, $95.8\%$, and $93.2\%$ even though $\kappa/g_{0}=0.05$. In addition,
it is evident that the fidelity in Fig. 8(b) can reach values similar to that
without taking account of edge states in Fig. 5. When $\gamma/g_{0}=0.01$, the
fidelities of 11-, 18-, and 25-body GHZ states maintain $53.2\%$, $45.2\%$,
and $33.6\%$, respectively. With regard to $\gamma$, the fidelity of the ideal
state decreases more slightly and displays a more robust result than Scheme A
because of the suppression of excitation populations in qutrits $A_{1}$ and
$A_{N}$ during the whole evolution.

Figure 8: (a) and (c) for Scheme B and Scheme C, respectively: Time evolution
of population for $|\Phi_{0}\rangle$, $|\Phi_{ideal}\rangle$,
$|\varphi_{0}\rangle$, $|\varphi_{ideal}\rangle$, edge states $|l_{\rm
B(C)}\rangle$ and $|r_{B(C)}\rangle$ for the size of superconducting qutrit-
resonator chain $L=2N-1=49$. (b) and (d) for Scheme B and Scheme C,
respectively: Fidelity for the GHZ states versus the decay rate $\gamma$ or
$\kappa$ of the superconducting qutrit-resonator chain for different number of
qutrits, where $\gamma$ ($\kappa$) in the legend represents decoherence
involving only the qutrit (resonator) decay. Parameters for (a) and (b):
$J^{\prime}_{1(2)}=20J_{1(2)}$, $\Delta_{12}=400g_{0}$, and
$\Omega_{1(2)}=20g_{0}$.

###  IV.2 Scheme C for suppressing excitation of all qutrits

As shown in Figs. 6(c) and (d), for the qutrit $A_{1}$, the transition between
$|e\rangle\leftrightarrow|L\rangle$ is coupled resonantly to the resonator
$B_{1}$ with coupling strength $J_{1}$. When $N$ is odd (even), the transition
between $|e\rangle\leftrightarrow|R\rangle~{}(|L\rangle$) in the last qutrit
$A_{N}$ is coupled resonantly to the resonator $B_{N-1}$ with the coupling
strength $J_{2}$. Also, one classical field drives the transition resonantly
between $|e\rangle\leftrightarrow|P\rangle$ for qutrit $A_{1}$ with Rabi
frequency $\tilde{\Omega}_{1}$. Under the condition of $N$ being odd (even),
the other classical field with Rabi frequency $\tilde{\Omega}_{2}$ drives
resonantly the transition $|e\rangle\leftrightarrow|L\rangle~{}(|R\rangle)$ in
qutrit $A_{N}$. Therefore, the interaction Hamiltonian involving the first and
last qutrits can be written as

$\begin{split}H_{1^{\prime}}&=J_{1}b_{1}|e\rangle_{1}\langle
L|+\tilde{\Omega}_{1}|e\rangle_{1}\langle P|+\rm{H.c.},\\\
H_{N^{\prime}}&=J_{2}b_{N-1}|e\rangle_{N}\langle
m|+\tilde{\Omega}_{2}|e\rangle_{N}\langle n|+\rm{H.c.},\end{split}$ (18)

where $m=R~{}(L)$ and $n=L~{}(R)$ when $N$ is odd (even). And the interaction
Hamiltonian of other cells still keeps consistent with Eq. (3). The shapes of
the coupling strengths are engineered as Gauss functions

$\begin{split}\tilde{\Omega}_{1}=J_{2}=g_{0}\exp{[-(t-3\tau)^{2}/\tau^{2}]},\\\
\tilde{\Omega}_{2}=J_{1}=g_{0}\exp{[-(t-2\tau)^{2}/\tau^{2}]},\end{split}$

where $J_{1}$ and $J_{2}$ are reverse with respect to Eq. (7).

We take $N=25$ as an example, and when the chain is initially in the left edge
state $|l_{\rm C}\rangle=|\phi_{1}\rangle=|PLR\cdots
R\rangle_{A_{25}}\otimes|000\cdots 0\rangle_{B_{24}}$. The system evolves in
the finite subspace $\\{|\phi_{n}\rangle\\}$

$\begin{split}&|\phi_{1}\rangle=|PLR\cdots R\rangle_{A_{25}}\otimes|000\cdots
0\rangle_{B_{24}},\\\ &|\phi_{2}\rangle=|eLR\cdots
R\rangle_{A_{25}}\otimes|000\cdots 0\rangle_{B_{24}},\\\
&|\phi_{3}\rangle=|LLR\cdots R\rangle_{A_{25}}\otimes|100\cdots
0\rangle_{B_{24}},\\\ &|\phi_{4}\rangle=|LeR\cdots
R\rangle_{A_{25}}\otimes|000\cdots 0\rangle_{B_{24}},\\\ &\quad\vdots\\\
&|\phi_{49}\rangle=|LRL\cdots R\rangle_{A_{25}}\otimes|000\cdots
1\rangle_{B_{24}},\\\ &|\phi_{50}\rangle=|LRL\cdots
e\rangle_{A_{25}}\otimes|000\cdots 0\rangle_{B_{24}},\\\
&|\phi_{51}\rangle=|LRL\cdots R\rangle_{A_{25}}\otimes|000\cdots
0\rangle_{B_{24}}.\end{split}$ (19)

Then we plot the distribution of the zero energy mode on component states
$|\phi_{n}\rangle$ in Fig. 7(b). Distinctly, both the two ends of evolutionary
states $|\phi_{1}\rangle$ and $|\phi_{51}\rangle$ are populated with the
maximal distributions in the regions of $g_{0}t\in[0,1800)$ and
$g_{0}t\in(2200,4000]$, respectively. In comparison between Figs. 2(d) and
7(b), the system degree of freedom in the evolution subspace changes from
$2N-1$ to $2N+1$ due to the introductions of classical drives on the two
extremity qutrits of the chain. In Fig. 7, the distribution and spacing of
bright and dark fringes are different from that of Scheme A or Scheme B. The
bright fringes in Scheme A or Scheme B are distributed in the excitations of
qutrits, while in Scheme C they are distributed in the excitations of
resonators. Therefore, in Scheme C excitations of all qutrits are suppressed.

So as to obtain a large-scale GHZ states, the initial state of the chain is
prepared in Eq. (15). The first state component $|G\rangle$ does not evolve,
while the second state component $|l^{\prime}_{B}\rangle$ evolves into
$(-1)^{N}|r^{\prime}_{B}\rangle$. Hence, the final state is

$\displaystyle\begin{split}|\Psi_{F}\rangle=&\frac{1}{\sqrt{2}}\left(|G\rangle+(-1)^{N}|r^{\prime}_{B}\rangle\right).\end{split}$
(20)

which has a minus sign difference from Eq. (16). In Fig. 8(c), we plot the
time evolution of populations for the initial state
$|\varphi_{0}\rangle=|\Phi_{0}\rangle$ , the ideal state
$|\varphi_{ideal}\rangle=\frac{1}{\sqrt{2}}(|RLR\cdots
R\rangle_{A_{25}}-|LRL\cdots L\rangle_{A_{25}})\otimes|000\cdots
0\rangle_{B_{24}}$, two edge states $|l_{C}\rangle=|l^{\prime}_{B}\rangle$ and
$|r_{\rm C}\rangle=|r^{\prime}_{B}\rangle$ for generating a 25-body GHZ state.
As expected, the population of $|\varphi_{ideal}\rangle$
($|\varphi_{0}\rangle$) is close to unity (0.25) at the time $g_{0}t=3600$ and
remains stable. As for the two edge states, the population in Scheme C has the
same climate as that in Scheme A and Scheme B. The result reveals that the
above theoretical analysis is correct and feasible. Figure 8(d) shows the
relationship between the fidelity and the decay rate $\gamma$ or $\kappa$. As
the decay rates increase, the fidelity of the ideal GHZ state decreases. As
opposed to the loss mechanisms of Scheme A and Scheme B, the decay rate
$\kappa$ of resonators has a greater influence on the fidelity. When
$\kappa/g_{0}=0.01$, the fidelity is reduced to $56.1\%$ for the 11-body GHZ
state. In contrast among the six lines, the fidelity is relatively immune to
the decay of qutrits. The fidelity can retain at $95.9\%$, $95.0\%$, and
$93.6\%$ with $\gamma/g_{0}=0.05$, which indicates that Scheme C displays the
most robust performance to resist the losses of qutrits in the superconducting
qutrit-resonator chain among the three schemes.

In Scheme C, the fidelity of the ideal large-scale GHZ state is affected least
by the losses of qutrits, compared with Scheme A and Scheme B. However, the
cost is that the losses of resonators have a greatest influence on the
fidelity, which indicates that the Scheme C requires resonators to hold a long
coherence time. While according to the interaction of Scheme A and Scheme B,
it is required that the qutrits have a long coherence time, but without a
strict requirement on the quality factor of resonators. Thus, corresponding to
the performance of experimental devices in the superconducting qutrit-
resonator chain, we can choose different schemes to realize large-scale GHZ
states with high fidelity in the case of loss mechanisms by adjusting the
energy level structure of qutrits and coupling strengths.

Figure 9: Fidelity of the GHZ state with the scale of entanglement, $N$, under
the feasible coupling strengths (e.g., $g_{0}/2\pi=50$ MHz, 10 MHz and 1 MHz)
and coherence times of qutrits and resonators in Scheme A. We choose coherence
time of the qutrits and resonators as $\tau_{a}=\tau_{b}=1$ ms.

##  V Scale of GHZ states

In experiment, by designing a $\pi$-phase difference across the Josephson
junction in circuit to restrain the energy relaxation induced by quasiparticle
dissipation, one can obtain a qutrit with coherence time over 1 ms Pop  _et
al._ (2014). As for a resonator, the coherence time of the photons in the
resonator can be much longer Gu  _et al._ (2017). Up to now, the
superconducting resonator lifetimes between 1 ms and 10 ms have been reported
Reagor  _et al._ (2013, 2016); Axline  _et al._ (2016). Generally, the typical
feasible coupling strength can be modulated in the range of 1 MHz to 50 MHz
Mundada  _et al._ (2019), providing a considerable adjustability in
experiment.

In this section, we discuss an accessible GHZ state scale $N$ by taking into
account the experimentally available coherence times of qutrits and resonators
under the condition of feasible coupling strengths. For convenience, we take
Scheme A as an example. Figure 9 shows the relationship between fidelities of
GHZ states and the scale of entanglement $N$ with feasible $g_{0}$ and
coherence times of qutrits ($\tau_{a}=1/\gamma=1$ ms) and resonators
$\tau_{b}=1/\kappa=1$ ms). Obviously, the fidelity exhibits the decreasing
tendency with increasing $N$. Regarding to feasible coupling strengths, the
fidelity declines at different rates. Under the condition of $g_{0}/2\pi=50$
MHz, fidelities of 10- to 50-body GHZ states keep above $90\%$. Even if the
scale of a GHZ state is $N=150$, its fidelity would still reach $50.60\%$. As
for $g_{0}/2\pi=10$ MHz, the fidelity of 67-body GHZ state stays above $50\%$.
The fidelity can be over $90\%$ with $N$ less than 23. However, the GHZ state
fidelity has the fastest rate of decline in the case of $g_{0}/2\pi=1$ MHz,
where the fidelity of a GHZ state can achieve $50\%$ with the maximum
entanglement scale $N=20$. Under the existing experimental conditions, the
higher fidelity of large-scale GHZ states can be generated with the higher
value of coupling strengths.

##  VI Experimental consideration and prospective improvement

###  VI.1 Device and initial state

Benefiting from the rapid development in circuit-QED technologies, the
circuit-QED system provides us an excellent experimental platform to realize
large-scale GHZ states proposed in our work. We can construct a circuit-QED
system via arranging the transmission line resonator and the superconducting
qutrits in the space, as shown in Fig. 1(b). According to the existing
circuit-QED technology, the qubit-resonator chain systems with on the order of
10–20 qubits have been demonstrated Kelly  _et al._ (2015); Kandala  _et al._
(2017); Neill  _et al._ (2018); Otterbach  _et al._ (2017); Wei  _et al._
(2020); Mooney  _et al._ (2019). Simultaneously, a chain of 72 superconducting
resonators coupled via transmons can be realized Fitzpatrick  _et al._ (2017).
The scale of qutrit-resonator chain manufacturing in a metal chip is generally
from micron to millimeter DiCarlo  _et al._ (2009); Clerk  _et al._ (2020);
Niemczyk  _et al._ (2010); Blais  _et al._ (2020). In general, the flux qutrit
is a superconducting circuit made up of Josephson junctions or/and capacitance
Manucharyan  _et al._ (2009). While the flux qutrit can be operated at any
applied external flux through the flux qutrit loop. The resonator is composed
by a linear inductance in parallel with a capacitance, which can be fabricated
from a NbN film deposited on a sapphire substrate Mahashabde  _et al._ (2020).
The coupler is replaced by two Josephson junctions with a SQUID loop to
realize tunable coupling: the magnetic flux that threads this loop determines
its effective inductance $E_{J}(\phi)\sim E_{J}(0)\cos(2\pi\phi/\Phi_{0})$
García-Ripoll  _et al._ (2020). Josephson junctions can be designed by the
metallic chip (Al/AlOx/Al) using the electron-beam lithography Lecocq  _et
al._ (2011); Niemczyk  _et al._ (2010). In experiment, the circuit-QED device
is usually operated in a single-shot liquid 3He with a base millikelvin
temperatures and heavily filtered cryogenic microwave lines DiCarlo  _et al._
(2009); Blais  _et al._ (2020).

In the above schemes, specified initial states are needed to prepare in Eqs.
(9) and (15) to realize $N$-body GHZ states. These specified initial states
denote $A_{1}$ in the superposition $(|R\rangle+|e(P)\rangle)/\sqrt{2}$,
$B_{1}$ in $|0\rangle$, $A_{2}$ in $|L\rangle$, $\cdots$, $A_{N}$ in
$|R\rangle$ ($|L\rangle$) when $N$ is odd (even). In experiment, the
preparation of the initial states can be conducted through two steps: (i) As
reported recently in Ref. Vool  _et al._ (2018), the qutrit holding a
$\Lambda$-type structure can be cooled to one of its two lowest energy
eigenstates ($|R\rangle$ or $|L\rangle$) by resonator decay through
spontaneous Raman scattering so as to obtain the state $|RLR\cdots
R(L)\rangle_{A_{N}}$. (ii) A $\pi/2$ pulse is applied to the qutrit $A_{1}$ to
create a superposition state of $(|R\rangle+|e(P)\rangle)/\sqrt{2}$ Wang  _et
al._ (2019b). The resonator keeps in the zero photon Fock state. Finally, the
qutrit-resonator chain can be prepared in $1/\sqrt{2}(|RLR\cdots
R(L)\rangle_{A_{N}}+|P(e)LR\cdots R(L)\rangle_{A_{N}})\otimes|000\cdots
0\rangle_{B_{N-1}}$.

###  VI.2 Tunable couplings

In order to achieve high-fidelity GHZ states, tunable couplings among qutrits
and resonators are required. Tunable couplers of both varieties have been
realized in several experiments: between two tunable qubits Chen  _et al._
(2014); Harrabi  _et al._ (2009), between a qubit and a resonator Allman  _et
al._ (2014); Srinivasan  _et al._ (2011), and between resonators Wang  _et
al._ (2011); Zakka-Bajjani  _et al._ (2011). Particularly, a direct tunable
coupler is realized by a tunable circuit element between the qutrit and the
resonator, e.g., a flux-biased direct-current SQUID to generate strong
resonant and nonresonant tunable interactions between a qubit and a lumped-
element resonator Allman  _et al._ (2014).

In this work, we adopt a direct tunable coupler replaced by SQUID between
qutrits and resonators. The coupling strengths $J_{1}$ and $J_{2}$ can be
tunable through adopting controlled voltage pulses generated by an AWG to tune
the flux threading the SQUID loop DiCarlo  _et al._ (2009). For example, the
resonator $B_{N}$ is coupled resonantly to transitions
$|e\rangle\leftrightarrow|m\rangle$ ($m\in\\{R,L\\}$). The coupling strength
$J_{1(2)}$ can be expressed by Yang  _et al._ (2003, 2004b)

$J_{1(2)}=\frac{1}{L}\sqrt{\frac{\omega_{B_{N}}}{2\mu_{0}\hbar}}\langle
e|\Phi|m\rangle\int_{S}\textbf{B}_{B_{N}}(\vec{r},t)\cdot d\textbf{S},$ (21)

where S is the surface bounded by the loop of SQUID, $\omega_{B_{N}}$ the
resonator frequencies of $B_{N}$. Accordingly, $\textbf{B}_{B_{N}}(\vec{r},t)$
is the the magnetic components of the $B_{N}$ resonator mode. For a
standingwave resonators,
$\textbf{B}_{B_{N}}(\vec{r},t)=\mu_{0}\sqrt{2/V_{N}}\cos k_{N}z_{N}$ ($k_{N}$
is the wave number of $B_{N}$ resonator, $V_{N}$ and $z_{N}$ are the $B_{N}$
resonator volume and the $B_{N}$ resonator anxis). In this case, a modulating
field can be added when a qutrit works on its optimal frequency point Reagor
_et al._ (2018), and thus does not cause the shortening of qutrit coherent
times.

###  VI.3 Accelerating adiabatic process

Over the past decade, techniques of shortcuts to adiabaticity (STA) Chen  _et
al._ (2010); Guéry-Odelin  _et al._ (2019) receive a lot of attention, because
STA can accelerate adiabatic processes but remain the robustness of adiabatic
processes. Recently, a fast quantum state transfer from the left edge state to
the right edge state in a topological SSH chain with next-to-nearest-neighbor
(NNN) interaction was presented, which provides the simplest one-dimensional
lattice with protected edge state D’Angelis  _et al._ (2020). The idea of this
approach is based on an engineering of NNN interactions between the sites of
the chain, which exactly cancels nonadiabatic couplings due to an imperfect
condition of adiabatic evolution. Inspired by counterdiabatic driving methods
Berry (2009); Demirplak and Rice (2003), the quantum chain without limitation
of the adiabaticity constraint, combined with a dynamical control of NNN
interactions, is thus governed by the following Hamiltonian $H(t)$:

$\displaystyle H(t)$ $\displaystyle=$ $\displaystyle H_{0}(t)+H_{c}(t),$
$\displaystyle H_{0}(t)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N-1}t_{2}(t)|B_{n}\rangle\langle
A_{n}|+t_{1}(t)|A_{n+1}\rangle\langle B_{n}|+\rm{H.c.},$ $\displaystyle
H_{c}(t)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N-1}i\alpha_{n}(t)|A_{n+1}\rangle\langle
A_{n}|+\rm{H.c.},$ (22)

where $H_{c}(t)$ is the control Hamiltonian literally canceling the
nonadiabatic couplings. The time-dependent NNN hoppings $\alpha_{n}(t)$ is
only to cancel nonadiabatic transitions from the time-dependent eigenvector
$|\phi_{0}(t)\rangle=\mathcal{N}\sum^{N}_{n=1}(\frac{-t_{2}}{t_{1}})^{n-1}|A_{n}\rangle$,
where $\mathcal{N}$ is the normalization constant.

Analogously, with the help of such a method for speeding up the adiabatic
state transfer in the SSH model, it is possible to find a control Hamiltonian,
added to the initial Hamiltonian, that literally cancels the nonadiabatic
couplings in the topological model. Therefore, it is of great potential to
realize fast preparation of a large-scale GHZ state by engineering the NNN
interaction between qutrits, which may constrain the impact of systematic
decoherence and thus enhance the fidelity of generating large-scale GHZ
states.

In addition, an alternative method, Floquet-engineering STA (FESTA) Petiziol
_et al._ (2018); Claeys  _et al._ (2019), may also be applied in speeding up
the generation of a large-scale GHZ state. FESTA is an effective succedaneum
of counterdiabatic driving methods, which does not require an additional
control Hamiltonian but is combined with a periodic driving component to
oscillate the initial Hamiltonian Petiziol  _et al._ (2018). This Floquet-
engineering periodic driving component will form an effective connection
between transferred two states so as to offset exactly nonadiabatic couplings
Petiziol  _et al._ (2020). FESTA has been demonstrated for quantum state
transfer in spin chains in a recent experiment Zhou  _et al._ (2019).
Therefore, it is of great potential to use FESTA in our model for accelerating
the adiabatic generation of GHZ states by replacing the adiabatic pulses with
the Floquet-engineering oscillating pulses.

##  VII Conclusion

In conclusion, we have proposed a model of a superconducting qutrit-resonator
chain, and the topological edge state with zero energy is analytically
derived. Along this topological zero-energy mode, a state transfer from one
extremity qutrit to the other of the chain can be implemented, accompanied
with state flips of all intermediate qutrits. Three schemes are shown for
generating large-scale GHZ states that are protected by the topological zero-
energy mode and thus hold great robustness against disorder of the qutrit-
resonator coupling strengths. The effect of losses induced by decay of qutrits
and resonators on entanglement fidelity is investigated, and the results
indicate that the three schemes meet different performance requirements of
experimental devices. Furthermore, we study the accessible entanglement scale
when taking into account the experimentally available coherence times of
qutrits and resonators, and find that with the maximum of the qutrit-resonator
coupling strengths $g_{0}/2\pi=50$ MHz, it is possible to achieve a $50$-body
GHZ state with the fidelity $F>0.9$, and a $150$-body GHZ state with $F>0.5$.
Finally, we discuss the experimental consideration of generating GHZ states,
including the physical implementation of the model, preparation of the initial
states, and tunable couplings in circuit-QED chain, and also show the
potential to accelerate adiabatic process of generating GHZ states. The
present work is expected to be helpful for promoting the experimental
improvement of large-scale GHZ states.

## ACKNOWLEDGEMENTS

This work was supported by National Natural Science Foundation of China (NSFC)
(Grant No.11675046), Program for Innovation Research of Science in Harbin
Institute of Technology (Grant No. A201412), and Postdoctoral Scientific
Research Developmental Fund of Heilongjiang Province (Grant No. LBH-Q15060).

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