# Arbitrary entangled state transfer via a topological qubit chain

Chong Wang  School of integrated circuits, Tsinghua University, Beijing,
100084, China  Frontier Science Center for Quantum Information, Beijing, China
Linhu Li  Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing
& School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus),
Zhuhai 519082, China  Jiangbin Gong<EMAIL_ADDRESS>Department of Physics,
National University of Singapore, Singapore 117542, Singapore  Yu-xi Liu
<EMAIL_ADDRESS>School of integrated circuits, Tsinghua
University, Beijing, 100084, China  Frontier Science Center for Quantum
Information, Beijing, China

(August 27, 2024)

###### Abstract

Quantum state transfer is one of the basic tasks in quantum information
processing. We here propose a theoretical approach to realize arbitrary
entangled state transfer through a qubit chain, which is a class of extended
Su-Schrieffer-Heeger models and accommodates multiple topological edge states
separated from the bulk states. We show that an arbitrary entangled state,
from $2$-qubit to $\mathcal{N}$-qubit, can be encoded in the corresponding
edge states, and then adiabatically transferred from one end to the other of
the chain. The dynamical phase differences resulting from the time evolutions
of different edge states can be eliminated by properly choosing evolution
time. Our approach is robust against both the qubit-qubit coupling disorder
and the evolution time disorder. For the concreteness of discussions, we
assume that such a chain is constructed by an experimentally feasible
superconducting qubit system, meanwhile, our proposal can also be applied to
other systems.

##  I introduction

There are many possible platforms to realize quantum computation (Steane,
1998; Preskill, 2018; Bennett and DiVincenzo, 2000), e.g., cold atoms (Weiss
and Saffman, 2017; Negretti  _et al._ , 2011; Briegel  _et al._ , 2000;
García-Ripoll  _et al._ , 2005), trapped ions (García-Ripoll  _et al._ , 2005;
Häffner  _et al._ , 2008; Benhelm  _et al._ , 2008; Bruzewicz  _et al._ ,
2019; Cirac and Zoller, 2000; Lanyon  _et al._ , 2013; García-Ripoll  _et al._
, 2003; Pachos and Walther, 2002; Feng, 2002), and superconducting quantum
circuits (Jeffrey  _et al._ , 2014; Gambetta  _et al._ , 2017; Brecht  _et
al._ , 2016; Devoret  _et al._ , 2004; You and Nori, 2005; Yan  _et al._ ,
2019; Song  _et al._ , 2017; Neill  _et al._ , 2018; Huang  _et al._ , 2020;
Krantz  _et al._ , 2019; You and Nori, 2003; Osborn  _et al._ , 2007; Pechal
_et al._ , 2018). On these platforms, quantum state transfer (QST) in a
controllable way (Bennett and DiVincenzo, 2000; Bouwmeester and Zeilinger,
2000) is one of crucial requirements. Though long-distance quantum
communication has been widely achieved in optical fibers and free-space (Zhu
_et al._ , 2017; Liu  _et al._ , 2019; Huo  _et al._ , 2018; Chen  _et al._ ,
2021a; Liao  _et al._ , 2018), it is still very important to find a promising
way for transferring quantum states through solid-state devices or condensed
matter (Bose, 2003; Christandl  _et al._ , 2004; Balachandran and Gong, 2008;
Song and Sun, 2005; Yung, 2006; Yung and Bose, 2005; Wu  _et al._ , 2009a, b).
A number of QST protocols have been proposed for different solid-state medium
(Bienfait  _et al._ , 2019; Sete and Eleuch, 2015; Vermersch  _et al._ , 2017;
He  _et al._ , 2017; Maring  _et al._ , 2017; Northup and Blatt, 2014; Cirac
_et al._ , 1997; Shi  _et al._ , 2005; Zwick  _et al._ , 2011; Kandel  _et
al._ , 2021). In recent years, QST via a spin chain has attracted extensive
attentions (Bose, 2003; Christandl  _et al._ , 2004; Balachandran and Gong,
2008) and many technologies have already been developed (Shi  _et al._ , 2005;
Zwick  _et al._ , 2011; Kandel  _et al._ , 2021; Yao  _et al._ , 2011; Bose,
2007; Mei  _et al._ , 2018; Longhi, 2019; D’Angelis  _et al._ , 2020; Wang
_et al._ , 2018).

Perfect QST can be realized through well designed spin chain with invariable
couplings (Zhang  _et al._ , 2005; Petrosyan  _et al._ , 2010; Gualdi  _et
al._ , 2008). Meanwhile, it can also be realized by precisely modulating the
spin-spin couplings (Wang  _et al._ , 2020; Lyakhov and Bruder, 2006; Zwick
_et al._ , 2014; Benjamin, 2001). For example, quantum states can be
transferred by simply applying a sequence of SWAP operations implemented by π
pulses between the pairs of nearest neighboring sites (Petrosyan  _et al._ ,
2010), which needs only the spin-spin couplings to be switched on and off
periodically (Benjamin and Bose, 2003). Nevertheless, these known methods
require the accurate design of the system Hamiltonian, thus are usually less
robust against disorders and imperfections in large-scale implementations. To
overcome this issue, adiabatic QST protocols have been widely studied
(Balachandran and Gong, 2008; Tan  _et al._ , 2020; Demirplak and Rice, 2003;
Gong and Rice, 2004a, b; Eckert  _et al._ , 2007; Ho and Gong, 2012; Tian,
2012; Agundez  _et al._ , 2017; Sandberg  _et al._ , 2016), as the QST
exploiting the adiabatic theorem (Messiah, 1962; Berry, 1984) is independent
of the protocol operation details so long as the evolution of the system is
slow enough.

Our QST method here is based on the spectral and dynamical features of a
topological qubit chain. For topological insulators, robust conducting edge
states are guaranteed by the nontrivial topology of bulk bands (Hasan and
Kane, 2010; Qi and Zhang, 2011; Asbóth  _et al._ , 2016; Luo, 2019). These
edge states are insensitive to smooth changes in the system parameters unless
a topological phase transition occurs (Hasan and Kane, 2010; Qi and Zhang,
2011). Such robustness based on topological protection provides topological
quantum systems a great potential for quantum information and quantum
computing (Nayak  _et al._ , 2008; Freedman  _et al._ , 2003; Stern and
Lindner, 2013; Bomantara and Gong, 2018a, b, 2020). The Su-Schrieffer-Heeger
(SSH) chain (Su  _et al._ , 1979) is the simplest model of the topological
insulators and can be realized by a qubit chain with staggered couplings when
the qubit chain is restricted to the subspace of single excitation.

Building on the concepts of topological edge states and adiabatic QST via a
qubit chain, protocols for efficient QST (Yao  _et al._ , 2013; Mei  _et al._
, 2018; Palaiodimopoulos  _et al._ , 2021; Longhi, 2019; D’Angelis  _et al._ ,
2020; Wang  _et al._ , 2018) have also been proposed, with more robustness to
disorder due to the underlying topological protection. With specific design of
topological qubit chains, quantum states can be encoded in edge states of the
systems, and transferred from one end of the chain to the other by
adiabatically altering couplings between qubits (Mei  _et al._ , 2018;
Palaiodimopoulos  _et al._ , 2021; Longhi, 2019; D’Angelis  _et al._ , 2020).
However, most of these available proposals focus on single qubit state
transfer, and multi-qubit state transfer with arbitrary entanglement is still
a challenging task. Recently, a more advanced protocol for QST via the so-
called Floquet topological edge modes was proposed (Tan  _et al._ , 2020). In
this protocol, some entangled states are encoded in the edge states of
quasienergy zero and $\pi$ modes, and the high-fidelity transfer of entangled
states can be achieved over a long distance. However, this method requires
additional dynamical modulation on the time-periodic couplings between the
qubits and may pose new experimental challenges.

In our method, arbitrary entangled state can be encoded in topological edge
states, which are supported by a generalized SSH chain. As the parameters of
the system are slowly altered, the edge states can be adiabatically
transferred from one end of the chain to the other. Thus the entangled state
can be transferred along this adiabatic passage. It is well known that the
adiabatic evolution leads to two phases, i.e, geometric phase and dynamical
phase. The geometric phase can be easily wiped out with a chosen gauge in non-
closed adiabatic evolution (Berry, 1984), and the dynamical phase can be
eliminated when the evolution time is carefully chosen. Thus, the quantum
phases between different components encoding an entangled state can be well-
controlled or recovered, and the entangled state can be truly transferred.

For the concreteness of discussions, we assume that the qubit chain is formed
by the superconducting quantum circuits, which have been developing rapidly
(Jeffrey  _et al._ , 2014; Gambetta  _et al._ , 2017; Brecht  _et al._ ,
2016). In particular, superconducting qubit chains have been widely studied
for simulating many-body quantum physics (Houck  _et al._ , 2012; Paraoanu,
2014; Roushan  _et al._ , 2017). There are several advantages to use
superconducting qubits as quantum simulators. First, superconducting qubits
are highly coherent with long coherence time ($10\sim 100\mu s$) (Paik  _et
al._ , 2011; Novikov  _et al._ , 2016). Second, most of the parameters of
superconducting qubits are highly controllable (Chen  _et al._ , 2014; Geller
_et al._ , 2015; Barends  _et al._ , 2015; Reuther  _et al._ , 2010; Baust
_et al._ , 2015), thus we can perform rather arbitrary operation on the
systems. Third, the superconducting quantum circuits have high scalability and
designability. One quantum chip can support a large number of controllable
qubits with different connecting manners (Arute  _et al._ , 2019). These
advantages make superconducting quantum circuits one of the best platforms to
perform quantum simulation and quantum computing. Moreover, the topological
chain constructed by superconducting qubits has already been experimentally
demonstrated (Cai  _et al._ , 2019). Thus, for the state-of-the-art,
superconducting qubit circuits are very suitable to construct such a chain for
QST. The system parameters considered in this work are all based on the
existing experiments of Xmon qubits (Chen  _et al._ , 2014; Geller  _et al._ ,
2015; Barends  _et al._ , 2015).

The paper is organized as follows. In Section II, we introduce a generalized
SSH model, in which each unit cell contains three qubits, and the topology of
the model is characterized by winding number. We also use Xmon qubits as an
example to show how to form such generalized model. In Section III, we derive
the edge states of the Hamiltonian given in Section II for the generalized SSH
model, and then show that arbitrary two-qubit entangled state can be encoded
in these edge states and transferred from one end of the chain to the other
through adiabatic process. The exact dynamical solution based on adiabatic
theorem is also given. Furthermore, we illustrate the robustness of our
proposal against disorder from two parts, i.e., the coupling strength and
evolution time. In Section IV, we generalize QST from the entangled state of
the two-qubit to those of $\mathcal{N}$-qubit with $\mathcal{N}\geq 3$. In
particular, the QST of three-qubit state is carefully analyzed. In Section VI,
we further discuss our proposal and analyze the experimental feasibility, and
finally summarize our results.

##  II generalized ssh model with three qubits in each unit cell

Figure 1:  (a) Schematic diagram of a one-dimensional qubit chain. Each unit
cell hosts three qubits, labelled as $A_{1,m}$, $A_{2,m}$ and $B_{m}$
separately (The first subscript of $A$ denotes the intracell index. The
subscript of $B$ and the second subscript of $A$ denote the intercell index).
The coupling strength $g$ between $A_{1,m}$ and $A_{2,m}$ is uniform along the
chain, whereas the hopping between $A_{1,m}$ and $B_{m}$ is staggered, denoted
as $v$ and $w$. Each qubit can also be labeled by one unique numerical index
$x$ in order of $A_{1,1}A_{2,1}B_{1}A_{1,2}\cdots B_{M-1}A_{1,M}A_{2,M}$. (b)
Realization of the qubit chain with Xmon qubits. Each qubit contains three
basic superconducting circuits elements, i.e, the josephson junction, the
capacitance and the inductance. Two adjacent qubits are coupled through a
Josephson coupler. The qubit-qubit coupling can be tuned via an external
magnetic flux $\Phi_{\mathrm{ext}}$ with a dc control. The Josephson junctions
labeled $\mathrm{L_{J}}$ are each double junctions threaded by additional
fluxes (not shown) that tune the qubit frequencies. Therefore, both the qubit
frequencies and the couplings in the qubit chain are tunable.

The SSH model (Su  _et al._ , 1979) is one of the simplest examples hosting
one-dimensional topological phases. This model and its various extensions have
been widely used to study different physical phenomena (Pérez-González  _et
al._ , 2018; Li  _et al._ , 2014; Li and Miroshnichenko, 2019; Xu  _et al._ ,
2020; Atala  _et al._ , 2013; Li  _et al._ , 2015; Meier  _et al._ , 2016;
Sirker  _et al._ , 2014). In the standard SSH model, each unit cell has two
sublattices. By contrast, for one class of extended SSH models, each unit cell
contains 3 or more sublattices, thus called the SSH3 or SSHN models (Xie  _et
al._ , 2019; He and Chien, 2020). As schematically shown in Fig. 1(a), we
first consider a extended SSH3 model consisting of a qubit chain with $M$ unit
cells. Hereafter, we use the unit cell number $M$ to denote the length of the
chain. Each unit cell contains three sublattices labelled as $A_{1,m}$,
$A_{2,m}$, and $B_{m}$ with $m=1,2,\cdots,M$. The sublattice $B_{M}$ of the
$M$th unit cell is removed from the right end of the chain. Sublattices
$A_{1,m}$ and $B_{m}$ are analogous to those in the standard two-band SSH
model with staggered coupling strengths $v$ and $w$. An extra sublattice
$A_{2,m}$ is coupled to $A_{1,m}$ with the coupling strength $g$ ($g>0$).
Thus, the Hamiltonian of such a qubit chain is ($\hbar=1$)

$\displaystyle H$ $\displaystyle=$
$\displaystyle\stackrel{{\scriptstyle[}}{{m}}=1]{M-1}{\sum}\left(v\sigma_{A_{1,m}}^{+}\sigma_{B_{m}}^{-}+w\sigma_{A_{1,m+1}}^{+}\sigma_{B_{m}}^{-}+\textrm{H.c.}\right)$
(1) $\displaystyle+$
$\displaystyle\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}\left(g\sigma_{A_{1,m}}^{+}\sigma_{A_{2,m}}^{-}+\textrm{H.c.}\right).$

with $m$ denoting the index of each unit cell. The ladder operators in the
$m$th unit cell is given by $\sigma_{I}^{+}=|e_{I}\rangle\langle g_{I}|$
($I=A_{1,m},A_{2,m},B_{m}$), with $|e_{I}\rangle$ and $|g_{I}\rangle$ denoting
the excited and ground states, and the operator $\sigma_{I}^{-}$ is the
Hermitian conjugate of the operator $\sigma_{I}^{+}$.

We note that our proposal can be applied to any kinds of controllable qubit
systems. However, for the concreteness of the studies, we use superconducting
qubit circuits, e.g., Xmon qubits, to construct our theoretical model. This
$X$-shape qubit has several advantages, e.g., high-coherence, fast tunable
coupling, and easy connection (Chen  _et al._ , 2014; Geller  _et al._ , 2015;
Barends  _et al._ , 2015), thus is more suitable for our proposal. Our
proposal is not limited to the Xmon qubit, and can also be applied to other
types of superconducting qubits, e.g., transmon or flux qubit. For our setup,
as shown in Fig. 1(b), two Xmon qubits are connected with each other by a
Josephson junction coupler (Chen  _et al._ , 2014; Geller  _et al._ , 2015).
An extra magnetic flux bias $\Phi_{\rm ext}$ is applied to tune the effective
linear inductance of the coupler junction. Thus, the coupling constant between
these two qubits is tunable with controllable extra magnetic flux $\Phi_{\rm
ext}$ (see details in Appendix A).

In the following studies, we only consider the single-excitation of the chain,
thus the Hamiltonian can be rewritten as

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{M-1}\left(v|\mathcal{A}_{1,m}\rangle\langle\mathcal{B}_{m}|+w|\mathcal{A}_{1,m+1}\rangle\langle\mathcal{B}_{m}|+\textrm{H.c.}\right)$
(2) $\displaystyle+$
$\displaystyle\sum_{m=1}^{M}\left(g|\mathcal{A}_{1,m}\rangle\langle\mathcal{A}_{2,m}|+\textrm{H.c.}\right)$

in the single-excitation subspace
{$|\mathcal{A}_{1,m}\rangle,|\mathcal{A}_{2,m}\rangle,|\mathcal{B}_{m}\rangle$},
with $|\mathcal{A}_{1,m}\rangle=\sigma_{A_{1,m}}^{+}|G\rangle$,
$|\mathcal{A}_{2,m}\rangle=\sigma_{A_{2,m}}^{+}|G\rangle$ and
$|\mathcal{B}_{m}\rangle=\sigma_{B_{m}}^{+}|G\rangle$. Here $|G\rangle$
denotes that all qubits in the chain are in the ground state, i.e.,
$|G\rangle=|g_{A_{1,1}}g_{A_{2,1}}g_{B_{1}}\cdots g_{A_{2,M}}\rangle$, which
is written as $|G\rangle=|gg\cdots g\rangle$ for simplicity.

For the standard two-band SSH model, the topologically nontrivial phase is
characterized by a nonzero winding number. Two topological in-gap edge modes
are degenerate at zero energy in the thermodynamic limit (Asbóth  _et al._ ,
2016). However, for a finite lattice size, the two edge states hybridize due
to finite-size effect and so that their energy eigenvalues are shifted by an
exponentially small amount. Dynamics-wise, this edge-state hybridization would
then induce Rabi oscillation between two topological edge modes if the initial
state is prepared by exciting the leftmost or rightmost qubit only (Nie and
Liu, 2020). To obtain a steady edge state in a finite-size system, one may
remove one edge qubit from the standard SSH chain (Mei  _et al._ , 2018; Cai
_et al._ , 2019). In this case, this imperfect SSH chain only supports one
edge state. This is useful for our considerations here since we are
considering the same platform. Indeed, applying this idea of eliminating one
edge qubit from an extended SSH model, we can likewise engineer steady edge
states in the extended SSH setting, as shown below.

For our extended SSH3 model shown in Fig. 1(a), the Hilbert space of the
Hamiltonian is enlarged by the extra qubits $A_{2,m}$ compared with the
standard SSH model. The edge states are expected to form from two renormalized
branches of the qubit chain, thus resulting in the upper and the lower edge
states with positive and negative eigenenergies, as discussed further in
Appendix B. To see the topological aspect of these edge states, we first
consider the case with $g=0$, where sublattices $A_{1,m}$ and $B_{m}$ are
decoupled from $A_{2,m}$. In this case, the imperfect SSH model with an odd
number of qubits is formed, where $v$ and $w$ are the intra- and inter- cell
couplings. In the standard SSH model, edge states appear when the coupling
strength at the edge is weaker than the coupling next to it. Similarly, for
the imperfect SSH model, a weaker coupling strength appears at the left
(right) edge and generates an edge state there when $v<w$ ($v>w$). Therefore,
in the presence of an odd number of sites, the system always has one edge
state in the topologically nontrivial regime (Mei  _et al._ , 2018). That is,
in our extended SSH3 model, either the left or the right edge state
corresponds to a winding number of $1$ for the Fourier transformed Hamiltonian
(in momentum space), with unit cells defined either as sites
$(A_{1,m},A_{2,m},B_{m})$ or as sites $(B_{m},A_{1,m+1},A_{2,m+1})$ in Fig.
1(a) respectively. When $g=0$, the edge state is only located at the
sublattice $A_{1,m}$, which returns to the case of the standard SSH chain.
When $g>0$, the edge states are expected to occupy both sublattices $A_{1,m}$
and $A_{2,m}$, because they take the role of the $A$-type sublattice in the
standard SSH chain (see Appendix C). With this understanding, we can see that
the edge states in our extended SSH model do originate from the topological
edge states in the standard SSH model, and hence should be robust to local
disorder.

Below we show how such edge states in the extended SSH chain can be used to
robustly transfer arbitrary entangled states. We here emphasize that the
parameters used for the following numerical simulations are taken from
superconducting qubit circuits, e.g., Xmon qubit circuits (Chen  _et al._ ,
2014; Geller  _et al._ , 2015; Barends  _et al._ , 2015).

##  III 2-qubit entanglment state transfer

###  III.1 Edge states of the qubit chain

In our extended SSH model, there are $\mathcal{L}=3M-1$ qubits in the chain
when one $B$-type qubit at the right end of the chain is removed. Let us first
give an ansatz that the resultant edge state in our system exclusively
occupies sublattices $A_{1,m}$ and $A_{2,m}$ (see appendix B), i.e., the
associated wavefunction can be written as

$|\varPsi_{\rm
edge}\rangle=\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}\lambda^{m}\left(a\sigma_{A_{1,m}}^{+}+b\sigma_{A_{2,m}}^{+}\right)|G\rangle.$
(3)

As energy-eigenstates of the system’s Hamiltonian, the wavefunction in Eq. (3)
must satisfy the stationary Schrödinger equation

$H|\varPsi_{\rm edge}\rangle=E|\varPsi_{\rm edge}\rangle.$ (4)

Combining Eq. (1) with Eq. (3), we have

$\displaystyle
E\left(a\sigma_{A_{1,m}}^{+}+b\sigma_{A_{2,m}}^{+}\right)|G\rangle$
$\displaystyle=a\left(v+w\lambda\right)\sigma_{A_{2,m}}^{+}|G\rangle$ (5)
$\displaystyle+g\left(b\sigma_{A_{1,m}}^{+}+a\sigma_{A_{2,m}}^{+}\right)|G\rangle.$

It is straightforward to get $v+w\lambda=0$, i.e., $\lambda=-v/w$, and the
coefficients $a$ and $b$ satisfy the following equation as

$\left(\begin{array}[]{cc}0&g\\\
g&0\end{array}\right)\left(\begin{array}[]{c}a\\\
b\end{array}\right)=E\left(\begin{array}[]{c}a\\\ b\end{array}\right).$ (6)

Thus, the energy eigenvalues of the edge states are given by $E_{\pm}=\pm g$,
with coefficients ($a,b$) being ($1/\sqrt{2},1/\sqrt{2}$) or
($1/\sqrt{2},-1/\sqrt{2}$) respectively. This explicit edge state solution
indicates that there are two edge states with the form of the Bell state
$|\chi_{m,\pm}\rangle=\left(|\mathcal{A}_{1,m}\rangle\pm|\mathcal{A}_{2,m}\rangle\right)/\sqrt{2}$
in each unit cell. The wavefunctions corresponding to the edge states hence
take the following form (unnormalized)

$|\varPsi_{\pm}\rangle=\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}\lambda^{m}\left(\frac{\sigma_{A_{1},m}^{+}\pm\sigma_{A_{2},m}^{+}}{\sqrt{2}}\right)|G\rangle,$
(7)

where both the upper (labeled as $+$) and lower (labeled as $-$) edge states
only occupy the $A$-type qubits. When $\left|\lambda\right|=v/w\ll 1$, i.e.,
$v\ll w$, the edge states are mostly localized in the left end of the chain as

$|\varPsi_{\pm}\rangle\approx|L_{\pm}\rangle=\frac{1}{\sqrt{2}}\left(\sigma_{A_{1,1}}^{+}\pm\sigma_{A_{2,1}}^{+}\right)|G\rangle.$
(8)

In the limit of $v=0$, the qubit chain degenerates into $M-1$ trimers and an
additional dimer, an exact edge state $|L_{\pm}\rangle$ can be obtained at the
left end of the chain. When $\left|\lambda\right|=v/w\gg 1$, i.e., $v\gg w$,
the edge states are mostly localized in the right end of the chain as

$|\varPsi_{\pm}\rangle\approx|R_{\pm}\rangle=\frac{1}{\sqrt{2}}\left(\sigma_{A_{1,M}}^{+}\pm\sigma_{A_{2,M}}^{+}\right)|G\rangle.$
(9)

In the limit of $w=0$, an exact edge state $|R_{\pm}\rangle$ can be obtained
at the right end of the chain.

In Fig. 2(a), as an example, the energy spectrum of a $14$-qubit chain with
$5$ unit cells is plotted for
$\left|\lambda\right|=v/w\in\left[0,\infty\right]$. It clearly shows that the
two topological edge modes (color-dashed lines) exist in the gaps of three
bulk bands. The bulk band in the middle of two edge modes is 4-fold degenerate
as shown in Fig 2(b). As the parameter $\lambda$ changes from $0$ to $\infty$,
the localized topological edge states as a function of $\lambda$ also change
their qualitatively behavior, namely, from being localized at the left end
(red) to being localized at the right end (green). Meanwhile, the eigenvalues
$E_{\pm}$ corresponding to these two edge states keep constants all along.

###  III.2 QST process for arbitrary two-qubit entangled states

Figure 2: Two-qubit state transfer via a 14 qubits chain with $5$ unit cells,
i.e., $M=5$. The system parameters are $J=5g$ and $g/2\pi=10$MHz. (a) The
energy spectrum for an extended SSH3 model. Black-solid lines represent the
bulk states and color-dashed lines represent the edge states. Three bulk bands
are divided by two edge states, the middle bulk is $4$-fold degenerate. As
$\omega t$ changes from $0$ to $\pi$, edge states are transferred from the
left end (red) to the right end (green). (b) Schematics for eigenstates
distribution at $\omega t=\pi/6$. Each point represents one eigenstate. Blue
and red dots represent bulk and edge states, respectively. (c) The variation
of $|\langle\Phi_{r}(t)|\dot{{\Phi}}_{l}(t)\rangle|$ with the time evolution.
Here $\Phi_{l}(t)$ takes the lower edge state, i.e., $l=5$. (d) Time
evolutions of the target state occupations
$\left|\langle\varPsi_{F}|\varPsi\left(t\right)\rangle\right|$ for different
initial states. Red-solid and blue-dashed curves denote theoretical solutions.
Red dots and blue stars denote the numerical simulations. The red-solid curve
and red dots correspond to the initial state
$\left(|\mathcal{A}_{1,1}\rangle+|\mathcal{A}_{2,1}\rangle\right)/\sqrt{2}$.
Meanwhile, the blue-dashed curve and blue stars correspond to the initial
state $|\mathcal{A}_{1,1}\rangle$. (e) Time-dependent population distribution
on each qubit in the chain when the initial state is prepared to single-qubit
state $|\mathcal{A}_{1,1}\rangle$. (f) Time-dependent population distribution
on each qubit in the chain when the initial state is prepared to two-qubit
Bell state
$\left(|\mathcal{A}_{1,1}\rangle+|\mathcal{B}_{2,1}\rangle\right)/\sqrt{2}$.

According to the quantum adiabatic theorem, if the parameters of the qubit
chain in Eq. (1) can be changed slowly enough, it will remain in its
instantaneous eigenstate. At $v=0$, the leftmost two qubits $A_{1,1}$ and
$A_{2,1}$ are isolated and the edge states are
$|L_{\pm}\rangle=|\chi_{1,\pm}\rangle|gg\cdots g\rangle$. At $w=0$, the
rightmost two qubits $A_{1,M}$ and $A_{2,M}$ are isolated and the edge states
are $|R_{\pm}\rangle=|gg\cdots g\rangle|\chi_{M,\pm}\rangle$. It is obvious
that the coupling strengths

$v=J\left[1-\cos\left(\omega t\right)\right],w=J\left[1+\cos\left(\omega
t\right)\right],$ (10)

can be changed slowly, then the state of the left end can be transferred to
the right end of the chain. Where $\omega$ and $J$ are the frequency and
strength of the control field, respectively. For example, as shown in Fig.
2(a), if the system is initially prepared to the state $|L_{\pm}\rangle$ for
the time $t=0$, then this state adiabatically evolves from $|L_{\pm}\rangle$
to $|R_{\pm}\rangle$ when the time $t$ slowly varies from $0$ to $\pi/\omega$.
The adiabatic following is feasible here because the considered state is the
system’s edge state, which is gapped from the bulk states.

In above, we analyze QST when the edge state is the initial state. Let us now
consider the general situation that the system is initially prepared to an
arbitrary entangled state

$|\varPsi_{\rm
in}\rangle=\left(\alpha\sigma_{A_{1,1}}^{+}+\beta\sigma_{A_{2,1}}^{+}\right)|G\rangle,$
(11)

at the left end of the chain, which can be decomposed as

$|\varPsi_{\rm
in}\rangle=\frac{\alpha+\beta}{\sqrt{2}}|L_{+}\rangle+\frac{\alpha-\beta}{\sqrt{2}}|L_{-}\rangle.$
(12)

For such more general situation involving the superpositions of two edge
states, one must carefully analyze the quantum phases. In an adiabatic
process, if the system is initially prepared to an eigenstate
$|\varPhi\left(0\right)\rangle$, then the final state at the time $t$ is given
as

$|\psi_{\rm
ad}\left(t\right)\rangle=e^{ir\left(t\right)}e^{i\theta\left(t\right)}|\varPhi\left(t\right)\rangle,$
(13)

where $\theta\left(t\right)=-\int_{0}^{t}E(t^{\prime})dt^{\prime}$ is the
dynamical phase and
$r\left(t\right)=i\int_{0}^{t}\langle\varPhi\left(t^{\prime}\right)|\dot{\varPhi}\left(t^{\prime}\right)\rangle
dt^{\prime}$ is the geometric phase, which can be gauged out unless the
evolution path is closed. The normalization of
$|\varPhi\left(t^{\prime}\right)\rangle$ implies that
$\langle\varPhi\left(t^{\prime}\right)|\dot{\varPhi}\left(t^{\prime}\right)\rangle$
is imaginary, which guarantees that $r\left(t\right)$ is real (Berry, 1984).
In our protocol, with the chosen gauge as shown in Eq. (7),
$|\varPsi_{\pm}\rangle$ only contains real parameters along
$\lambda\left(t\right)=\left[1-\cos\left(\omega
t\right)\right]/\left[1+\cos\left(\omega t\right)\right]$, thus
$\langle\varPhi\left(t^{\prime}\right)|\dot{\varPhi}\left(t^{\prime}\right)\rangle$
is real, which guarantees that $r\left(t\right)$ is imaginary. Therefore, the
geometric phase $r\left(t\right)$ must be zero and is naturally gauged out.

We thus only need to consider the dynamical phase associated with the
adiabatic process. If we adiabatically change the parameters $v$ and $w$ to
the final time $t$, following the quantum adiabatic theorem, the initial state
in Eq. (12) should evolve to the state

$\displaystyle|\varPsi\left(t\right)\rangle$ $\displaystyle=$
$\displaystyle\frac{\alpha+\beta}{\sqrt{2}}|\varPsi_{+}(t)\rangle
e^{-i\int_{0}^{t}E_{+}dt^{\prime}}$ (14) $\displaystyle+$
$\displaystyle\frac{\alpha-\beta}{\sqrt{2}}|\varPsi_{-}(t)\rangle
e^{-i\int_{0}^{t}E_{-}dt^{\prime}}.$

As discussed above, $E_{\pm}$ in our model are both constants due to the
nature of the topological edge state. As such the final state at the time $t$
is

$|\varPsi\left(t\right)\rangle=\frac{\alpha+\beta}{\sqrt{2}}|\varPsi_{+}(t)\rangle
e^{-igt}+\frac{\alpha-\beta}{\sqrt{2}}|\varPsi_{-}(t)\rangle e^{igt}.$ (15)

Indeed, by solely observing the values of $E_{\pm}=\pm g$, we find that the
phase factors $e^{-igt}$ and $e^{igt}$ for two involved edge states (from
upper and lower branch respectively) have the same period $T=2\pi/g$.
Therefore, if the time evolution takes the dynamical period $T$, the dynamical
phase difference between the two involved edge states in Eq. (15) will be
zero. We can call this dynamical period as one evolution circle. If the total
adiabatic protocol time is always chosen to be a multiple of evolution
circles, then the concern with dynamical phases can be lifted.

As shown in Fig. 2(a), if the control field applied to the coupling strength
involves from $t_{0}=0$ to $t_{f}=\pi/\omega$, then the edge state is
transferred from the left end to the right end. In this case, Eq. (15) becomes

$|\varPsi_{f}\rangle=\frac{\alpha+\beta}{\sqrt{2}}|R_{+}\rangle
e^{-i\pi\frac{g}{\omega}}+\frac{\alpha-\beta}{\sqrt{2}}|R_{-}\rangle
e^{i\pi\frac{g}{\omega}}.$ (16)

Here we choose the evolution time $t_{f}$ to be a multiple of evolution
circles, i.e., $t_{f}/T=g/2\omega=n$. $n$ should be a large integer number to
satisfy the adiabatic condition. Thus, the dynamical phases become zero and
the state in Eq. (16) is involved to

$|\varPsi_{F}\rangle=\left(\alpha\sigma_{A_{1,M}}^{+}+\beta\sigma_{A_{2,M}}^{+}\right)|G\rangle.$
(17)

Here, the subscript $F$ denotes the state transferred perfectly, i.e., the
final state. Therefore, an arbitrary two-qubit entangled state can be encoded
by two edge states and perfectly transferred from the left end to the right
end via an adiabatic passage. Notice that all along the protocol, only the
$A_{1,m}$\- and $A_{2,m}$-type qubits are occupied by the edge states, and the
$B_{m}$-type qubits serve as the invariable medium. Thus the qubits $A_{1,m}$
and $A_{2,m}$ can be considered as the transport qubits, and the qubits
$B_{m}$ can be considered as the mediated qubits.

We here have two remarks. First, for the special case of two-qubit state
transfer, the evolution time can be half of the evolution cycle defined above.
In this case, the final state acquires a global phase factor $-1$, which does
not affect the task of entangled state transfer. Second, one may worry about
the robustness of such adiabatic protocol as we need a precise timing. This is
unnecessary because along the way, the eigenvalues of the edge states are
pinned at special constant values due to topological features and hence we
still expect to have robust protocols.

###  III.3 Analysis on adiabatic condition and two examples of QST

We now analyze the condition of the adiabatic evolutions. The adiabatic
approximation requires a small changing rate of the Hamiltonian
$\dot{H}\left(t\right)$ and a large energy gap $\left|E_{r}-E_{l}\right|$
between the $r$th and $l$th eigenstates. For our protocol, if we assume that
the $l$th eigenstate $|\varPhi_{l}\left(t\right)\rangle$ is the edge state,
then the adiabatic condition is given by

$\left|\langle\varPhi_{r}\left(t\right)|\dot{\varPhi}_{l}\left(t\right)\rangle\right|=\left|\frac{\langle\varPhi_{r}\left(t\right)|\dot{H}\left(t\right)|\varPhi_{l}\left(t\right)\rangle}{E_{r}\left(t\right)-E_{l}\left(t\right)}\right|\ll
1,$ (18)

where $E_{r}\left(t\right)$ is the instantaneous eigenenergy corresponding to
the instantaneous state $|\varPhi_{r}\left(t\right)\rangle$ for the time $t$.
The eigenstates are sorted according to the corresponding eigenenergies
(lowest to highest), and the $5$th eigenstate in Fig. 2(a) is the lower edge
state. Here we make the 14-qubit chain evolve 5 evolution circles, i.e.,
$\omega=0.1g$. As shown in Fig. 2(c), for the lower edge state ($l=5$), the
adiabatic conditions are checked to be satisfied with $r=3,4,6,7,8,9$. For the
bulk state between two edge states ($r=6,7,8,9$), the results are not
continuous due to the numerical instablity of degenerate eigensolutions of the
system.

To interpret our protocol, we choose the final state occupation
$\left|\langle\varPsi_{F}|\varPsi\left(t\right)\rangle\right|$ as the
dynamical indicator. In Fig. 2(d), we compare the analytical and numerical
results of this indicator. The analytical solution for $|\Psi(t)\rangle$ is
given in Eq. (15), while the numerical solution is calculated with ordinary
differential equation (ODE) solver for small and fixed step size. Here,
$\omega=0.1g$, and the evolving time is $t_{f}=0.5$$\mu$s, i.e., exact $5$
times of the dynamical period ($T=2\pi/g=0.1$$\mu s$). We typically choose two
initial states to be transferred. The first one is single-qubit state

$|\varPsi_{\rm
in}^{\left(1\right)}\rangle=\sigma_{A_{1,1}}^{+}|G\rangle\equiv\frac{\sqrt{2}}{2}\left(|L_{+}\rangle+|L_{-}\rangle\right)$
(19)

which can be obtained from Eq. (12) by setting $\alpha=1$ and $\beta=0$. The
other one is two-qubit Bell state

$|\varPsi_{\rm
in}^{\left(2\right)}\rangle=\frac{\sqrt{2}}{2}\left(\sigma_{A_{1,1}}^{+}+\sigma_{A_{2,1}}^{+}\right)|G\rangle\equiv|L_{+}\rangle$
(20)

with $\alpha=\beta=1/\sqrt{2}$. For the first one of single-qubit state
transfer, the analytical evolution of the state is given as

$|\Psi^{\left(1\right)}(t)\rangle=\frac{\sqrt{2}}{2}\left(|\varPsi_{+}\rangle
e^{-igt}+|\varPsi_{-}\rangle e^{igt}\right)$ (21)

The analytical solution (blue-dashed line) and numerical simulation (blue
stars) for the dynamical indicator
$\left|\langle\varPsi_{F}|\varPsi\left(t\right)\rangle\right|$ are plotted as
a function of the evolution time $t$ in Fig. 2(d). In this case, the indicator
$\left|\langle\varPsi_{F}|\varPsi\left(t\right)\rangle\right|$ has a periodic
variation due to the dynamical phase difference between the two edge states.
The analytical result agrees well with numerical one, this confirms that the
chosen parameters have fulfilled the adiabatic conditions. For two-qubit Bell
state transfer, the adiabatic evolution of the state is

$|\varPsi^{\left(2\right)}\left(t\right)\rangle=|\varPsi_{+}\rangle e^{-igt}.$
(22)

In this case, the dynamical phase can be gauged out as one global phase, so
the dynamical indicator does not oscillate and increases smoothly with the
time. Again, the analytical result (red line) agrees well with numerical (red
dots) one.

The first case for single-qubit state transfer of Eq. (21) is further analyzed
in Fig. 2(e), where we show the population distribution $|\langle
P_{m}|\Psi^{\left(1\right)}(t)\rangle|$
($P_{m}\in\\{|\mathcal{A}_{1,m}\rangle,|\mathcal{A}_{2,m}\rangle,|\mathcal{B}_{m}\rangle\\}$)
on each site of the state $|\Psi^{\left(1\right)}(t)\rangle$ during the
adiabatic protocol. The state distribution shifts from the left to the right
with rapid oscillations, due to coherent effect caused by the dynamical phase
difference between the two involved edge states as mentioned above. However,
for the second case shown in Eq. (22), Fig. 2(f) demonstrates the smooth
transfer of the state $|\Psi^{\left(2\right)}(t)\rangle$ without any
oscillation.

###  III.4 Robustness analysis for two-qubit state transfer

Any realistic implementation of the theoretical protocol unavoidably involves
disorders from many aspects, e.g., the environmental effect, the time
inaccuracy of the control field applied for the adiabatic evolution,
nonuniform of the prepared qubits and imprecise couplings between the qubits
in the chain. Here, we consider two main imperfections: one is the disorder of
the qubit couplings and the other one is inaccuracy of the evolution time to
achieve perfect state transfer.

The first kind of the disorders can be analyzed by modelling it as external
perturbation terms in our system Hamiltonian, i.e.,

$\displaystyle\delta H$
$\displaystyle=\stackrel{{\scriptstyle[}}{{m}}=1]{M-1}{\sum}\left(\delta\mu_{A_{1,m}}\sigma_{A_{1,m}}^{+}\sigma_{B_{m}}^{-}+\delta\mu_{B_{m}}\sigma_{A_{1,m+1}}^{+}\sigma_{B_{m}}^{-}+\textrm{H.c.}\right)$
$\displaystyle+\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}\left(\delta\mu_{A_{2,m}}\sigma_{A_{1,m}}^{+}\sigma_{A_{2,m}}^{-}+\textrm{H.c.}\right),$
(23)

where $\delta\mu_{A_{1,m}}$, $\delta\mu_{A_{2,m}}$, and $\delta\mu_{B_{m}}$
are the disorder coefficients, and assumed to satisfy the the Gaussian
distribution as $\sim\exp[(-\delta\mu)^{2}/2\xi^{2}]$, with $\xi$ being the
standard deviation of the disorder in the coupling strength, i.e., the
coupling disorder strength. Note that all these disorders are related to the
control field. Thus, with the variation of the adiabatic parameters, it is
more realistic to assume these disorders to be time-dependent, i.e., the
temporal noises. If the initial state is prepared to $|\varPsi_{\rm
in}\rangle=\left(\alpha\sigma_{A_{1},1}^{+}+\beta\sigma_{A_{2},1}^{+}\right)|G\rangle$,
then the adiabatic evolution can be derived as

$|\psi\left(t\right)\rangle=U\left(t\right)|\varPsi_{\rm
in}\rangle=\mathcal{T}e^{-i\int_{0}^{t}H\left(t^{\prime}\right)+\delta
H\left(t^{\prime}\right)dt^{\prime}}|\varPsi_{\rm in}\rangle,$ (24)

where $\mathcal{T}$ is the time order operator. The numerical simulation of
this process can be written as $U\left(t\right)=\mathcal{T}\prod
e^{-i\left[H\left(t^{\prime}\right)+\delta
H\left(t^{\prime}\right)\right]\Delta t}$ ($\Delta t\ll T$). When $t=t_{f}$,
the state described in Eq. (24) evolves to $|\psi_{f}\rangle$. Thus, the
fidelity is given as

$F=\left|\langle\varPsi_{F}|\psi_{f}\rangle\right|,$ (25)

where $|\varPsi_{F}\rangle$ is the perfectly transferred state given in Eq.
(17).

Figure 3:  Robustness of our protocol for two-qubit state transfer. (a) The
right edge target-state occupations (fidelity $F$) with different evolving
circles. As the qubit number of the chain becomes larger, the evolving circles
demanded for the adiabatic evolution increase from 5 to 10. For a qubit chain
containing 23 qubits, i.e., 8 unit cells, it needs at least 10 evolving
circles to guarantee the adiabaticity. (b) The distribution of fidelities for
one settled state transfer process with 100 repetitions where $\xi=0.5g$ and
$M=4$. (c) The average fidelities of two-qubit entangled state transfer with
the coupling disorder. The numbers of unit cells are 2, 4, 6, and 8 for each
curve, separately. (d) The average fidelities of two-qubit entangled state
transfer with the imperfection of the evolution time.

To verify the protocol robustness against these disorders, we first determine
the proper evolution circles $n$ under the ideal condition without disorder.
As shown in Fig. 3(a), we have numerically calculated the fidelity $F$ given
in Eq. (25) for the right edge state transferred with different evolution
circles and different lengths of the chain. These results help us to determine
what extent we are working in the adiabatic regime. Indeed, the fidelity rises
rapidly with the increases in the number of evolution circles. As shown in
Fig. 3(a), when the number of unit cells of the chain varies from $2$ to $8$,
the necessary numbers of evolution circles to achieve good fidelity increase
from $5$ to $10$. Therefore, to guarantee the adiabatic condition and obtain
good fidelity for the transferred state, we choose $n=10$ as the number of
evolution circles for the following discussions, i.e., $t_{f}=10\,T=1\mu s$.

In our numerical simulations, we do $100$ repetitions of the adiabatic
evolution for a given time $t_{f}$, and each repetition has its different
random choice of disorder. We here consider the average fidelity
$\mathcal{F}=\overline{F}$ over these $100$ calculations. In Fig. 3(b), we
plot the distribution of fidelities for $100$ repetitions of one chosen state
transfer protocol. Here, the number of unit cells is taken as $M=4$, and the
coupling disorder strength $\xi$ is taken as $\xi=0.5g$. The distribution of
fidelities for each simulation run is highly concentrated around the average
fidelity, so the average fidelity here is a proper representation for the
protocol fidelity.

In Fig 3(c), the effect of the coupling disorder on the protocol fidelities is
shown for different lengths of the chain, i.e., $M=2,4,6,8$, respectively. For
each length of the chain, we find that the average fidelity $\mathcal{F}$ for
$100$ repetition calculations decreases from $1$ to $0.92$, as the coupling
disorder strength $\xi$ changes from $10^{-3}g$ to $g$. We also find that
there is a plateau near $\mathcal{F}=1$ for each case when $\xi$ is taken from
the range $\xi\in\left[10^{-3}g,10^{-1}g\right]$. Therefore, for our protocol,
the fidelity $\mathcal{F}$ will remain above $99\%$ as long as the coupling
disorder strength $\xi$ is less than $0.1g$, which should be experimentally
accessible.

The second kind of the disorders is the inaccuracy of protocol execution time.
Unlike a usual adiabatic process, our protocol demands the total evolution
time be exact multiple of the dynamical period. However, the control
inaccuracy for the evolution time may lead to the imperfection of the target
state. This kind of imperfection can be examined by an external perturbation
time $\delta t$, which satisfies the Gaussian distribution as $\delta
t\sim\exp[(-\delta t)^{2}/2\eta^{2}]$. $\eta$ is the standard deviation of the
evolution time. In this case, the modified transferred state can be derived as

$|\psi_{f}\rangle=U\left(t_{f}+\delta t\right)|\varPsi_{\rm
in}\rangle=\mathcal{T}e^{-i\int_{0}^{t_{f}+\delta
t}H\left(t^{\prime}\right)dt^{\prime}}|\varPsi_{\rm in}\rangle,$ (26)

with the time inaccuracy $\delta t$. Figure 3(d) depicts the impact of the
evolution time imperfection. We find that the average fidelity $\mathcal{F}$
for each chain decreases from $1$ to $0.94$ as the time disorder strength
$\eta$ changes from $10^{-3}T$ to $10^{-1}T$. For our protocol, the fidelity
will remain above $99\%$ when the time disorder strength $\eta$ is less than
$0.01T$.

Therefore, we conclude that our protocol is robust against both the qubit
coupling disorder and the inaccuracy of the protocol execution time.

##  IV Extended protocols for $\mathcal{N}$-qubit state transfer

Figure 4: Schematic of the extended SSH chain for $3$-qubit and
$\mathcal{N}$-qubit state transfer. Each unit cell hosts $4$ to
$\mathcal{N}+1$ qubits. The edge states are localized among the transport
qubits from $A_{1,m}$ to $A_{\mathcal{N},m}$ with $m=1,\cdots,M$. The rest
qubits are mediated qubits labeled as $B_{m}$ with $m=1,\cdots,M-1$. $v$
denotes the coupling between qubits $A_{1,m}$ and $B_{m}$, $w$ denotes the
coupling between qubits $A_{1,m+1}$ and $B_{m}$.

Our protocol for arbitrary two-qubit entangled state transfer through extended
SSH chain can be easily generalized to $\mathcal{N}$-qubit state transfer. For
$\mathcal{N}$-qubit transfer process, as schematically shown in Fig. 4, the
extended SSH model has $\mathcal{N}+1$ sites ($\mathcal{N}$ transport qubits
and $1$ mediated qubit) in each unit cell, and the whole qubit chain with $M$
unit cells has $\mathcal{L}=\left(\mathcal{N}+1\right)M-1$ qubits. With the
increase of the site number in each unit cell, more edge states emerge. Thus
it is not an easy task to find a proper evolution time to cancel all dynamical
phase differences between these edge states in our protocol. As such, the
couplings between the transport qubits should not simply be the constant $g$.
To this end, we may consider the following modified Hamiltonian

$\displaystyle H$ $\displaystyle=$
$\displaystyle\stackrel{{\scriptstyle[}}{{m}}=1]{M-1}{\sum}\left(v\sigma_{A_{1,m}}^{+}\sigma_{B_{m}}^{-}+w\sigma_{A_{1,m+1}}^{+}\sigma_{B_{m}}^{-}+\textrm{H.c.}\right)$
(27) $\displaystyle+$
$\displaystyle\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}\left(g_{1}\sigma_{A_{1,m}}^{+}\sigma_{A_{2,m}}^{-}+g_{2}\sigma_{A_{2,m}}^{+}\sigma_{A_{3,m}}^{-}+\cdots\right.$
$\displaystyle+$
$\displaystyle\left.g_{\mathcal{N}-1}\sigma_{A_{\mathcal{N}-1,m}}^{+}\sigma_{A_{\mathcal{N},m}}^{-}+\textrm{H.c.}\right).$

In each unit cell, these $\mathcal{N}$ transport quibts form an array with
coupling constants $g_{1},\cdots,g_{\mathcal{N}-1}$, and all of the mediated
qubits $B_{m}$ are used to couple the arrays of transport qubits with coupling
constants $v$ and $w$, respectively.

###  IV.1 $3$-qubit state transfer

Let us now further illustrate our proposal for $3$-qubit entangled state
transfer, i.e., $\mathcal{N}=3$ as an example. In this case, the qubit chain
contains $\mathcal{L}=4M-1$ qubits and the corresponding Hamiltonian is given
by Eq. (27) with $\mathcal{N}=3$ as

$\displaystyle H$ $\displaystyle=$
$\displaystyle\stackrel{{\scriptstyle[}}{{m}}=1]{M-1}{\sum}\left(v\sigma_{A_{1,m}}^{+}\sigma_{B_{m}}^{-}+w\sigma_{A_{1,m+1}}^{+}\sigma_{B_{m}}^{-}+\textrm{H.c.}\right)$
(28) $\displaystyle+$
$\displaystyle\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}\left(g\sigma_{A_{1,m}}^{+}\sigma_{A_{2,m}}^{-}+g\sigma_{A_{2,m}}^{+}\sigma_{A_{3,m}}^{-}+\textrm{H.c.}\right).$

The edge states are only localized in all $A$-type qubits (i.e., $A_{1,m}$-,
$A_{2,m}$-, and $A_{3,m}$-type) and can be expanded as

$|\varPsi_{\rm
edge}\rangle=\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}\lambda^{m}\left(a\sigma_{A_{1,m}}^{+}+b\sigma_{A_{2,m}}^{+}+c\sigma_{A_{3,m}}^{+}\right)|G\rangle.$
(29)

Substituting this equation into eigen-energy function $H|\varPsi_{\rm
edge}\rangle=E|\varPsi_{\rm edge}\rangle$, we have

$\displaystyle E$
$\displaystyle\left(a\sigma_{A_{1,m}}^{+}+b\sigma_{A_{2,m}}^{+}+c\sigma_{A_{3,m}}^{+}\right)|G\rangle$
$\displaystyle=g\left(b\sigma_{A_{1,m}}^{+}+a\sigma_{A_{2,m}}^{+}+c\sigma_{A_{2,m}}^{+}+b\sigma_{A_{3,m}}^{+}\right)|G\rangle$
$\displaystyle+a\left(v\sigma_{B_{m}}^{+}+w\lambda\sigma_{B_{m}}^{+}\right)|G\rangle.$
(30)

Here, for this special case of 3-qubit entangled state transfer, we have
assumed that the coupling constant $g_{1}$ between $A_{1,m}$-qubit and
$A_{2,m}$-qubit equals to that $g_{2}$ between $A_{2,m}$-qubit and
$A_{3,m}$-qubit, i.e., $g_{1}=g_{2}=g$.

It is straightforward to obtain $\lambda=-v/w$, and the coefficients $a$, $b$,
and $c$ satisfy the following eigen-equation

$\left(\begin{array}[]{ccc}0&g&0\\\ g&0&g\\\
0&g&0\end{array}\right)\left(\begin{array}[]{c}a\\\ b\\\
c\end{array}\right)=E\left(\begin{array}[]{c}a\\\ b\\\ c\end{array}\right).$
(31)

Solving Eq. (31), we can obtain three eigenvalues $E_{\pm}=\pm\sqrt{2}g$ and
$E_{0}=0$, corresponding to three eigenstates
$\left(1/2,\pm\sqrt{2}/2,1/2\right)$ and
$\left(1/\sqrt{2},0,-1/\sqrt{2}\right)$, respectively. These three eigenvalues
are also eigenenergies of three edge states. Thus, three edge states are
constructed by the eigenstates, i.e.,
$|\chi_{m,\pm}\rangle=\left(|\mathcal{A}_{1,m}\rangle\pm\sqrt{2}|\mathcal{A}_{2,m}\rangle+|\mathcal{A}_{3,m}\rangle\right)/2$
and
$|\chi_{m,0}\rangle=\left(|\mathcal{A}_{1,m}\rangle-|\mathcal{A}_{3,m}\rangle\right)/\sqrt{2}$
associated with the $m$th unit cell. As shown in Eq. (29), then these edge
states can be given as

$\displaystyle|\varPsi_{\pm}\rangle$
$\displaystyle=\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}\lambda^{m}\left(\frac{\sigma_{A_{1,m}}^{+}\pm\sqrt{2}\sigma_{A_{2,m}}^{+}+\sigma_{A_{3,m}}^{+}}{2}\right)|G\rangle\,$
$\displaystyle|\varPsi_{0}\rangle$
$\displaystyle=\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}\lambda^{m}\left(\frac{\sigma_{A_{1,m}}^{+}-\sigma_{A_{3,m}}^{+}}{\sqrt{2}}\right)|G\rangle.$
(32)

When $\left|\lambda\right|\ll 1$, i.e., $v\ll w$, the edge states are mainly
localized at the left end of the chain, and when $\left|\lambda\right|\gg 1$,
i.e., $v\gg w$, the edge states are mainly localized at the right end of the
chain. In particular, when $v=0$, these edge states are written as
$|L_{\pm}\rangle=|\chi_{1,\pm}\rangle|gg\cdots g\rangle$ and
$|L_{0}\rangle=|\chi_{1,0}\rangle|gg\cdots g\rangle$, supported by the
transport qubits on the left end of the chain. However, when $w=0$, the edge
states are $|R_{\pm}\rangle=|gg\cdots g\rangle|\chi_{M,\pm}\rangle$ and
$|R_{0}\rangle=|gg\cdots g\rangle|\chi_{M,0}\rangle$, supported by the
transport qubits on the right end of the chain.

Figure 5:  $3$-qubit state transfer with extended SSH$4$ chain. The parameters
are taken as $J=5g$ and $g/2\pi=10$MHz. (a) The energy spectrum of a qubit
chain with 6 unit cells when $\omega t$ changes from $0$ to $\pi$. Edge states
can be transferred from left edge (red) to right edge (green) along the dashed
energy levels. (b) Schematics for eigenstate distribution of the SSH$4$ chain
in a topological nontrivial regime ($v\ll w$) when $\omega t=\pi/6$. Four bulk
bands (blue dots), are divided by three edge states (red dots). (c) Target-
state occupation probabilities
$F(nT)=\left|\langle\varPsi_{F}|\varPsi\left(nT\right)\rangle\right|$ as a
function of the number of evolution circles $n$. Each point represents a
complete adiabatic evolution from $t=0$ to $t=\pi/\omega$. Various colors from
red to green represent different lengths of the chain, where $M=2,4,6,8$. (d)
Time evolution of the whole qubit-chain state with initial state prepared to a
three-qubit W state
$\left(|\mathcal{A}_{1,1}\rangle+|\mathcal{A}_{2,1}\rangle+|\mathcal{A}_{3,1}\rangle\right)/\sqrt{3}$
when the total evolution time is $20$ evolution circles $n$. The color from
dark-red to bright-yellow represents the population distribution of the state
on each qubit site. (e) The average fidelities of 3-qubit W-state transfer
with different coupling disorder strength. $M=2,4,6,8$ for each curve,
separately. (f) The average fidelities of $3$-qubit W-state transfer with
different execution time disorder.

As discussed above, we can change the coupling constants slowly as
$v=J\left[1-\cos\left(\omega t\right)\right]$ and $w=J\left[1+\cos\left(\omega
t\right)\right]$, then the edge states of the system will adiabatically evolve
from the left end to the right end of the chain. As shown in Fig. 5(a), using
a qubit chain with $M=6$ unit cells (i.e., $23$ qubits) as an example, we plot
the variations of the instantaneous eigeneneries from $t=0$ to $\pi/\omega$.
We find that there are four bands of bulk states, separated by three
topological edge states. The bulk bands outside of the edge states are not
degenerate, however, each bulk band inside the gap of the edge states has
five-fold degenerate. This has been further illustrated in Fig. 5(b) by
arranging $23$ eigenenstates by the corresponding eigenenergies (from the
lowest to the highest), when the time $t$ is taken as $t=\pi/6\omega$. Figure
5(a) also shows if the system is prepared to one of the left edge states
($|L_{\pm}\rangle$ and $|L_{0}\rangle$) at $t=0$, then the state will evolve
to the corresponding right edge state ($|R_{\pm}\rangle$ or $|R_{0}\rangle$)
when $t_{f}=\pi/\omega$.

More generally, if the initial state is prepared to an arbitrary 3-qubit
entangled state at the left end of the chain as

$|\varPsi_{\rm
in}\rangle=\left(\alpha\sigma_{A_{1,1}}^{+}+\beta\sigma_{A_{2,1}}^{+}+\gamma\sigma_{A_{3,1}}^{+}\right)|G\rangle,$
(33)

which can be rewritten as

$|\varPsi_{\rm
in}\rangle=\frac{\alpha+\sqrt{2}\beta+\gamma}{2}|L_{+}\rangle+\frac{\alpha-\sqrt{2}\beta+\gamma}{2}|L_{-}\rangle+\frac{\alpha-\gamma}{\sqrt{2}}|L_{0}\rangle,$
(34)

by using the left edge states, then the state will adiabatically evolve to

$\displaystyle|\varPsi\left(t\right)\rangle$ $\displaystyle=$
$\displaystyle\frac{\alpha+\sqrt{2}\beta+\gamma}{2}|\varPsi_{+}\left(t\right)\rangle
e^{-i\int_{0}^{t}E_{+}dt^{\prime}}$ (35) $\displaystyle+$
$\displaystyle\frac{\alpha-\sqrt{2}\beta+\gamma}{2}|\varPsi_{-}\left(t\right)\rangle
e^{-i\int_{0}^{t}E_{-}dt^{\prime}}$ $\displaystyle+$
$\displaystyle\frac{\alpha-\gamma}{\sqrt{2}}|\varPsi_{0}\left(t\right)\rangle
e^{-i\int_{0}^{t}E_{0}dt^{\prime}},$

at the moment $t$. As we learn from the case of the two-qubit state transfer,
$E_{\pm}=\pm\sqrt{2}g$ and $E_{0}=0$ here are still constant during the
adiabatic protocol, thus the final state at $t_{f}=\pi/\omega$ is

$\displaystyle\left|\varPsi_{f}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{\alpha+\sqrt{2}\beta+\gamma}{2}|R_{+}\rangle
e^{-i2\pi\frac{g}{\sqrt{2}w}}$ (36) $\displaystyle+$
$\displaystyle\frac{\alpha-\sqrt{2}\beta+\gamma}{2}|R_{-}\rangle
e^{i2\pi\frac{g}{\sqrt{2}w}}$ $\displaystyle+$
$\displaystyle\frac{\alpha-\gamma}{\sqrt{2}}|R_{0}\rangle.$

Again, let us consider the evolution time to be exact integral multiples of
the dynamical period $T=2\pi/\sqrt{2}g$, i.e., $t_{f}/T=g/\sqrt{2}\omega=n$
($n\gg 1$), then the final state becomes

$|\varPsi_{F}\rangle=\left(\alpha\sigma_{A_{1,M}}^{+}+\beta\sigma_{A_{2,M}}^{+}+\gamma\sigma_{A_{3,M}}^{+}\right)|G\rangle.$
(37)

Clearly then, as time $t$ changes from $0$ to $\pi/\omega$, arbitrary
$3$-qubit entangled state can be transported from the left end to the right
end of the chain.

Same as the case of $2$-qubit state transfer, we choose different numbers of
evolution circles to examine the adiabaticity. In Fig. 5(c), target-state
occupation probabilities
$F(t)=\left|\langle\varPsi_{F}|\varPsi\left(t\right)\rangle\right|$ are
plotted as a function of the number of evolution circles when a W-state is
transferred from the left edge to the right one for different lengths of the
chain. We find that the evolution circles to achieve high fidelity increase
with the length $M$ of the chain when the qubit number in the unit cell is
given. For example, Fig. 5(c) shows that $20$ evolving circles are required to
achieve fidelity one when $M=8$, however $5$ evolving circles are enough to
achieve fidelity one when $M=2$. For a specific case shown in Fig. 5(d), the W
state is shown to be transferring from the left to the right with a rapid
oscillation within the execution time (20 evolution circles). In Fig. 5(e), we
have also evaluated the average fidelities of the 3-qubit W-state transfer
with different disorder strengths $\xi$ for the coupling strengths. There is
also a plateau at $\xi\in\left[10^{-3}g,10^{-1}g\right]$ for each qubit chain.
However, different from the two-qubit transfer, these plateaus are pinned at
different values of fidelity with the increase of the qubit number. We find
that the average fidelity for all the qubit chains considered here is far
beyond $96\%$ so long as the disorder strength is $\xi<0.1g$. Meanwhile, the
effect of the disorder of the time evolution is presented in Fig. 5(f). The
average fidelity $\mathcal{F}$ is also good enough for state transfer as the
disorder strength $\eta$ is less than $0.01T$. Figs. 5(e) and (f) clearly show
that our proposal is also promising for transferring 3-qubit state along a
long qubit chain.

###  IV.2 $\mathcal{N}$-qubit state transfer

Figure 6:  (a) Spectrum of the qubit chain for 4-qubit state transfer. Here,
$M=5$ and $\mathcal{L}=24$. The red-dashed lines represent the edge states and
the black-solid lines represent the bulk states. (b) Schematics for
eigenstates distribution corresponding to (a) in a topologically nontrivial
regime ($v\ll w$) when $\omega t=\pi/6$. Four red dots denote four edge
states. The blue dots denote the bulk bands. The four blue dots in line
between two red dots denote a bulk band with $4$-fold degenerate. (c) Spectrum
of the qubit chain for 5-qubit state transfer. Here, $M=5$ and
$\mathcal{L}=29$. (d) Schematics for eigenstates distribution corresponding to
(c) in a topologically nontrivial regime ($v\ll w$) when $\omega t=\pi/6$.
Five red dots denote five edge states. The blue dots denote the bulk bands.
The four dots in line between two red dots denote a bulk band with $4$-fold
degenerate.

We have extended our protocol from 2-qubit state transfer to 3-qubit state
transfer. Analogously, we can extend our proposal to $\mathcal{N}$-qubit state
case. Similar to Eq. (3) for the two-qubit entangled state transfer, the edge
states corresponding to the Hamiltonian in Eq. (27) do not occupy the mediated
qubits $B$ and can be expanded as

$|\varPsi_{\rm
edge}\rangle=\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}\lambda^{m}\left(\chi_{1}\sigma_{A_{1,m}}^{+}+\cdots+\chi_{\mathcal{N}}\sigma_{A_{\mathcal{N},m}}^{+}\right)|G\rangle.$
(38)

Substituting Eq. (27) and Eq. (38) into the Schrödinger equation
$H|\varPsi_{\rm edge}\rangle=E|\varPsi_{\rm edge}\rangle$, then we can have

$\displaystyle E$
$\displaystyle\left(\chi_{1}\sigma_{A_{1,m}}^{+}+\chi_{2}\sigma_{A_{2,m}}^{+}+\cdots+\chi_{\mathcal{N}}\sigma_{A_{\mathcal{N},m}}^{+}\right)|G\rangle$
$\displaystyle=\left(g_{1}\chi_{2}\sigma_{A_{1,m}}^{+}+\cdots+g_{\mathcal{N}-1}\chi_{\mathcal{N}}\sigma_{A_{\mathcal{N}-1,m}}^{+}\right)|G\rangle$
$\displaystyle+\left(g_{1}\chi_{1}\sigma_{A_{2,m}}^{+}+\cdots+g_{\mathcal{N}-1}\chi_{\mathcal{N}-1}\sigma_{A_{\mathcal{N},m}}^{+}\right)|G\rangle$
$\displaystyle+\chi_{1}\left(v\sigma_{B_{m}}^{+}+w\lambda\sigma_{B_{m}}^{+}\right)|G\rangle.$
(39)

From Eq. (39), it is straightforward to get $\lambda=-v/w$, and the
coefficients $\chi_{1}$, $\chi_{2},\cdots,\chi_{\mathcal{N}}$ satisfy the
following equation

$\left(\begin{array}[]{ccccc}0&g_{1}&0&0&0\\\ g_{1}&0&g_{2}&0&0\\\
0&g_{2}&0&\ddots&0\\\ 0&0&\ddots&\ddots&g_{\mathcal{N}-1}\\\
0&0&0&g_{\mathcal{N}-1}&0\end{array}\right)\left(\begin{array}[]{c}\chi_{1}\\\
\chi_{2}\\\ \chi_{3}\\\ \vdots\\\
\chi_{\mathcal{N}}\end{array}\right)=E\left(\begin{array}[]{c}\chi_{1}\\\
\chi_{2}\\\ \chi_{3}\\\ \vdots\\\ \chi_{\mathcal{N}}\end{array}\right).$ (40)

Solving this eigen-equation, we can obtain the eigenenergies of the edge
states as $E_{1},\cdots,E_{\mathcal{N}}$. The explicit eigenstates can also be
derived, and then edge states can be obtained. Similar to Eq. (10), when the
time $t$ is adiabatically changed from $0$ to $\pi/\omega$, the coupling
constant $v$ ($w$) is changed from $2J$ ($0$) to $0$ ($2J$), and the edge
states initially at the left end of the chain can be adiabatically transferred
to the right end. We know that an arbitrary $\mathcal{N}$-qubit entangled
state can be represented by these edge states, thus $\mathcal{N}$-qubit
entangled state can also be perfectly transferred from the left end to the
right end of the chain if the total evolution time $t_{f}=\pi/\omega$ is exact
multiple of all the dynamical periods corresponding to all eigenenergies
$E_{1},\cdots,E_{\mathcal{N}}$. That is, there must be a least common period
of all dynamical periods
$T_{1}=2\pi/E_{1},\cdots,T_{\mathcal{N}}=2\pi/E_{\mathcal{N}}$ corresponding
to the $\mathcal{N}$ edge states. This common period can only exist for
specific set of parameters $\left\\{g_{1},\cdots,g_{\mathcal{N}-1}\right\\}$.

It seems that the generalization is straightforward, however the main problem
is how to engineer the proper series of parameters
$\left\\{g_{1},\cdots,g_{\mathcal{N}-1}\right\\}$ to get a common period. It
is highly challenging to find a general solution for any $\mathcal{N}$,
because the solution varies case by case. However, for given qubit states, we
can always engineer the coupling constants between the transport qubits such
that these states can be transferred through the extended SSH chain. Below,
two examples for $\mathcal{N}=4$ and $\mathcal{N}=5$ are further shown in Fig.
6. At $\mathcal{N}=4$, the parameters $\left\\{g_{1},g_{2},g_{3}\right\\}$ can
be set as $\left\\{\sqrt{2}g,g,\sqrt{2}g\right\\}$ and the corresponding
eigenenergies are $\left\\{2g,g,-g,-2g\right\\}$. In Figs. 6(a) and (b), we
show the spectrum of the qubit chain for $4$-qubit state transfer with a
number of $24$ qubits. Five bulk bands are divided by four edge states, and
the three bulk bands between edge states are $4$-fold degenerate. The least
common oscillation period of these edge states is $2\pi/g$, which can be taken
as the ideal time evolution circle. At $\mathcal{N}=5$, the parameters
$\left\\{g_{1},g_{2},g_{3},g_{4}\right\\}$ can be set as
$\left\\{g,2g,2g,g\right\\}$ and the corresponding eigenenergies are
$\left\\{3g,g,0,-g,-3g\right\\}$. The spectrum of the qubit chain in this case
is plotted in Figs. 6(c) and (d) for the total number $29$ of the qubits in
the chain. Six bulk bands are divided by five edge states, and the four bulk
bands between edge states are $4$-fold degenerate. The least common period is
also $2\pi/g$. Similarly, for the case of $\mathcal{N}$-qubit state transfer,
we can find a proper set of parameters to get a common period as the time
evolution circle. By use of such time evolution cycles, the involved edge
states will not suffer from mutual dynamical phase differences at the end of
our adiabatic protocol, and hence an arbitrary $\mathcal{N}$-qubit entangled
state can be transferred from the left end to the right end.

##  V discussions

###  V.1 General discussions

We have proposed a QST approach along an extended SSH qubit chain. However,
several issues should be further discussed. First, we consider only the
neighbor-couplings and on-site potentials are not included as shown in Eq.
(1). This is because the on-site potentials only result in a global dynamical
phase if all the qubits are tuned to resonate with each other, and thus this
global phase can be dropped out. Second, we note that the gap between the edge
states and the bulk states decreases with the increase in length of the qubit
chain. This indicates that the adiabatic condition is highly demanded for very
large systems. However, as shown in Fig. 5(c), for an extended SSH4 chain
consisting of $8$ unit cells, the adiabatic condition is still well met for an
execution time of only 20 evolution circles. When the length of the chain
becomes much larger, one possible solution is hinted in Ref. (Tan  _et al._ ,
2020). That is, our QST protocol realized by one-step adiabatic evolution can
be decomposed into multi-step process to achieve a better performance.

To verify the topological protection of our QST protocol, one may compare our
approach with QST protocols independent of topology. Using mirror symmetry,
one can think of an obvious non-topological QST protocol in a qubit chain as
follows. At the beginning, the leftmost qubit is decoupled from other part of
the chain, and at the end the couplings of the qubit chain is slowly changed
to the reversed form, i.e., the rightmost qubit is decoupled with other
qubits. With this adiabatic process, the quantum state initially prepared to
the leftmost qubit can be transferred from the left end to the right end. Ref.
(Palaiodimopoulos  _et al._ , 2021) has shown that such topologically-
unrelated QST protocol is not robust to disorder. Hence, our topology-based
protocol does have an advantage, accounting for the fairly good fidelity
presented above in the presence of disorder.

###  V.2 Discussions for implementations using superconducting qubit circuits

In principle, our proposal can be implemented in various platforms, e.g., cold
atoms (Du  _et al._ , 2016), trapped ions (Harty  _et al._ , 2014), or coupled
waveguides (Shen  _et al._ , 2020; Chen  _et al._ , 2021b). However, with the
significant development in recent years, superconducting qubit circuits, e.g.,
transmon or Xmon qubits (Chen  _et al._ , 2014), seem to be a more promising
platform (Koch  _et al._ , 2007). Also, a fast and high-fidelity transfer of
arbitrary single-qubit state in a chain of superconducting qubits has been
achieved in experiment recently (Li  _et al._ , 2018).

The good scalability and flexible tunability for the couplings make our
extended SSH chain easy to be realized by the superconducting qubit circuits.
In particular, the coupling strengths can be tuned from topologically trivial
to non-trivial regime. Thus topological and non-topological phenomena can be
studied in one quantum circuits. Moreover, the coherent time is a very
important for realizing our proposal. As shown in Appendix A, the coupling
coefficient $g$ can be chosen as $g/2\pi=10$Mhz by using the parameters of
current superconducting qubit circuits for realizing our proposal. Therefore,
for two-qubit state transfer, one evolution circle can be $T=2\pi/g=0.1\mu s$.
The total evolution time is 10 evolution circles, i.e., $t_{f}=1\mu s$. For
three-qubit state transfer, one evolution circle can be
$T=2\pi/\sqrt{2}g=0.0707\mu s$. The total evolution time is 20 evolution
circles, i.e., $t_{f}=1.414\mu s$. In superconducting qubit circuits, the
coherence time of single qubit is about $10\sim 100\mu s$ (Paik  _et al._ ,
2011; Novikov  _et al._ , 2016), which is much longer than the adiabatic time
in our proposal. Meanwhile, steady topological edge states have already been
observed in superconducting qubit circuits, which can last more than $1\mu s$
(Cai  _et al._ , 2019). Therefore, the decoherence effect is not expected to
be troublesome in our protocol.

To make sure that the adiabatic evolution time for our proposal is indeed
integral multiple of the evolution circle, one needs to measure the exact
value of $g$, which is equivalent to determining the eigenenergy of edge
states. This can be achieved by the reflection spectrum of a weak probe light
through a waveguide coupled to the extended SSH chain (see Appendix D). The
reflection peaks of the input weak signal can reveal the energy spectrum of
the qubit chain with appropriate parameters, and the value of coupling $g$ can
be obtained with the energy shift between the edge states.

##  VI Conclusion

In summary, we have proposed an experimentally feasible approach for
transferring arbitrary entangled state through an extended SSH chain. The
entangled states are encoded in the edge states of a class of extended SSH
chains, and then they are transported via an adiabatic protocol. Due to the
topological protection of the edge states, our protocol is robust against the
the temporal noise caused by the imperfection in the control field. We have
numerically confirmed this robustness against two kinds of disorders, i.e, the
coupling strength disorder and the execution time disorder. Compared with most
contemporary studies realizing QST in qubit chains, this work represents an
exciting advance that a general scenario is proposed to achieve QST of
arbitrary $\mathcal{N}$-qubit entangled state. Our proposal is easy to be
realized by using superconducting qubit circuits and the parameters required
in our protocol are estimated according to the recent experiments. Given the
feasibility and tunability of the proposed protocal, our idea can also be
extended for realizing QST in two dimensional quantum networks.

##  VII ACKNOWLEDGMENTS

We thank Wei Nie for helpful discussions. J.G. acknowledges funding support by
the Singapore NRF Grant No. NRF-NRFI2017-04 (WBS No. R-144-000-378- 281).
Y.X.L. is supported by the Key-Area Research and Development Program of
GuangDong Province under Grant No. 2018B030326001, the National Basic Research
Program (973) of China under Grant No. 2017YFA0304304, and the NSFC under
Grant No. 11874037.

##  Appendix A Xmon qubit chain with tunable couplings

For our state transfer protocol, the feasibility comes from the precise tuning
of the coupling between qubits in the topological qubit chain. We manage to
realize such setup with superconducting qubits. A superconducting qubit chain
with tunable couplings is presented in Fig. 1(a). The details of the coupler
circuit are schematically shown in Fig. 1(b), where two Xmon qubits are
coupled by a tunable Josephson junction coupler. Such coupling scheme has been
experimentally realized in Ref. (Chen  _et al._ , 2014) and theoretically
analyzed in Ref. (Geller  _et al._ , 2015).

Figure 7:  (a) The change of $\delta$ versus $\phi_{\text{ext}}$. The
parameters we use here are $L_{\rm g}=300$ pH and $L_{\rm T}=1$ nH. (b)
Coupling strength varies with $\phi_{\text{ext}}$. The parameters are $L_{\rm
J}=8$ nH and $\omega_{q}=5.5$ GHz. Other parameters are the same as above.

The junction connecting two qubits provides a tunable effective inductance
$L_{\text{eff}}$ to tune the coupling, and a magnetic flux bias $\Phi_{\rm
ext}$ is used to tune the coupler’s effective inductance
$L_{\text{eff}}=L_{\text{T}}/\cos\delta$ (Geller  _et al._ , 2015), where
$\delta$ is the phase difference across the coupler. For the coupler loop, we
have $\Phi=(\Phi_{0}/2\pi)\delta$, where $\Phi$ is the total magnetic flux and
$\Phi_{0}=h/2e$. $L_{\text{T}}=\Phi_{0}/2\pi I_{c}$ is the zero-bias
inductances of the Josephson coupler and $I_{c}$ is the critical current of
the coupler junction. The circuit flux has the relation $\Phi=\Phi_{\rm
ext}-2L_{\text{g}}I_{c}\sin\delta$ and therefore we can have

$\phi_{\rm
ext}=\delta+\left(\frac{2L_{\text{\text{g}}}}{L_{\text{T}}}\right)\sin\delta$
(41)

where $\phi_{\rm ext}=2\pi\Phi_{\rm ext}/\Phi_{0}$. In the the weakly coupled
limit the effective coupling strength is approximately (Chen  _et al._ , 2014)

$g=-\frac{\omega_{q}}{2}\frac{M}{L_{\textrm{J}}+L_{\textrm{g}}},$ (42)

where $M=L_{\textrm{g}}^{2}/\left(2L_{\textrm{g}}+L_{\textrm{eff}}\right)$ is
the mutual inductance and $\omega_{q}$ is the qubit frequency. Thus the
coupling strength can be finally given by (Geller  _et al._ , 2015)

$\displaystyle g$ $\displaystyle=$
$\displaystyle-\frac{L_{\text{g}}^{2}\cos\delta}{2\left(L_{\text{J}}+L_{\text{g}}\right)\left(L_{\text{T}}+2L_{\text{g}}\cos\delta\right)}\omega_{q}.$
(43)

In Fig. 7(a), we present the relation between $\Phi_{\rm ext}$ and $\delta$.
The phase $\delta$ can be tuned from $-2\pi$ to $2\pi$ when the external
magnetic flux $\Phi_{\text{ext}}$ is continuously changed. In Fig. 7(b), we
show how the coupling between qubits can be tuned with $\Phi_{\text{ext}}$. As
$\phi_{\text{ext}}$ is changing from $-2\pi$ to $2\pi$, the coupling can be
tuned from $-20$MHZ to $60$MHZ, which meet the requirement of the system
parameters to realize our proposal.

##  Appendix B A straightforward diagram for exact solution of the edge states

As for standard SSH model, there are only approximated solutions for
hybridized edge states (Asbóth  _et al._ , 2016), but for our imperfect SSH
model, one can obtain the exact solution of the edge states (Mei  _et al._ ,
2018). To illustrate this process straightforwardly, let us rewrite the qubit
array with specific set of basis. As shown in Fig. 1, we here analyze
$2$-qubit state transfer, and the formula can be extended to $3$-qubit and
$\mathcal{N}$-qubit cases.

Figure 8: Schematic diagram of the renormalization process on qubit array with
orthogonal basis of transport qubits. (a) For each unit cell of the qubit
chain, the renormalization leads to two on-site potentials as $g$ and $-g$,
due to the coupling strength $g$ between the transport qubits $A_{1,m}$ and
$A_{2,m}$. (b) Renormalization of the whole qubit array. All the transport
qubits are decoupled with each other, and the hopping amplitudes are staggered
through the mediated qubits $B_{m}$. The qubit array now can be distinguished
into two branches by the on-site energy: the upper branch with the hybrid
qubit $A_{1,m}+A_{2,m}$ and the lower branch with the hybrid qubit
$A_{1,m}-A_{2,m}$.

For 2-qubit state transfer, the Hamiltonian of the qubit chain is

$\displaystyle H$
$\displaystyle=\stackrel{{\scriptstyle[}}{{m}}=1]{M-1}{\sum}\left(v\sigma_{A_{1,m}}^{+}\sigma_{B_{m}}^{-}+w\sigma_{A_{1,m+1}}^{+}\sigma_{B_{m}}^{-}+\textrm{H.c.}\right)$
$\displaystyle+\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}\left(g\sigma_{A_{1,m}}^{+}\sigma_{A_{2,m}}^{-}+\textrm{H.c.}\right),$
(44)

same as Eq. (1). As shown in Fig. 8(a), for a single unit cell of the qubit
chain, the Hamiltonian is

$\displaystyle H_{\text{cell}}$
$\displaystyle=v\sigma_{A_{1,m}}^{+}\sigma_{B_{m}}^{-}+w\sigma_{A_{1,m+1}}^{+}\sigma_{B_{m}}^{-}+\textrm{H.c.}$
$\displaystyle+g\sigma_{A_{1,m}}^{+}\sigma_{A_{2,m}}^{-}+g\sigma_{A_{1,m+1}}^{+}\sigma_{A_{2,m+1}}^{-}+\textrm{H.c.}.$
(45)

In the single-excitation subspace, the Hamiltonian in Eq.(45) can be rewritten
as

$\displaystyle H_{\text{cell}}$
$\displaystyle=v|\mathcal{A}_{1,m}\rangle\langle\mathcal{B}_{m}|+w|\mathcal{A}_{1,m+1}\rangle\langle\mathcal{B}_{m}|+\textrm{H.c.}$
$\displaystyle+g|\mathcal{A}_{1,m}\rangle\langle\mathcal{A}_{2,m}|+g|\mathcal{A}_{1,m+1}\rangle\langle\mathcal{A}_{2,m+1}|+\textrm{H.c.}.$
(46)

The term of
$g|\mathcal{A}_{1,m}\rangle\langle\mathcal{A}_{2,m}|+\textrm{H.c.}$ in matrix
form is $\left(\begin{array}[]{cc}0&g\\\ g&0\end{array}\right)$, and can be
renomorlized as $\left(\begin{array}[]{cc}g&0\\\ 0&-g\end{array}\right)$ with
basis of
$|\chi_{m,+}\rangle=\left(|\mathcal{A}_{1,m}\rangle+|\mathcal{A}_{2,m}\rangle\right)/\sqrt{2}$
and
$|\chi_{m,-}\rangle=\left(|\mathcal{A}_{1,m}\rangle-|\mathcal{A}_{2,m}\rangle\right)/\sqrt{2}$.
In the basis
{$|\mathcal{\chi}_{m,+}\rangle,|\mathcal{\chi}_{m,-}\rangle,|\mathcal{B}_{m}\rangle$},
the Hamiltonian $H_{\text{cell}}$ of the single unit cell can be rewritten as

$\displaystyle H_{\text{cell}}^{R}$ $\displaystyle=$ $\displaystyle
v\frac{|\chi_{m,+}\rangle+|\chi_{m,-}\rangle}{\sqrt{2}}\langle\mathcal{B}_{m}|+\textrm{H.c.}$
(47) $\displaystyle+$ $\displaystyle
w\frac{|\chi_{m+1,+}\rangle+|\chi_{m+1,-}\rangle}{\sqrt{2}}\langle\mathcal{B}_{m}|+\textrm{H.c.}$
$\displaystyle+$ $\displaystyle
g|\chi_{m+1,+}\rangle\langle\chi_{m+1,+}|-g|\chi_{m+1,-}\rangle\langle\chi_{m+1,-}|$
$\displaystyle+$ $\displaystyle
g|\chi_{m,+}\rangle\langle\chi_{m,+}|-g|\chi_{m,-}\rangle\langle\chi_{m,-}|.$

The coupling between $A_{1,1}$ and $A_{2,1}$ in the unit cell leads to two on-
site potentials $g$ and $-g$. The total Hamiltonian of the qubit chain in such
new basis is

$\displaystyle H^{R}$ $\displaystyle=$
$\displaystyle\stackrel{{\scriptstyle[}}{{m}}=1]{M-1}{\sum}\left(v\frac{|\chi_{m,+}\rangle+|\chi_{m,-}\rangle}{\sqrt{2}}\langle\mathcal{B}_{m}|+\textrm{H.c.}\right)$
(48) $\displaystyle+$
$\displaystyle\stackrel{{\scriptstyle[}}{{m}}=1]{M-1}{\sum}\left(w\frac{|\chi_{m+1,+}\rangle+|\chi_{m+1,-}\rangle}{\sqrt{2}}\langle\mathcal{B}_{m}|+\textrm{H.c.}\right)$
$\displaystyle+$
$\displaystyle\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}g\left(|\chi_{m,+}\rangle\langle\chi_{m,+}|-|\chi_{m,-}\rangle\langle\chi_{m,-}|\right).$

That is, as shown in Fig. 8(b), the renormalized qubit chain can be divided
into two different branches by on-site potential. The upper branch’s transport
qubits now are $A_{1,m}+A_{2,m}$ and lower branch’s transport qubits are
$A_{1,m}-A_{2,m}$. Two branches have the same mediated qubits $B_{m}$. The
edge states for the upper and lower branches can be written as

$\displaystyle|\varPsi_{\pm}\rangle$
$\displaystyle=\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}\lambda^{m}|\chi_{m,\pm}\rangle$
$\displaystyle=\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}\lambda^{m}\left(\frac{|\mathcal{A}_{1,m}\rangle\pm|\mathcal{A}_{2,m}\rangle}{\sqrt{2}}\right),$
(49)

which is the same as Eq. (7). The extension of this analysis for $3$-qubit and
$\mathcal{N}$-qubit state transfer is straightforward, thus we will not
elaborate here.

##  Appendix C Topological invariant for our extended SSH model

Figure 9:  Topological invariant and edge states for our extended SSH model.
(a) SSH model with an even number of qubits. When $v<w$, the qubits at two end
are insulated and edge states appear. When $v>w$, all qubits are included in
bulk states and edge states disappear. (b) SSH model with an odd number of
qubits. we can see the system always has an edge state whether if $v<w$ or
$v>w$. (c) Energy spectrum for extended SSH model with complete $3M$ qubits.
Two edge states are denoted with blue- and red- lines. Topological phase
transition occurs with the increasing of $v$, and edge states join the bulk
band when $v\gg w$. (d) Energy spectrum for our extended SSH model with $3M-1$
qubits, i.e., lacking of one qubit at the right end of the chain. The system
always has two edge states with the variation of couplings. (e) Winding number
in $d_{x}$-$d_{y}$ plane. When $v<w$ ($v>w$), the centre of the integral
circle is located at $v$ ($w$) on $d_{x}$ axis. (f) Edge states for our
extended SSH model, i.e.,
$\left(|\mathcal{A}_{1,1}\rangle\pm|\mathcal{A}_{2,1}\rangle\right)/\sqrt{2}$.

In the standard SSH model with an even number of qubits, edge states appear
when the coupling strength at the edge is weaker than the coupling at bulk. As
shown in Fig. 9(a), if $v<w$, the qubits at two ends are isolated from bulk
and form two edges. If $v>w$, all qubits are included in bulks. For our
extended SSH$3$ model, with $g=0$, sublattices $A_{1,m}$ and $B_{m}$ are
decoupled from $A_{2,m}$, forming the SSH model with an odd number of qubits.
The Hamiltonian of such SSH model is ($A_{1,m}$ is redefined as $A_{m}$)

$\displaystyle H_{\text{SSH}}$
$\displaystyle=\stackrel{{\scriptstyle[}}{{m}}=1]{M-1}{\sum}\left(v\sigma_{A_{m}}^{+}\sigma_{B_{m}}^{-}+w\sigma_{A_{m+1}}^{+}\sigma_{B_{m}}^{-}+\text{H.c.}\right).$
(50)

As shown in Fig. 9(b), if $v<w$, the weaker coupling strength is $v$ at the
left edge. Otherwise, if $w<v$, the weaker coupling strength is $w$ at the
right edge.

For a standard SSH model which can be obtained by setting $g=0$ of the SSH$3$
model, edge states are supported by the topological invariant defined in the
phase space. We first assume $v<w$ and the unit cell is $(A_{m},B_{m})$. Due
to the translation invariance of the bulk, we can make Fourier transforms to
the vectors $|\mathcal{A}_{m}\rangle$ and $|\mathcal{B}_{m}\rangle$ as

$|\mathcal{A}_{k}\rangle=\frac{1}{\sqrt{M}}\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}e^{imk}|\mathcal{A}_{m}\rangle,$
(51)
$|\mathcal{B}_{k}\rangle=\frac{1}{\sqrt{M}}\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}e^{imk}|\mathcal{B}_{m}\rangle,$
(52)

for $k\in\left\\{\delta_{k},2\delta_{k},\cdots,M\delta_{k}\right\\}$ with
$\delta_{k}=2\pi/M$. Here $k$ is the wavenumber of the first Brillouin zone
(Asbóth  _et al._ , 2016). The bulk momentum-space Hamiltonian
$H\left(k\right)$ is defined as

$H\left(k\right)=\underset{\imath,\jmath\in\left\\{\mathcal{A}_{k},\mathcal{B}_{k}\right\\}}{\sum}\langle\imath|H_{\text{SSH}}|\jmath\rangle|\imath\rangle\langle\jmath|.$
(53)

By choosing a fixed momentum $k$, we can get the matrix form of the bulk
momentum-space Hamiltonian $h\left(k\right)$ as

$h\left(k\right)=\left(\begin{array}[]{cc}0&v+we^{-ik}\\\
v+we^{ik}&0\end{array}\right).$ (54)

Thus the total Hamiltonian can be written as
$H\left(k\right)=\sum_{k}\varPsi_{k}^{\dagger}h\left(k\right)\varPsi_{k}$,
with
$\varPsi_{k}^{\dagger}=\left(|\mathcal{A}_{k}\rangle,|\mathcal{B}_{k}\rangle\right)$
and we have

$h\left(k\right)=d_{x}\left(k\right)\sigma_{x}+d_{y}\left(k\right)\sigma_{y},$
(55)

where $d_{x}\left(k\right)=v+w\cos k$, $d_{y}\left(k\right)=w\sin k$. The
winding number as the topological invariant hence can be defined in
$d_{x}$-$d_{y}$ plane as

$\text{Winding}=\frac{1}{2\pi
i}\int_{-\pi}^{\pi}\frac{d}{dk}\log\boldsymbol{h}\left(k\right),$ (56)

where
$\boldsymbol{h}\left(k\right)=d_{x}\left(k\right)+id_{y}\left(k\right)=v+we^{ik}$.

Meanwhile, if $w<v$, the unit cell is $(B_{m},A_{m+1})$. Thus the bulk
momentum-space Hamiltonian of the system becomes

$h\left(k\right)=\left(\begin{array}[]{cc}0&w+ve^{-ik}\\\
w+ve^{ik}&0\end{array}\right),$ (57)

and we have
$\boldsymbol{h}\left(k\right)=d_{x}\left(k\right)+id_{y}\left(k\right)=w+ve^{ik}$
where $d_{x}\left(k\right)=w+v\cos k$, $d_{y}\left(k\right)=v\sin k$.

Fig. 9(e) shows that the winding number of this SSH model with an odd number
of qubits is always 1 whether $v<w$ or $v>w$. The wavefunctions of two edge
states are presented in Fig. 9(f).

##  Appendix D Energy spectrum detecting with Input-output theory

To estimate the dynamical period in our QST protocol, one can measure the
eigen-energy spectrum of edge states with input-output theory for our extended
SSH chain. Specifically, for the $2$-qubit state transfer as discussed above,
the energy difference between the two edge states of the chain is $2g$.
Meanwhile, for the $3$-qubit state transfer protocol, the energy difference
between the three edge states of the chain is $\sqrt{2}g$. To measure the
energy spectrum of the extended SSH chain, we assume that a weak probe field
is applied to the qubit chian via a transmission line. Thus, the energy
spectrum can be detected by the transmission or reflection of the probe light.
This setup is schematically shown in Fig. 10(a).

Figure 10: (a) Schematic diagram for the SSH qubit array detected by a
microwave transmission line. The transmission line here is coupled to the
qubit chain with a set of josephson couplers, and the coupling strength
between each qubit and the waveguide is tunable by the coupler. The
transmission line can be seen as a long waveguide with infinite microwave
modes. The frequency of input probe light is tunable and we can get the energy
spectrum of the qubit chain by detecting the transmission of the probe light.
(b) Inhomogeneous couplings between the waveguide and qubits for the extended
SSH chain in Fig. 2(a). (c) Energy spectrum of the qubit chain for 2-qubit
state transfer with waveguide-qubit couplings in (b). (d) Inhomogeneous
couplings between the waveguide and qubits for the extended SSH chain in Fig.
5(a). (e) Energy spectrum of the qubit chain for 3-qubit state transfer with
waveguide-qubit couplings in (d).

For the convenience of derivation, we here consider a generic qubit chain with
arbitrary qubit-qubit couplings. The Hamiltonian of this qubit chain can be
written as

$H_{\textrm{qubits}}=\sum_{x=1}^{\mathcal{L}}\omega_{x}\sigma_{x}^{+}\sigma_{x}^{-}+\sum_{x,y}^{x\neq
y}\left(\Omega_{x,y}\sigma_{x}^{+}\sigma_{y}^{-}+\textrm{H.c.}\right)$ (58)

Here $\omega_{x}$ is the frequency of the $x$th qubit, and $\Omega_{x,y}$
($x,y\in\left\\{1,\cdots,\mathcal{L}\right\\}$) is the coupling strength
between $x$th and $y$th qubits. When coupled with the transmission line, the
Hamiltonian of whole system will become

$\displaystyle H$
$\displaystyle=H_{\text{qubits}}+\omega_{c}a^{\dagger}a+\stackrel{{\scriptstyle[}}{{x}}=1]{\mathcal{L}}{\sum}G_{x}\left(\sigma_{x}^{+}a+\sigma_{x}^{-}a^{\dagger}\right)$
$\displaystyle+\varepsilon\left(ae^{i\omega_{p}t}+a^{\dagger}e^{-i\omega_{p}t}\right),$
(59)

where $\omega_{c}$ is the frequency of waveguide mode, $G_{x}$ is the coupling
strength between waveguide and $x$th qubit, and $\varepsilon$ is the driving
strength of the probe light. After rotating-wave approximation by applying a
unitary operator
$U=\exp\left(\omega_{p}a^{\dagger}at+\stackrel{{\scriptstyle[}}{{x}}=1]{\mathcal{L}}{\sum}\omega_{p}\sigma_{x}^{+}\sigma_{x}^{-}t\right)$,
we get the effective Hamiltonian as

$\displaystyle H_{eff}$
$\displaystyle=\stackrel{{\scriptstyle[}}{{x}}=1]{\mathcal{L}}{\sum}\triangle_{x}\sigma_{x}^{+}\sigma_{x}^{-}+\stackrel{{\scriptstyle[}}{{x}},y]{x\neq
y}{\sum}\left(\Omega_{x,y}\sigma_{x}^{+}\sigma_{y}^{-}+\text{H.c.}\right)$
$\displaystyle+\triangle_{c}a^{\dagger}a+\stackrel{{\scriptstyle[}}{{x}}=1]{\mathcal{L}}{\sum}G_{x}\left(\sigma_{x}^{+}a+\sigma_{x}^{-}a^{\dagger}\right)+\varepsilon\left(a+a^{\dagger}\right),$
(60)

where $\triangle_{x}=\omega_{x}-\omega_{p}$ and
$\triangle_{c}=\omega_{c}-\omega_{p}$ are the detuning frequency for qubits
and waveguide, respectively. The waveguide naturally has infinite modes and
only the mode resonating with the probe light has contribution to the
Hamiltonian, i.e., $\omega_{c}=\omega_{p}$. Thus we can have
$\triangle_{c}=0$. We here consider the low-excitation limit, so the
evolutions of operators’ average value can be derived as

$\displaystyle\dot{\left\langle a\right\rangle}$
$\displaystyle=-\left(\kappa+i\triangle_{c}\right)\left\langle
a\right\rangle-i\stackrel{{\scriptstyle[}}{{x}}=1]{\mathcal{L}}{\sum}G_{x}\left\langle\sigma_{x}^{-}\right\rangle+\varepsilon$
(61) $\displaystyle\dot{\left\langle\sigma_{x}^{-}\right\rangle}$
$\displaystyle=-i\triangle_{x}\left\langle\sigma_{x}^{-}\right\rangle-
iG_{x}\left\langle
a\right\rangle-i\stackrel{{\scriptstyle[}}{{y}}=1]{\mathcal{L}}{\sum}\Omega_{x,y}\left\langle\sigma_{y}^{-}\right\rangle$
$\displaystyle-\Gamma_{x}\left\langle\sigma_{x}^{-}\right\rangle,$ (62)

where $\kappa$ is decay rate for the waveguide and $\Gamma_{x}$ is the decay
of $x$th qubit. The evolution equations above can be written into matrix form
as

$\displaystyle\dot{\left\langle a\right\rangle}$ $\displaystyle=$
$\displaystyle-\left(\kappa+i\triangle_{c}\right)\left\langle
a\right\rangle-i\boldsymbol{G}^{T}\boldsymbol{\sigma}+\varepsilon$ (63)
$\displaystyle\dot{\left\langle\boldsymbol{\sigma}\right\rangle}$
$\displaystyle=$
$\displaystyle-i\left(\boldsymbol{\triangle}+\boldsymbol{\Omega}-i\boldsymbol{\Gamma}\right)\left\langle\boldsymbol{\sigma}\right\rangle-i\boldsymbol{G}\left\langle
a\right\rangle,$ (64)

where
$\boldsymbol{\sigma}=\left(\left\langle\sigma_{1}^{-}\right\rangle,\left\langle\sigma_{2}^{-}\right\rangle,\cdots\left\langle\sigma_{\mathcal{L}}^{-}\right\rangle\right)^{T}$,
$\boldsymbol{\triangle}=\textrm{Diag}\left(\triangle_{1},\triangle_{2},\cdots,\triangle_{\mathcal{L}}\right)$,
$\boldsymbol{\Gamma}=\textrm{Diag}\left(\Gamma_{1},\Gamma_{2},\cdots,\Gamma_{\mathcal{L}}\right)$
, $\boldsymbol{G}=\left(G_{1},G_{2},\cdots,G_{\mathcal{L}}\right)^{T}$ and
$\boldsymbol{\Omega}$ has the matrix form of the coupling terms in
$H_{\text{qubits}}$, i.e., $\boldsymbol{\Omega}_{x,y}=\Omega_{x,y}$. Here,
notice that for our measurement setup as shown in Fig. 10(a), the transmission
line only couples to the top half of the chain. This feature can be reflected
by the coupling coefficient $G_{x}$ as shown in Fig. 10(b) and (d).

When the system is stable, i.e., $\dot{\left\langle
a\right\rangle}=\dot{\boldsymbol{\sigma}}=0$, we can get the steady solution
of the equations as

$\displaystyle\boldsymbol{\sigma}$
$\displaystyle=-i\boldsymbol{M}^{-1}\boldsymbol{G}\left\langle a\right\rangle$
(65) $\displaystyle\left\langle a\right\rangle$
$\displaystyle=\frac{\varepsilon}{\kappa+i\triangle_{c}+\boldsymbol{G}^{T}\boldsymbol{M}^{-1}\boldsymbol{G}}$
(66)

where
$\boldsymbol{M}=i\boldsymbol{\triangle}+i\boldsymbol{\Omega}+\boldsymbol{\Gamma}$.
From the input-output theory, the transmission of the probe field is

$\displaystyle t_{p}$ $\displaystyle=1-\frac{\kappa}{\varepsilon}\left\langle
a\right\rangle$
$\displaystyle=1-\frac{\kappa}{\kappa+i\triangle_{c}+\boldsymbol{G}^{T}\boldsymbol{M}^{-1}\boldsymbol{G}},$
(67)

and therefore the reflection of the probe field is

$r_{p}=\frac{\kappa}{\kappa+i\triangle_{c}+\boldsymbol{G}^{T}\boldsymbol{M}^{-1}\boldsymbol{G}}.$
(68)

For the 2-qubit state transfer protocol with 14 qubits chain in Fig. 2(a), all
the qubits are resonating with each other, i.e., $\omega_{x}=\omega_{q}$ for
$x=1,2,...,\mathcal{L}$. Therefore we have
$\boldsymbol{\triangle}=\textrm{Diag}\left(\triangle_{q},\triangle_{q},\triangle_{q},\cdots,\triangle_{q}\right)$
and $\triangle_{q}=\omega_{q}-\omega_{p}$. $\boldsymbol{\Omega}$ is the matrix
form of the Hamiltonian shown in Eq. (2). The decay of the waveguide is
$\kappa=2.5g$ and all the qubits have the same decay rate as $\gamma=0.01g$.
The coupling strengths $\boldsymbol{G}$ between qubits and waveguide are shown
in Fig. 10(b). From Eq. (68) we can get the reflection spectrum of the qubit
chain for different probe light as shown in Fig. 10(c), where the vertical
coordinates represent the detuning frequency $\triangle_{q}$ and also the
eigen-energy of the system. Two edge states are clearly shown in the figure
and we can get the energy shift as $2g$. Another case we show here is the 23
qubits chain for 3-qubit state transfer, and we can also get the energy shift
as $2\sqrt{2}g$ in Fig. 10(e).

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