# Robust beam splitter with fast quantum state transfer through a topological
interface

Jia-Ning Zhang1 Jin-Xuan Han2 Jin-Lei Wu3<EMAIL_ADDRESS>Jie Song2 Yong-
Yuan Jiang1,2,4,5,6<EMAIL_ADDRESS>1Department of Optoelectronics Science,
Harbin Institute of Technology, Weihai 264209, China  2School of Physics,
Harbin Institute of Technology, Harbin 150001, China  3School of Physics and
Microelectronics, Zhengzhou University, Zhengzhou 450001, China
4Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan
030006, China  5Key Laboratory of Micro-Nano Optoelectronic Information
System, Ministry of Industry and Information Technology, Harbin 150001, China
6Key Laboratory of Micro-Optics and Photonic Technology of Heilongjiang
Province, Harbin Institute of Technology, Harbin 150001, China

###### Abstract

The Su-Schrieffer-Heeger (SSH) model, commonly used for robust state transfers
through topologically protected edge pumping, has been generalized and
exploited to engineer diverse functional quantum devices. Here, we propose to
realize a fast topological beam splitter based on a generalized SSH model by
accelerating the quantum state transfer (QST) process essentially limited by
adiabatic requirements. The scheme involves delicate orchestration of the
instantaneous energy spectrum through exponential modulation of nearest
neighbor coupling strengths and onsite energies, yielding a significantly
accelerated beam splitting process. Due to properties of topological pumping
and accelerated QST, the beam splitter exhibits strong robustness against
parameter disorders and losses of system. In addition, the model demonstrates
good scalability and can be extended to two-dimensional crossed-chain
structures to realize a topological router with variable numbers of output
ports. Our work provides practical prospects for fast and robust topological
QST in feasible quantum devices in large-scale quantum information processing.

Keywords: quantum state transfer, beam splitter, topological router.

##  I Introduction

In large-scale quantum information processing, information encoded in quantum
states needs to be transmitted in a coherent manner between different nodes
within a quantum network Cirac  _et al._ (1997); Matsukevich and Kuzmich
(2004); Christandl  _et al._ (2004); Banchi  _et al._ (2011); Wang and Clerk
(2012); Tan  _et al._ (2020). In the last few years, great efforts have been
devoted into exploring the optimal protocol for achieving efficient state
transfer in the simplest and most common one-dimensional spin-1/2 chain, and
the results can be further applied to incorporate various quantum systems such
as quantum dots Ouyang and Awschalom (2003); Greentree  _et al._ (2004); Chen
_et al._ (2012); He  _et al._ (2017); Kandel  _et al._ (2021), coupled
waveguides Perez-Leija  _et al._ (2013); Chapman  _et al._ (2016); Liu  _et
al._ (2022), superconducting circuits Wei  _et al._ (2008); Mei  _et al._
(2018); Li  _et al._ (2018), and coupled-cavity arrays Almeida  _et al._
(2016); Qin and Nori (2016). However, due to the existence of inevitable
manufacturing imperfections within the devices and decoherence effect induced
by the environment, the reliability of the quantum information transmission
may be significantly reduced Balachandran and Gong (2008); Yang  _et al._
(2010); Nikolopoulos (2013); Ashhab (2015); Keele and Kay (2022). Therefore,
it is of urgent need to improve the fidelity of quantum state transfer (QST)
and circumvent the impact of different sources of disorder and decoherence.

Recently, the discovery of topological insulators opens up new prospects for
efficient and robust quantum information processing Niu  _et al._ (1985);
Moore (2010); Qi and Zhang (2010); Chiu  _et al._ (2016); Wen (2017). Owing to
their nontrivial topological energy band structures in momentum space which
are inequivalent to traditional insulators, topological insulators hold
simultaneously insulating bulk states and conducting edge states which can be
characterized by topological invariants rooted in global geometric properties
of the system Hasan and Kane (2010); Qi and Zhang (2011); Bansil  _et al._
(2016); Ozawa  _et al._ (2019). These conducting edge states localized at the
boundary of the system are inherently protected by the energy gap, and as a
consequence, are immune to mild manufacturing defects or environmental
perturbations and able to propagate along the boundary unidirectionally
without generating back scattering Seo  _et al._ (2010); Lang and Büchler
(2017); Dlaska  _et al._ (2017). These prominent features make topological
edge states a promising candidate for not only robust QST Tan  _et al._
(2020); Mei  _et al._ (2018); Lemonde  _et al._ (2019); Cao  _et al._ (2021);
Cheng  _et al._ (2022); Qi  _et al._ (2020a, b) but also quantum computing
Stern and Lindner (2013); Sarma  _et al._ (2015); He  _et al._ (2019a) and
quantum entanglement Monkman and Sirker (2020); Dai  _et al._ (2022); Han  _et
al._ (2021, 2022). As one of the most commonly studied one-dimensional models
enabling nontrivial topological edge states, the Su-Schrieffer-Heeger (SSH)
model possesses structural simplicity and rich forms of edge states. Based on
the topological edge channel in the SSH lattice, the topologically protected
QST between different nodes has been extensively explored and corresponding
topological quantum devices has been proposed, including topological beam
splitters Qi  _et al._ (2020c, 2021a), topological routers Qi  _et al._
(2021b); Zheng  _et al._ (2022), topological lasers St-Jean  _et al._ (2017);
Parto  _et al._ (2018); Longhi (2018); Qi  _et al._ (2020d); Harder  _et al._
(2021); Ezawa (2022), and so on. The key point of QST based on the topological
edge channel is the adiabatic evolution of the channel state. The speed of QST
needs to be sufficiently slow so as to avoid nonadiabatic transitions between
the channel and bulk states during the whole transfer process, which results
in topological quantum devices with total evolution time usually too long to
be feasibly implemented in actual physical systems. Accelerated adiabatic
pumping of topological edge state can be realized by designing the topological
edge channel reasonably, and several protocols of QST based on the SSH model
have been proposed Longhi (2019); D’Angelis  _et al._ (2020); Brouzos  _et
al._ (2020); Palaiodimopoulos  _et al._ (2021). However, these techniques,
elaborately orchestrated to improve the transmission efficiency and enhance
robustness against disorder and decoherence within the quantum system, focus
mainly on the topologically protected state transfer from one single node to
the other, and are rarely combined with specific topological quantum devices,
which are not conducive to the construction of large-scale quantum networks.

In this work, we propose to realize fast and robust QST in a symmetrical
topological beam splitter based on an odd-sized SSH model with alternating
onsite energies and a topological interface, for which the exponential
modulation of nearest neighbor coupling strengths and onsite energies account
for accelerating the transfer process. The introduction of alternating onsite
energies and the topological interface opens up a topological channel, through
which the input state initially prepared at the interface site can be
transferred to two end sites with equal probabilities and phases. We propose
exponential modulation of nearest neighbor coupling strengths and onsite
energies to accelerate the state transfer process whose speed is intrinsically
limited by adiabatic requirements. The effect of different exponential
parameters on the performance of scheme are examined, and the optimal
exponential parameters for chains of different sizes are shown. Furthermore,
we investigate the robustness of the topological beam splitter by taking into
consideration the impact of diagonal and off-diagonal disorders and losses of
system. In addition, we prove the scalability of the symmetrical beam splitter
and generalize the model to two-dimensional crossed-chain structures that can
be employed to implement a topological router whose number of outports can be
adjusted conveniently by cross-linking different numbers of identical even-
sized SSH chains via one mutual site. Finally, we stress that fast and robust
QST in the proposed beam splitter and router can be realized in
superconducting circuit devices under current experimental conditions, which
has numerous potential applications in efficient quantum information
processing and the construction of large-scale quantum networks.

##  II Physical model and engineering of topological pumping

###  II.1 Topologically protected edge states for the generalized SSH model

Schematic illustration of the generalized SSH model is shown in Fig. 1, which
describes a one-dimensional dimerized lattice composed of $N$ unit cells. The
Hamiltonian of system reads as ($\hbar=1$)

$H=\sum_{n}V_{a}a_{n}^{\dagger}a_{n}+V_{b}b_{n}^{\dagger}b_{n}+(J_{1}a_{n}^{\dagger}b_{n}+J_{2}a_{n+1}^{\dagger}b_{n}+{\rm
H.c.}),$ (1)

where the first two terms represent onsite energies of two types of sites
while the rest terms represent the nearest coupling between two adjacent
sites. Here, $a_{n}~{}(a_{n}^{\dagger})$ and $b_{n}~{}(b_{n}^{\dagger})$ are
the annihilation (creation) operators of a particle at the $n$th a\- and
b-type sites with onsite energies $V_{a}$ and $V_{b}$, and $J_{1}$ and $J_{1}$
are the respective intracell and intercell coupling coefficients assumed to be
real and positive.

Figure 1: Diagrammatic sketch of the generalized SSH model composed of $N$
unit cells. Each unit cell contains a pair of a\- (blue dot) and b-type
(purple dot) sites with onsite energies $V_{a}$ and $V_{b}$. Double lines and
single lines denote the intracell coupling strength $J_{1}$ and intercell
coupling strength $J_{2}$ between two adjacent sites, respectively.

For periodic boundary conditions (PBC), we can use the Bloch theorem and
rewrite the bulk Hamiltonian as

$H_{\text{bulk}}=\sum_{\mathrm{n}=1}^{N}V_{a}a_{n}^{\dagger}a_{n}+V_{b}b_{n}^{\dagger}b_{n}+(J_{1}a_{n}^{\dagger}b_{n}+J_{2}a_{m+1}^{\dagger}b_{n}+{\rm
H.c.}),$ (2)

with $m=n\bmod N$. After performing a Fourier transformation
$a_{n}=\frac{1}{\sqrt{N}}\sum_{k}e^{ikn}a_{k}$ and
$b_{n}=\frac{1}{\sqrt{N}}\sum_{k}e^{ikn}b_{k}$, with the wavenumber
$k\in\left\\{\frac{2\pi}{N},\frac{4\pi}{N},\cdots,\frac{2N\pi}{N}\right\\}$
being from the first Brillouin zone, the Hamiltonian can be moved into the
momentum space, represented by $H_{\text{bulk
}}=\sum_{k}V_{a}a_{k}^{\dagger}a_{k}+V_{b}b_{k}^{\dagger}b_{k}+\left[\left(J_{1}+J_{2}e^{-ik}\right)a_{k}^{\dagger}b_{k}+\text{
H.c. }\right]$. For each wavenumber $k$, the bulk Hamiltonian in the momentum
space under the basis of $(a_{k},b_{k})^{T}$ can be expressed as

$H_{k}=\left(\begin{array}[]{cc}V_{a}&J_{1}+J_{2}e^{-ik}\\\
J_{1}+J_{2}e^{ik}&V_{b}\end{array}\right).$ (3)

Figure 2: (a)-(c) Energy spectrum of SSH lattice in the momentum space for
three settings of the coupling strengths (a) $J_{1}=1,J_{2}=0.6$; (b)
$J_{1}=1,J_{2}=1$; (c) $J_{1}=0.6,J_{2}=1$. (d)-(f) Winding of the bulk
momentum-space Hamiltonian for the three settings as the wavenumber runs
across the Brillouin zone.

We first consider the case of the standard SSH model where there is no onsite
energy on the lattice sites. By diagonalizing $H_{k}$, the eigenvalues can be
obtained $E_{\pm}(k)=\pm\sqrt{J_{1}^{2}+J_{2}^{2}+2J_{1}J_{2}\cos k}$,
corresponding to eigenstates

$|\psi_{\pm}(k)\rangle=\frac{1}{\sqrt{2}}\left[E_{\pm}(k)/\left(J_{1}+J_{2}e^{ik}\right),1\right]^{T}.$
(4)

The eigenenergy spectrum of the system is divided into two bands, with an
energy gap of $2\Delta$ separating the lower and filled bands from the upper
and empty bands, with $\Delta=\min_{k}E(k)=\left|J_{1}-J_{2}\right|$. We plot
the dispersion relation for three choices of the coupling strengths in Figs.
2(a)-(c). As the coupling strengths ranges from $J_{1}>J_{2}$ to
$J_{1}<J_{2}$, the band gap is first closed and then reopened at the
boundaries of the first Brillouin zone. Introducing the Pauli matrices
$\bm{\sigma}=\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)$ as base vectors,
the Hamiltonian can be expressed in the form

$H(k)=\mathbf{d}\cdot\bm{\sigma},$ (5)

with $\mathbf{d}=\left(d_{x},d_{y},d_{z}\right)=\left(J_{1}+J_{2}\cos
k,J_{2}\sin k,V_{a}\right)$. As displayed in Figs. 2(d)-(f), $\mathbf{d}$ does
not enclose the origin as the wavenumber runs across the Brillouin zone for
$J_{1}>J_{2}$ while encloses the origin for $J_{1}<J_{2}$, which correspond to
two topological distinct phases, respectively. In the phase transition point
$J_{1}=J_{2}$, eigenstates of the bulk are available with arbitrarily small
energy, and the SSH model behaves like a conductor which can transport
electrons from one end of the chain to the other. Otherwise, the SSH model
behaves like an insulator. The encirclement of the Hamiltonian $H_{k}$ can be
characterized by the winding number $w$ defined as

$w_{\pm}=\frac{i}{\pi}\int_{-\pi}^{\pi}\left\langle\psi_{\pm}(k)\mid\partial_{k}\mid\psi_{\pm}(k)\right\rangle.$
(6)

$w=0$ indicates that $H_{k}$ does not enclose the origin, corresponding to the
topological trivial phase; $w=1$ indicates that $H_{k}$ encloses the origin,
corresponding to the topological nontrivial phase, in which according to the
body-boundary correspondence Hasan and Kane (2010), the SSH lattice exhibits
edge states in the bulk gap under the open boundary conditions (OBC). We
examine how the spectrum of an open chain changes as the intracell and
intercell coupling strengths are continuously modulated. For an even-sized SSH
model composed of $2N=40$ lattice sites, the topological nontrivial phase
hosts two edge states exponentially localized on the boundaries of the chain,
as shown in Figs. 3(a)-(c). By analytically solving the eigenvalue equation,
we get the eigenvalues
$E_{\text{even},\pm}=\pm\left|-J_{2}\frac{\left(-J_{1}/J_{2}\right)^{N}\left[\left(-J_{1}/J_{2}\right)^{2}-1\right]}{\left(-J_{1}/J_{2}\right)^{2N}-1}\right|$,
corresponding to eigenstates

$\left|\Psi_{\pm}\right\rangle=(|L\rangle\pm|R\rangle)/\sqrt{2},$ (7)

with

$\begin{array}[]{l}|L\rangle=|1,0,-J_{1}/J_{2},0,\cdots,0,\left(-J_{1}/J_{2}\right)^{n-1},0,\cdots\rangle,\\\
|R\rangle=|\cdots,0,\left(-J_{1}/J_{2}\right)^{N-n},0,\cdots,0,-J_{1}/J_{2},0,1\rangle,\end{array}$
(8)

denoting the ideal left and right edge states in thermodynamic limit (See
Appendix for more details). The degenerate gap modes of a finite-sized system
in the topological nontrivial phase takes a pair of almost-zero-energy
eigenvalues opposite to each other due to chiral symmetry, which makes
eigenstates take the superposition of the ideal left and right edge states and
localize at both ends of the system. For an odd-sized SSH model composed of
$2N+1=41$ lattice sites, there is always a zero-energy edge state with
eigenvector

$\left|\Psi_{0}\right\rangle=\left|1,0,-J_{1}/J_{2},0,\left(-J_{1}/J_{2}\right)^{2},\cdots,\left(-J_{1}/J_{2}\right)^{N}\right\rangle,$
(9)

with the localized position depending on the ratio $J_{1}/J_{2}$, as
demonstrated in Figs. 3(d)-(f).

Figure 3: (a) Energy spectrum of the SSH lattice with 40 sites with varying
intracell coupling $J_{1}$ but fixed intercell coupling $J_{2}=1$.
$J_{1}<1~{}(J_{1}>1)$ corresponds to the nontrivial (trivial) topological
phase. (b) Energy spectrum (upper panel) and distribution of the gap state
(lower panel) for $J_{1}=0.6$. (c) Energy spectrum (upper panel) and density
distribution of one bulk state (lower panel) for $J_{1}=2$. (d)-(f) Energy
spectra and distributions of the zero-energy edge state of the SSH lattice
with 41 sites with the same coupling strengths as those in (a)-(c),
respectively.

###  II.2 Symmetrical beam splitter via edge channel in the Rice-Mele model

We now consider the case of the Rice-Mele model Hu  _et al._ (2020) originated
from the standard SSH model by adding alternate onsite potentials
$V_{a}=-V_{b}$. For each wavenumber $k$, eigenvalues and corresponding
eigenstates can be obtained by analytically solving the eigenvalue equation

$\begin{array}[]{l}E_{\pm}(k)=\pm\sqrt{V_{a}^{2}+J_{1}^{2}+J_{2}^{2}+2J_{1}J_{2}\cos
k},\\\
\left|\psi_{\pm}(k)\right\rangle=N_{k}\left[\left(E_{\pm}(k)+V_{a}\right)/\left(J_{1}+J_{2}e^{ik}\right),1\right]^{T},\end{array}$
(10)

with $N_{k}$ being normalization factor. In the odd-sized Rice-Mele model,
there is always a gap state with eigenenergy $V_{a}$ and eigenstate
$\left|\psi_{V_{a}}\right\rangle=\left|1,0,-\frac{J_{1}}{J_{2}},0,\left(-\frac{J_{1}}{J_{2}}\right)^{2},0,\cdots,0,\left(-\frac{J_{1}}{J_{2}}\right)^{N}\right\rangle$
for which the localized position can be also modulated by tuning
$J_{1}/J_{2}$, and thus can be exploited as a topologically protected quantum
channel. Setting $J_{1}/J_{2}=0$ initially, $J_{1}/J_{2}=+\infty$ finally, and
continuously modulating the intracell and intercell coupling strengths, a
topologically protected state transfer can be realized from the left edge to
the right.

Figure 4: Schematic of SSH model with the size of $L=2N+1$ with alternate
onsite energies and an interface. Double and single lines denote the intracell
and intercell coupling strengths $J_{1}$ and $J_{2}$ between two adjacent
sites, respectively. Intracell (intercell) coupling strengths and alternate
onsite energies are mirror-symmetric with respect to the interface site. $N$
is an even number, so that the topological interface falls on an a-type site.

Inspired by topological edge pumping in the Rice-Mele model, a symmetrical
topological beam splitter with an equal phase can be obtained based on an odd-
sized SSH model with alternating onsite energies and a topological interface.
We consider a finite chain comprising of $L=2N+1$ sites, with $N$ being even,
i.e., structured by linking two even-sized SSH chains via one mutual a-type
site, as schematically shown in Fig. 4. Intracell (intercell) coupling
strengths and alternate onsite energies are mirror-symmetric with respect to
the interface site. The system can be described by the following interaction-
picture Hamiltonian

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{n}\left[V_{a}a_{n}^{\dagger}a_{n}+V_{b}b_{n}^{\dagger}b_{n}\right]+\left[\sum_{\mathrm{n}=1}^{N/2}(J_{1}a_{n}^{\dagger}b_{n}+J_{2}a_{n+1}^{\dagger}b_{n})\right.$
(11)
$\displaystyle\left.+\sum_{\mathrm{n}=N/2+1}^{N}(J_{2}a_{n}^{{\dagger}}b_{n}+J_{1}a_{n+1}^{{\dagger}}b_{n})+\text{
H.c.}\right].$

In the energy spectrum of this system there always exists a gap state with
eigenenergy $V_{a}$ and eigenvector

$\displaystyle\left|\psi_{V_{a}}\right\rangle$ $\displaystyle=$
$\displaystyle\left|1,0,-\frac{J_{1}}{J_{2}},0,\left(-\frac{J_{1}}{J_{2}}\right)^{2},0,\cdots,0,\left(-\frac{J_{1}}{J_{2}}\right)^{N/2},0,\right.$
$\displaystyle\left.\cdots,0,\left(-\frac{J_{1}}{J_{2}}\right)^{2},0,-\frac{J_{1}}{J_{2}},1\right\rangle,$

which is localized at the topological interface when $J_{1}/J_{2}=+\infty$ but
localized at both ends of the chain when $J_{1}/J_{2}=0$. Assisted by this
topological edge channel, a topologically protected QST from the interface
site to the two end sites with equal probabilities can be realized by
continuously modulating the intracell and intercell coupling strengths from
$J_{1}/J_{2}=+\infty$ to $J_{1}/J_{2}=0$. When regarding the interface site as
the input port and the two end sites as two output ports, the whole system is
equivalent to a symmetrical topological beam splitter, in which a particle
injected into the interface site can be transferred to the two endpoints of
the chain with equal probabilities. It is worth emphasizing that this beam
splitting process along the gap state $\left|\psi_{V_{a}}\right\rangle$ is
topologically protected by the band gap between the gap state and its adjacent
bulk eigenstates and is thus immune to scattering from inherent disorders and
local imperfections.

###  II.3 Choice of the modulating coupling strengths and analysis of energy
spectrum

The realization of the topological beam splitter is essentially based on the
adiabatic evolution of the gap state, which requires the system to be driven
slowly enough so that the initial state always evolves along the gap state
$\left|\psi_{V_{a}}\right\rangle$ during the transfer process. The topological
pumping based on the gap state is governed by the following time-dependent
Schrödinger equation

$i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle=H(t)|\Psi(t)\rangle$ (12)

where $|\Psi(t)\rangle$ can be expressed as

$|\Psi(t)\rangle=\sum_{n}a_{n}(t)e^{-iE_{n}(t)t}\left|\psi_{n}(t)\right\rangle,$
(13)

with $\left|\psi_{n}(t)\right\rangle$ and $E_{n}(t)$ obeying the instantaneous
eigenequation
$H(t)\left|\psi_{n}(t)\right\rangle=E_{n}(t)\left|\psi_{n}(t)\right\rangle$,
and $a_{n}(t)$ denotes the probability amplitude on the $n$th instantaneous
eigenstates. Substituting Eq. (13) into Eq. (12), we get

$\frac{\partial}{\partial t}a_{n}(t)=\sum_{m\neq
n}a_{m}(t)e^{it[E_{n}(t)-E_{m}(t)]}\frac{\langle\psi_{n}(t)|\frac{\partial
H(t)}{\partial t}|\psi_{m}(t)\rangle}{E_{m}(t)-E_{n}(t)}.$ (14)

In order to approach the adiabatic limit, we have

$\sum_{m\neq n}\frac{\left\langle\psi_{n}(t)\left|\frac{\partial
H(t)}{\partial
t}\right|\psi_{m}(t)\right\rangle}{\left|E_{m}(t)-E_{n}(t)\right|}\ll 1.$ (15)

To satisfy the adiabatic condition, the instantaneous energy difference
between the gap and bulk states should be large enough, and the derivative of
the Hamiltonian, which is directly related to the slope of the driving
function, should be sufficiently small. In order to enhance the speed and
efficiency of state transfer, we need to adjust the intracell and intercell
coupling strengths appropriately so that the system can be driven strongly
where the energy gap is wide but mildly when the energy gap is narrow.

Figure 5: Waveforms of coupling strengths and alternate onsite energy with (a)
$\alpha=2$, (c) $\alpha=6$, and (e) $\alpha=10$. (b), (d) and (f)
Instantaneous energy spectrum as a function of time for different exponential
parameters in (a), (c) and (e), respectively. The total evolution time is
chosen to unity and the size of chain to be 21. Figure 6: Distribution of the
gap state [magenta lines in Figs. 5(b), (d), and (f)] with eigenenergy $V_{a}$
with different values of (a) $\alpha=2$, (b) $\alpha=6$, and (c) $\alpha=10$.
(d) Minimum energy gap between the gap state and the nearest-neighbor bulk
states versus values of $\alpha$ with fixed chain size $L=21$.

Several protocols have been proposed to realize accelerated QST via
topological edge channel based on the SSH model Longhi (2019); D’Angelis  _et
al._ (2020); Brouzos  _et al._ (2020); Palaiodimopoulos  _et al._ (2021).
However, these techniques for accelerated adiabatic edge pumping are mainly
based on the standard SSH model where there is no onsite energy on the lattice
sites and focus on the topologically protected state transfer from one single
node to the other. For example, Palaiodimopoulos et al. Palaiodimopoulos  _et
al._ (2021) proposed exponential modulation of nearest-neighbor coupling in an
odd-sized SSH chain and achieved fast topological edge pumping. This approach,
unlike the shortcuts to adiabaticity Guéry-Odelin  _et al._ (2019) where
elaborately orchestrated counter-adiabatic terms in the Hamiltonian are
introduced to suppress unwanted excitations, only involves engineering of the
driving function. In this paper we adopt the exponential modulation of not
only the nearest neighbor coupling strengths but also onsite energies

$J_{1}=J_{0}\frac{1-e^{-\alpha\left(t^{\ast}-t\right)/t^{\ast}}}{1-e^{-\alpha}},$
(16a) $J_{2}=J_{0}\frac{1-e^{-\alpha t/t^{\ast}}}{1-e^{-\alpha}},$ (16b)
$V_{b}=-V_{a}=J_{0}\sqrt{\frac{J_{2}(2t)}{J_{0}}},$ (16c)

where $t^{\ast}$ denotes the total evolution time and $\alpha$ is a tunable
exponential modulation parameter. According to the definitions in Eq.
(16a-16c), $J_{1,2}$ satisfy $J_{1}/J_{2}=0$ and $J_{1}/J_{2}=+\infty$ at the
initial and end instants, respectively.

Taking the system with chain length of $L=21$ as an example, we select three
different values of exponential parameter and plot evolution of the coupling
strengths and alternating onsite energy in Figs. 5(a), (c), and (e). In
addition, in Figs. 5(b), (d), and (f), we show how the instantaneous eigen-
spectrum evolves over time. Referring to the gap states (magenta line) in
eigen-spectrum and the corresponding evolution of onsite energy, we identify
that the gap state $\left|\psi_{V_{a}}\right\rangle$ varies along the
topological channel in the symmetrical beam splitting process. This can be
further verified by distribution of the gap state during the evolution process
as depicted in Figs. 6(a)-(c), in which the state transfers from the interface
site to two-end sites with equal probabilities. Evolution of energy gap
between the gap state and the bulk adjacent state (green line) in Figs. 5(b),
(d), and (f) is shown in Figs. 5(a), (c), and (e) (green-dotted lines),
respectively. In the whole process of evolution, the moment of larger
(smaller) values of energy gap obviously corresponds to the larger (smaller)
slope of the driving function $J_{2}$. Besides, as illustrated in Fig. 6(d),
the minimum energy gap augments with the increase of parameter $\alpha$, which
is positively related to the slope of the driving function. As a result, the
exponential modulation of coupling strengths and alternate onsite energies is
a qualified candidate to achieve fast and efficient topologically protected
state transfer in the symmetrical topological beam splitter.

Figure 7: (a Fidelity as a function with respect to the final time of QST for
the cosine and exponential protocols. (b)-(e) Distribution of the gap state
during the evolution and the phase distribution of the evolved final state for
the cosine protocol in (b) and (c), and for the exponential protocol in (d)
and (e), respectively. Other parameters take $L=21$ and $\alpha=3.2$.

##  III Fast QST with high robustness and scalability

###  III.1 Fast QST in the topological beam splitter

The initial state of the system

$\displaystyle|\Psi_{i}\rangle$ $\displaystyle=$
$\displaystyle\left|\rho_{a,1}e^{i\phi_{a,1}},\rho_{b,1}e^{i\phi_{b,1}},\cdots,\rho_{a,N}e^{i\phi_{a,N}},\cdots,\right.$
$\displaystyle\left.\rho_{a,2N+1}e^{i\phi_{a,2N+1}},\rho_{b,2N+1}e^{i\phi_{b,2N+1}}\right\rangle.$
$\displaystyle=$
$\displaystyle\left|0,0,0,\cdots,0,1,0,\cdots,0,0,0\right\rangle$

is specified to set the interface site as the input port of the beam splitter.
To measure how faithfully the transfer from the interface site to two end
sites has occurred, we introduce fidelity defined as
$F=\left|\left\langle\Psi_{t}\mid\Psi\left(t^{\ast}\right)\right\rangle\right|^{2}$,
where

$\displaystyle|\Psi_{t}\rangle$ $\displaystyle=$
$\displaystyle\left|\rho_{a,1}^{\prime}e^{i\phi_{a,1}^{\prime}},\rho_{b,1}^{\prime}e^{i\phi_{b,1}^{\prime}},\cdots,\rho_{a,N}^{\prime}e^{i\phi_{a,N}^{\prime}},\cdots,\right.$
$\displaystyle\left.\rho_{a,2N+1}^{\prime}e^{i\phi_{a,2N+1}^{\prime}},\rho_{b,2N+1}^{\prime}e^{i\phi_{b,2N+1}^{\prime}}\right\rangle.$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left|1,0,0,\cdots,0,0,0,\cdots,0,0,1\right\rangle$

and $\Psi\left(t^{\ast}\right)$ denote the target state and the evolved state
at final time $t^{\ast}$, respectively. Consider the system with chain size
$L=21$. In order to see the speed of transfer, we plot in Fig. 7(a) the QST
fidelity of beam splitter versus the total transfer time to compare the
commonly-used cosine protocol (for example, protocols in Refs. Qi  _et al._
(2020c, 2021a)) and the exponential protocol with exponential parameter
$\alpha=3.2$, where parameters in the cosine protocol are
$J_{1}=\frac{J_{0}}{2}\left(1+\cos\frac{\pi t}{t^{\ast}}\right)$,
$J_{2}=\frac{J_{0}}{2}\left(1-\cos\frac{\pi t}{t^{\ast}}\right)$, and
$V_{b}=-V_{a}=J_{0}\sin\frac{\pi t}{t^{\ast}}$. For both protocols, as total
evolution time approaches infinity, the fidelity approaches unity, meaning
that an excitation imposed initially at the interface site can be perfectly
transferred along the chain to two-end sites with equal probabilities, which
satisfies the adiabatic approximation during the transfer process, so that the
system state always evolves along the gap state
$\left|\psi_{V_{a}}\right\rangle$ without leaking to others. Here we suppose
that the QST is successfully implemented if the fidelity is stabilized above
0.99. The implementation of the exponential protocol leads to a significantly
accelerated QST process which is about 10 times faster than its cosine
counterpart, since the fidelity is stabilized above 0.99 after
$t^{\ast}=100/J_{0}$ for the exponential protocol as compared to
$t^{\ast}=1080/J_{0}$ for the cosine protocol. The process of QST and the
phase distribution of the evolved final state for the cosine and exponential
protocols are illustrated in Figs. 7(b)-(e), indicating that both protocols
can achieve symmetrical topological beam splitting with equal phase by costing
sufficient transfer time, but obviously the exponential protocol is much
faster.

Figure 8:  Fidelity as a function of the transfer time for the exponential
protocol with different values of $\alpha$ with $L=21$.

###  III.2 Effect of different values of $\alpha$

The realization of fast QST via edge channel in the symmetrical topological
beam splitter is exemplified above by setting a fine-tuned exponential
parameter $\alpha=3$ in a chain of size $L=21$. We note in Sec. II.3 that for
different $\alpha$ in the exponential modulation, there are evident
differences in the slopes of the coupling functions and the corresponding
energy gaps between the gap state and its nearest-neighbor bulk state in the
instantaneous spectrum, leading to different effects on the QST process. To
give some quantitative results,in Fig. 8 we plot fidelity as a function of the
transfer time for the exponential protocol with different $\alpha$. Taking a
closer look at the fidelity curves of the exponential protocol, we notice the
existence of mild oscillations, which indicates that resonant processes are at
work in the QST process. When a smaller $\alpha$ is chosen, the driving
function is flattened and the minimum energy gap between the gap state and its
nearest-neighbor bulk state narrows down, as analytically investigated in Sec.
II.3, where the resonant processes are suppressed effectively and longer total
transfer time is required for successful symmetrical beam splitting.
Conversely, larger values of the exponential parameter $\alpha$ lead to a
steeper slope of the driving function and better separation between the gap
state and its nearest-neighbor bulk state. As a consequence, the resonant
processes are intensified and strong oscillations appear at the fidelity
curve, which takes longer time for the fidelity being stabilized at a
sufficiently large value. Therefore, it is a trade-off to set $\alpha=3$ for a
chain of size $L=21$, when the system is driven strongly enough to achieve
high-fidelity QST of beam splitter in relatively short time, yet mildly enough
to avoid strong oscillations of the fidelity curve.

Figure 9: Fidelity of QST versus varying $\alpha$ and $t^{\ast}$ for the
exponential protocol with (a) $L=21$, (b) $L=33$, (c) $L=45$, and (d) $L=57$.
The green and red solid lines represent 0.9 and 0.99 fidelity contour lines,
respectively. (e) Optimal exponential parameters and (f) the corresponding
total evolution time needed for 0.99-fidelity as a function of the size of the
chain. The scattering dots and the lines represent the numerical and cubic
polynomial fitting results, respectively.

Further, we investigate the fidelity of QST by varying $\alpha$ and the total
evolution time with $L=21,~{}33,~{}45$, and $57$, respectively, as illustrated
in Figs. 9(a)-(d). The 0.9 and 0.99 fidelity contour lines manifest intense
oscillations for larger $\alpha$, yet similar to Fig. 8, for smaller $\alpha$
the oscillations are suppressed substantially. We can always find the optimal
exponential parameter so as to reach a balance between accelerating the
symmetrical beam splitting process and avoiding excessive oscillations. For
instance, the optimal exponential parameters can be set as
$\alpha=3.2,~{}4.5,~{}5.5$, and $6.1$ for chain sizes $L=21,~{}33,~{}45$, and
$57$, respectively. As shown in Figs. 9(e)-(f), by selecting different chain
sizes and optimizing the best exponential parameters as well as corresponding
total evolution times for the final fidelity equaling to 0.99 as numerical
samples, $\alpha_{\text{optimal}}$ versus $L$ and
$\alpha_{\text{optimal}}^{0.99}$ versus $L$ can be fitted by cubic functions
$\alpha_{\text{optimal}}=1.2\times 10^{-5}L^{3}-0.0026L^{2}+0.22L-0.33$ and
$J_{0}t_{\text{optimal}}^{0.99}=0.00052L^{3}+0.059L^{2}-0.34L+68$,
respectively. The exponential parameter and corresponding evolution time
should be large enough to satisfy the adiabatic condition for a longer size of
the chain.

###  III.3 Robustness against disorders and loss of system

Due to the existence of manufacturing defects of systematic elements in
practice, perfect modulation of the coupling strengths and onsite energies is
almost unattainable. In this section, we examine robustness of QST in the beam
splitting process by introducing disorders both in coupling strengths and
onsite energies to discuss its effect on the performance of beam splitter. The
disorder in coupling strengths is generally addressed as off-diagonal
disorder, while the disorder in onsite energies as diagonal ones, depending on
its effect on the matrix representation of the Hamiltonian. We first consider
the case of symmetric distortion, in which the way each disorder realization
is imposed on the system parameters can be assumed as

$\displaystyle J_{1(2),n}^{i}\rightarrow J_{1(2),n}^{i}\left(1+\delta
J_{1(2)}^{i}\right),$ $\displaystyle V_{1(2),n}^{i}\rightarrow
V_{1(2),n}^{i}\left(1+\delta V_{1(2)}^{i}\right),$ (17)

where $\delta J_{1(2)}^{i}$ and $\delta V_{1(2)}^{i}$ are assumed to remain
constant during the QST process, but $\delta J_{1(2)}^{i}$ and $\delta
V_{1(2)}^{i}$ acquire random real values sampled from the interval
$[-\omega_{s},~{}\omega_{s}]$, where $\omega_{s}$ is termed the disorder
strength. We plot the mean fidelity of splitting QST versus total transfer
time for both kinds of disorders with moderate strength $\omega_{s}=0.4$ for
the cosine and exponential protocols with $L=21$ and $\alpha=3.2$ in Figs.
10(a) and (b), respectively. Each point corresponds to the mean value of
fidelity $\bar{F}=\frac{1}{M}\sum_{i=1}^{M}F_{i}$ averaged over $M=100$
disorder realizations for the sake of universality, while the error bars
correspond to the standard deviation. What we can immediately notice is that
both protocols are highly robust to the diagonal disorder, because the
$F$–$t^{\ast}$ curves almost coincide with the unperturbed curve. Besides, the
transfer process is insignificantly destroyed by non-diagonal disorder applied
in the both protocols. Rather, the main impact is longer total transfer time
required to achieve QST of beam splitter.

Figure 10: Impact of diagonal and off-diagonal disorders with $\omega_{s}=0.4$
on the fidelity for the (a) cosine and (b) exponential protocols. Each point
corresponds to the mean value of fidelity averaged over 100 disorder
realizations. Average fidelity as a function of the disorder strength for (c)
diagonal and (d) off-diagonal disorders for different protocols. Other
parameters take $L=21$ and $\alpha=3.2$.

In the presence of off-diagonal disorder, a longer total transfer time
$t^{\ast}$ is required to achieve high-fidelity QST for both protocols. In
Figs. 10(c) and (d), for different types of disorder for the cosine and
exponential protocols with $\alpha=3.2$ and $L=21$, we plot the average
fidelity ($M=100$) as a function of disorder strength. For the sake of
comparison, we set the total transfer time $t^{\ast}=1100/J_{0}$ so that the
QST of beam splitter can be implemented via both protocols when disorder
strength equals zero. Numerical results reveal strong robustness against
diagonal disorders for both protocols, because average fidelities in Fig.
10(c) always approach unity. However, we find that the exponential protocol
manifests stronger robustness than the cosine protocol against off-diagonal
disorder, as shown in Fig. 10(d) where the average fidelity of the former
decreases significantly when $\omega_{s}=0.7$, while it is $\omega_{s}=0.4$
that the average fidelity of the cosine protocol starts to declines
significantly. To sum up, the exponential protocol apparently outperforms the
cosine protocol in terms of robustness to off-diagonal disorder, while both
protocols are quite robust to diagonal disorder.

Figure 11: Average fidelity and phase difference of the evolved final state at
two-end sites versus disorder strength for asymmetric (a) diagonal and (b)
off-diagonal disorders for different protocols. Other parameters take $L=21$
and $\alpha=3.2$.

Diagonal and off-diagonal effects as depicted in Eq. (III.3) are mirror
symmetrical with respect to the topological interface. In the following, we
try to reveal the effects of asymmetric disorders. For asymmetric distortions
on coupling strengths and onsite energies, their effects are described by

$\begin{cases}J_{1,n}^{i}\rightarrow J_{1,n}^{i},J_{2,n}^{i}\rightarrow
J_{2,n}^{i}\left(1+\delta J_{1}^{i}\right),&n=1,\cdots,\frac{N}{2},\\\
J_{1,n}^{i}\rightarrow J_{1,n}^{i},J_{2,n}^{i}\rightarrow
J_{2,n}^{i}\left(1+\delta J_{2}^{i}\right),&n=\frac{N}{2}+1,\cdots,N,\\\
\end{cases}\\\ $ (18) $\begin{cases}V_{1,n}^{i}\rightarrow
V_{1,n}^{i},V_{2,n}^{i}\rightarrow V_{2,n}^{i}\left(1+\delta
V_{1}^{i}\right),&n=1,\cdots,\frac{N}{2},\\\ V_{1,n}^{i}\rightarrow
V_{1,n}^{i},V_{2,n}^{i}\rightarrow V_{2,n}^{i}\left(1+\delta
V_{2}^{i}\right),&n=\frac{N}{2}+1,\cdots,N,\\\ \end{cases}\\\ $ (19)

where $\delta J_{1(2)}^{i}$ and $\delta V_{1(2)}^{i}$ acquire random real
values in the interval $[-\omega_{s},~{}\omega_{s}]$. Taking the system of
chain size $L=21$ as an example, we plot in Fig. 11 the mean fidelity of the
topological beam splitter and phase difference of the evolved final state at
two end sites averaged over $M=100$ samples versus disorder strength
$\omega_{s}$ for the cosine and exponential protocols. We find that for both
protocols, when asymmetric diagonal disorders are imposed on the system, it
can still function as a robust symmetrical beam splitter with equal phase.
However, in terms of robustness against asymmetric diagonal disorders, the
evolved final state at the two ends may not have the same phase and amplitude,
and the deviations will be amplified when $\omega_{s}$ increases.

Figure 12: Final fidelity as a function of the loss rate $\gamma$ for the
cosine protocol with $t^{\ast}=1080/J_{0}$ and the exponential protocol with
$t^{\ast}=100/J_{0}$ and $\alpha=3.2$ under symmetrical losses. (b) Final
fidelity and phase difference of the evolved final state at two-end sites
versus loss rate $\gamma$ for both protocols under asymmetrical losses.

On the other hand, for high-efficiency QST in quantum networks, the systemic
loss is an important factor on infidelity. The effect of losses during the QST
of beam splitter can be considered by changing the Hamiltonian to a non-
Hermitian form

$H^{\prime}=H-i\sum_{n}\left[\gamma_{n}^{a}a_{n}^{\dagger}a_{n}+\gamma_{n}^{b}b_{n}^{\dagger}b_{n}\right],$
(20)

where $H$ is the lossless Hamiltonian, and $\gamma_{n}^{a,b}$ denotes the loss
rate of each type of sites. We first consider the impact of symmetrical loss.
For convenience, we assume $\gamma_{n}^{a}=\gamma_{n}^{b}=\gamma$. The
dynamics of system is governed by the non-Hermitian Lioville equation
$\dot{\rho}=-i\left(H^{\prime}\rho-\rho H^{\prime\dagger}\right)$. We plot in
Fig. 12(a) effects of symmetrical loss on the final fidelity of QST for a
chain of size $L=21$ for the cosine and exponential protocols with
$\alpha=3.2$ and transfer time fixed at $t^{\ast}=1080/J_{0}$ and
$t^{\ast}=100/J_{0}$, respectively, so that the QST can be successfully
implemented via both protocols when no loss exists. Evidently, the exponential
protocol manifests a notable improvement of the final fidelity compared to the
cosine protocol. The weakened damage of the loss effect to the QST process can
be attributed to the shorter accumulation time of decoherence during the QST
in the exponential protocol. We also demonstrate in Fig. 12(b) effects of
asymmetrical loss on the final fidelity and phase difference of the evolved
final state at two end sites. Here, we assume

$\displaystyle\gamma_{n}^{a}=\gamma,$
$\displaystyle\gamma_{n}^{b}=\gamma(1+\delta\gamma_{1}),\quad
n=1,\cdots,\frac{N}{2},$
$\displaystyle\gamma_{n}^{b}=\gamma(1+\delta\gamma_{2}),\quad
n=\frac{N}{2}+1,\cdots,N,$

where $\delta\gamma_{1(2)}$ acquire random real values in the interval
$[-0.1,~{}0.1]$. Apparently, numerical results for effect of asymmetrical loss
have few differences from its symmetrical counterpart.

###  III.4 Scalability

####  III.4.1 Size of chains

As noted above, the exponential protocol is a compelling alternative for fast
and robust QST in the symmetrical topological beam splitter, which exhibits a
remarkable improvement in speed of beam splitting and robustness against both
types of disorders compared to the commonly-used cosine protocol. In order to
verify more extensively the effect of exponential coupling strengths and
onsite energies, one crucial aspect determining the efficiency of QST is its
scalability. In the following, we focus on how the exponential protocol
behaves when the system size is altered. We show the phase diagram of the QST
in the parameter space $(t^{\ast},L)$ of the cosine and exponential protocols
in Figs. 13(a) and (b), respectively, with the exponential parameter fixed at
$\alpha=3.2$. The yellow (purple) areas indicate that the QST of beam splitter
can be implemented with fidelity over (below) 0.99. Evidently, the parameter
space can be divided into three regions according to different phase
boundaries, as demonstrated in Fig. 13(c). In region I, the symmetrical beam
splitter can be faithfully realized via both protocols. In region II, beam
splitting is faithfully implemented via only the exponential protocol, while
in region III neither protocols cannot work well. Consequently, we can choose
feasible modulation protocols according to different parameter design in the
system. We plot in Fig. 13(d) the total transfer time $t_{0.99}^{\ast}$ each
protocol takes to achieve beam splitting as a function of the size of the
system, where $t_{\text{cos}}^{0.99}$ versus $L$ for the cosine protocol can
be fitted by cubic function
$J_{0}t_{\text{cos}}^{0.99}=0.1L^{3}-0.46L^{2}+28L-260$. Obviously, the
symmetrical topological beam splitter modulated by both protocols needs a
longer total evolution time with the augmentation of chain size $L$.
Nevertheless, it is evident that the exponential protocol outperforms the
cosine protocol in terms of the transfer speed and manifests good scalability
within the range of length we have considered here.

Figure 13: Phase diagram of the QST in the parameter space $(t^{\ast},L)$ of
the (a) cosine and (b) exponential protocols. (c) The total phase diagram
derived from (a) and (b). (d) The transfer time $t_{0.99}^{\ast}$ that each
protocol takes to reach 0.99 fidelity as a function of the size of the system.
$\alpha=3.2$ is used here. Figure 14: Fidelity as a function of the chain size
with fixed loss parameter $\gamma=2.5\times 10^{-5}J_{0}$ for both protocols.
$\alpha=3.2$ is used here.

In the following, we take into consideration the inevitable loss of the
system. As shown in Fig. 14, we plot for both protocols the final fidelity as
a function of the chain size with fixed loss parameter $\gamma=2.5\times
10^{-5}J_{0}$. The parameter selection is made in accordance with current
technological capabilities in the superconducting circuit devices, where the
initial (end) coupling strengths between resonators and the decoherence rates
of photon in superconducting resonator are set to be $J_{0}/2\pi=100\rm{MHz}$
and $\gamma/2\pi=2.5\rm{kHz}$, respectively Schmidt and Koch (2013); Frunzio
_et al._ (2005); Underwood  _et al._ (2012). The exponential parameters are
chosen as $\alpha=1.2\times 10^{-5}L^{3}-0.0026L^{2}+0.22L-0.33$ for chains of
different sizes, and the total evolution times are set to be
$J_{0}t=0.1L^{3}-0.46L^{2}+28L-260$ for the cosine protocol and
$J_{0}t=0.00052L^{3}+0.059L^{2}-0.34L+68$ for the exponential protocol, which
ensures the 0.99-fidelity QST for both protocols in the absence of losses. The
results indicate that the exponential protocol not only exhibits stronger
robustness against environment-induced decoherence, but also manifests good
scalability, whereas the fidelity of the cosine protocol plummets with
increasing size.

Figure 15: Schematic illustration of the crossed-chain structure comprised of
four even-sized SSH chains connected to a mutual additional a-type site.

####  III.4.2 Number of chains

The symmetrical topological beam splitter based on an odd-sized SSH model with
alternating onsite energies and a topological interface can be regarded
structurally as a system composed of two even-sized SSH chains connected to a
mutual additional a-type site. Another vital direction for scalability is how
the exponential protocol behaves when the number of constituent chains in the
crossed-chain structure is altered. To this end, in Fig. 15 without loss of
generality we consider a crossed-chain structure comprised of $K=4$ even-sized
SSH chains connected to a mutual additional a-type site, which will be taken
as a typical example and analyzed extensively in the following. The
interaction of the crossed-chain structure formed by $L=4N+1$ sites can be
described by the following interaction-picture Hamiltonian

$\displaystyle H^{\prime}$ $\displaystyle=$
$\displaystyle\sum_{\sigma}\sum_{n}\left[V_{a}a_{n}^{\sigma\dagger}a_{n}^{\sigma}+V_{b}b_{n}^{\sigma\dagger}b_{n}^{\sigma}\right]+\sum_{\sigma}\sum_{\mathrm{n}=1}^{N/2}\left[J_{1}a_{n}^{\sigma\dagger}b_{n}^{\sigma}\right.$
(21) $\displaystyle\left.+J_{2}a_{n+1}^{\sigma\dagger}b_{n}^{\sigma}+\text{
H.c. }\right],$

where $a_{N/2+1}^{\sigma}=a_{N/2+1}=a_{0}$, $a_{n}^{\sigma}$ and
$b_{n}^{\sigma}$ are the amplitudes at the $n$th a\- and b-type sites in a
single SSH chain indexed by $\sigma$. If we regard the connecting site as the
input port and the $K$ end sites as $K$ output ports, the crossed-chain
structure is equivalent to a topological router, in which a particle injected
into the connecting site can be transferred to $K$ end sites with equal
probabilities. To verify the fast QST for the exponential protocol in
topological router with four output ports, we plot the fidelity versus the
total transfer time for the cosine and exponential protocols with $\alpha=3.2$
in Fig. 16(a). The QST process for the exponential protocol is still about 10
times faster than its cosine counterpart, since fidelity is stabilized above
0.99 after $t^{\ast}=91/J_{0}$ for the exponential protocol as compared to
$t^{\ast}=935/J_{0}$ for the cosine protocol. The process of QST and the
amplitude distribution of the evolved final state under the basis of

$\displaystyle C$ $\displaystyle=$
$\displaystyle\left(a_{1}^{1},b_{1}^{1},\ldots,a_{N/2}^{1},b_{N/2}^{1},a_{1}^{2},\ldots,b_{N/2}^{2},a_{1}^{3},\ldots,b_{N/2}^{3},a_{1}^{4},\right.$
$\displaystyle\left.\ldots,b_{N/2}^{4},a_{N/2+1}\right)$

for the exponential and cosine protocols are illustrated in Figs. 16(b)-(e),
implying that both protocols can achieve successful topological routing under
sufficient transfer time, but the former protocol is obviously faster.

Figure 16: (a) Final fidelity of four-outport router as a function of the
transfer time for the cosine and exponential protocols. (b)-(e) Distribution
of the gap state with energy eigenvalue of $V_{a}$ during the evolution and
amplitude distribution of the evolved final state for the exponential protocol
in (b) and (c), and the cosine protocol in (d) and (e), respectively. Other
parameters take $L=4N+1=41$ and $\alpha=3.2$. Figure 17: Using the exponential
protocol with $\alpha=3.2$ for the four-outport router, transfer time
$t_{0.99}^{\ast}$ as a function of (a) the size of each constituent chain with
$K=4$ and (b) the number of constituent chains with $N=10$.

We plot in Fig. 17(a) the transfer time $t_{0.99}^{\ast}$ as a function of the
size $N$ of each constituent chain. Total transfer time increases with the
augmentation of the size of each constituent chain for the four-chain
structure, which is consistent with the results in the two-chain beam
splitter. The transfer time $t_{0.99}^{\ast}$ as a function of the number of
constituent chains is illustrated in Fig. 17(b), where exponential parameter
is set as $\alpha=3.2$ and the size of each constituent chain is chosen to be
$N=10$. We can see that in general, total transfer time increases with the
augmentation of the number of constituent chains connected in the crossed-
chain structure, with mild fluctuations which can be attributed to inevitable
oscillation in the $F$–$t^{\ast}$ curves. Therefore, by modulating the number
of crossed linked chains, the number of outports can be adjusted conveniently.
Such good flexibility and scalability give the topological beam splitter and
topological router broad application prospects in quantum information
distribution and large-scale quantum information network construction.

Figure 18: Equivalent circuit of the coupled superconducting resonator system.
Circuit elements are used to model the microwave resonator $A_{n}$ ($B_{n}$)
and the coupler with the additional Josephson junction in a dilution
refrigerator (with a temperature $T\sim$ mK), which is placed in a magnetic
shield. The microwave resonator $A_{n}$ ($B_{n}$) is an $LC$ circuit composed
of a spiral inductor $L_{a}$ ($L_{b}$) and a capacitor $C_{a}$ ($C_{b}$). The
external classical filed can be attained independently via changing the
magnetic flux $\phi$ threading on the loop of coupler, which can add the FBL
to connect with an AWG by adopting controlled voltage pulses Peropadre  _et
al._ (2010); DiCarlo  _et al._ (2009).

##  IV Experimental Consideration and Anticipated Improvement

###  IV.1 Experiment consideration for superconducting circuit devices

This protocol for realizing fast and robust QST in a symmetrical topological
beam splitter is applicable to superconducting circuit devices, which benefits
from existing circuit-QED technologies.

We can construct a superconducting resonator chain to arrange alternately the
resonator $A_{n}$ and the resonator $B_{n}$ in one-dimensional space, whose
equivalent circuit of one unit cell is shown in Fig. 18. The resonator $A_{n}$
($B_{n}$) is composed by a spiral inductor $L_{a}$ ($L_{b}$) and a capacitor
$C_{a}$ ($C_{b}$) in analogy with harmonic oscillator, which has a single
mode. In terms of the capacitor charge $Q_{a}$ ($Q_{b}$) and the inductor
current $I_{a}$ ($I_{b}$), the Hamiltonian of oscillator is written as

$\displaystyle\hat{H}_{LC}$ $\displaystyle=$
$\displaystyle\frac{Q_{j}^{2}}{2C_{j}}+\frac{\Phi_{j}^{2}}{2L_{j}}$ (22)

where $\Phi_{j}$ is the flux through the inductor $L_{j}$, and $Q_{j}$ is the
charge on the capacitor $C_{j}$ ($j=a,b$). Based on the standard quantization
process of an $LC$ circuit Gu  _et al._ (2017), the Hamiltonian of the
resonator $A_{n}$ ($B_{n}$) can be further written as $H_{\rm{LC}}=\hbar
V_{j}j^{\dagger}j$ in terms of the creation and annihilation operators defined
by $j^{\dagger}=1/\sqrt{2\hbar
V_{j}}(Q_{j}/\sqrt{C_{j}}-i\Phi_{j}/\sqrt{L_{j}})$ and $j=1/\sqrt{2\hbar
V_{j}}(Q_{j}/\sqrt{C_{j}}+i\Phi_{j}/\sqrt{L_{j}})$, where
$V_{j}=1/\sqrt{L_{j}C_{j}}$ is the oscillator frequency. Thus, onsite energies
of two types of sites $A_{n}$ ($B_{n}$) can be engineered in a large range of
possible values by adjusting the parameters $L_{j}$ and $C_{j}$. In
experiment, the onsite energies of the resonator can be selectively controlled
by a DC bias voltage supply connected with the variable capacitor via a low-
pass filter. The dependence of its resonant frequency on the DC bias is
observed with no hysteresis, which is of great value for tunability He  _et
al._ (2019b).

Furthermore, a direct tunable coupler is realized by a tunable circuit element
between the resonators, e.g., a flux-biased direct-current superconducting
quantum interference device (SQUID) to generate strong resonant and
nonresonant tunable interactions between any two lumped-element resonators. In
this work, we adopt a direct tunable coupler replaced by a SQUID between
resonator $A_{n}$ and $B_{n}$, which is composed by the additional Josephson
junction $E_{J}$ Allman  _et al._ (2010, 2014). The flux $\phi$ threading the
SQUID loop gives rise to a circulating current
$I_{s}(\phi)=-I_{c}\sin(2\pi\phi/\phi_{0}$), where $\phi_{0}$ is the flux
quantum. Here, $\phi$ is the sum of the externally applied flux
$\phi_{\rm{ext}}$ and the flux generated by $I_{s}$, which can be represented
by $\phi=\phi_{\rm{ext}}+L_{s}I_{s}(\phi)$. Thus, the flux dependent coupling
between resonators is written as
$J_{1,2}=-\sqrt{\frac{V_{a}}{L_{a}}}\sqrt{\frac{V_{b}}{L_{b}}}\frac{L^{2}_{0}}{L(\phi)}$,
where $L_{0}$ is the inductance of the segment shared between resonator and
SQUID and the effective SQUID inductance with respect to external fluxes
$L(\phi)=\partial\phi_{\rm{ext}}/\partial I_{s}$ Schwarz  _et al._ (2016).
Therefore, the simplest way to tune the coupling strength $J_{1,2}$ is to
apply a control magnetic flux to this loop dynamically with
$\phi_{\rm{ext}}(\vec{x})=\int_{s}\textbf{B}(\vec{x},t)\cdot d\textbf{S}$, by
adding the external flux-bias line (FBL) to connect with an arbitrary
waveforms generator (AWG) by adopting controlled voltage pulses Peropadre  _et
al._ (2010); DiCarlo  _et al._ (2009), as shown in Fig. 18. Thus,
superconducting circuits possessing advantages of flexibility, scalability and
tunability Blais  _et al._ (2004); Clarke and Wilhelm (2008); You and Nori
(2005), providing an excellent platform for realizing fast and robust QST in a
symmetrical topological beam splitter with high fidelity.

###  IV.2 Possibility of further accelerating QST process in the symmetrical
beam splitter

As noted above, we have realized fast QST in the symmetrical beam splitter
through exponential modulation of the nearest-neighbor coupling strengths and
the onsite energies. The scheme acclerates the beam splitting process through
subtle control of the driving functions according to the instantaneous
eigenspetrum and is still limited by the adiabatic requirements. To further
accelerate the QST process, we can consider incorporating moderately
nonadiabatic resonant process between eigenstates into the adiabatic process.
For example, a fast topological edge pumping protocol in which quantum state
transfers rapidly from the left edge to the right was recently presented in an
SSH chain Brouzos  _et al._ (2020). The intracell and intercell nearest
neighbor coupling strenghts are governed by the following 3-step modulation
functions

$J_{1}=\begin{cases}J_{1}(0),&t\leq t^{*}-t_{op}\\\
\frac{(J_{1}(0)-J_{2}(0))t^{*}}{t_{op}}\left(1-\frac{t}{t^{*}}\right),&t>t^{*}-t_{op}\\\
\end{cases}$ (23a)
$J_{2}=\begin{cases}J_{2}(0)+\frac{(J_{1}(0)-J_{2}(0))t}{t_{op}},&t\leq
t_{op}\\\ J_{1}(0),&t>t_{op}\\\ \end{cases}\\\ $ (23b)

where $J_{1}(0)$ and $J_{2}(0)$ denote the intracell and intercell coupling
coefficients at the initial moment, $t^{*}$ is the total evolution time, and
$t_{op}$ is the time interval for the coupling strengths to increase
(decrease) from initial to terminal value (and vice versa). $J_{1}$ and
$J_{2}$ are of mirror symmetry, so $t_{op}$ is the only free parameter. It is
possible to find the best $t_{op}$ value through parameter optimization so as
to produce the fastest topological egde pumping.

Different from the case of the commonly-used trigonometric protocol in which
eigenenergy of the edge mode for an odd-sized SSH model remains constant
during the transfer process, for the 3-step protocol the instantaneous
eigenenergy are bended and its mean value is significantly increased.
Therefore, the timescales which may be considered inversely proportional to
the time average of eigenenergy of the edge state effectively decrease. The
key to this scheme is incorporating the nonadiabatic resonance transition,
whose dynamical evolution is closely related to the pulse area. Analogously,
with the help of such a 3-step modulation protocol, it is of great potential
to furthur speed up the QST in the symmetrical beam splitter proposed here.

##  V Conclusion

To sum up, we have proposed a protocol of fast and robust topological pumping
via edge channel through exponential modulation of the driving functions for
generating a symmetrical topological beam splitter and further for deriving a
topological router. We show both analytically and numerically that by
continuously modulating the intracell and intercell coupling strengths and
onsite energies, we can achieve topologically protected quantum state transfer
(QST) from the interface site towards end sites with equal probabilities.
Based on numerical analysis of the instantaneous energy spectrum of the
system, we confirm that the value of the instantaneous energy gap suitably
adapts to the slope of the driving functions, and then present numerical
evidence of accelerated adiabatic edge pumping in the symmetrical beam
splitter. Furthermore, we investigate how the selection of the exponential
parameter impact the QST process. The robustness of the topological beam
splitter is extensively discussed by taking into consideration the impact of
diagonal and off-diagonal disorders and systematic losses. In addition, we
prove the scalability in the size and number of chains for the symmetrical
beam splitter and topological router, respectively. Last but not least, we
propose superconducting circuit devices as a feasible platform to implement
fast and robust QST in the symmetrical beam splitter discussed in this
article. The scheme provides detailed assumption of topological beam splitter
and topological router assisted by fast and robust topological edge pumping,
which is expected to make substantial contribution to efficient quantum
information processing and the construction of large-scale quantum networks.

## Acknowledgements

The authors acknowledge the financial support by the National Natural Science
Foundation of China (Grant No. 62075048) and Natural Science Foundation of
Shandong Province of China (Grant No. ZR2020MF129).

## Appendix: Hybridized edge states for even-sized SSH model

For the even-sized SSH model, the matrix representation of the Hamiltonian on
a real-space basis reads

$H_{M}=\left(\begin{array}[]{cccccccc}0&J_{1}&0&0&\cdots&0&0&0\\\
J_{1}&0&J_{2}&0&\cdots&0&0&0\\\ 0&J_{2}&0&J_{1}&\cdots&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&0&\cdots&J_{2}&0&J_{1}\\\ 0&0&0&0&\cdots&0&J_{1}&0\end{array}\right).$
(A1)

In the following we prove that in the thermodynamic limit of $N\to\infty$, the
topological nontrivial phase hosts two edge states localized on the boundaries
of the chain. We first consider a semi-infinite long SSH lattice with left
boundry. In topological nontrivial phase, we assume that the SSH lattice
exhibits zero-energy eigenstate

$|L\rangle=|\psi_{a_{1}},\psi_{b_{1}},\psi_{a_{2}},\psi_{b_{2}},\cdots,\psi_{a_{n}},\psi_{b_{n}},\cdots\rangle,$
(A2)

where $\psi_{a_{n}}~{}(\psi_{b_{n}})$ are the amplitudes on lattice site
$a_{n}~{}(b_{n})$. Solving the eigenvalue equation $H_{M}|L\rangle=0$ ($H_{M}$
under right semi-infinite boundary condition), we get

$J_{1}\psi_{b_{1}}=0,$ (A3a)
$J_{1}\psi_{a_{n}}+J_{2}\psi_{a_{n+1}}=0,(n=1,2,\cdots),$ (A3b)
$J_{2}\psi_{b_{n}}+J_{1}\psi_{b_{n+1}}=0,(n=1,2,\cdots),$ (A3c)

The zero-energy edge state is derived as

$|L\rangle=|\psi_{a_{1}},0,\xi\psi_{a_{1}},0,\cdots,\xi^{n-1}\psi_{a_{1}},0,\cdots\rangle,$
(A4)

where $\xi=-J_{1}/J_{2}$ denotes the localization factor. Obviously, the edge
state is exponentially localized in the left side of the lattice in
topological nontrivial phase and only occupies the a-type sites. Similiarly,
for a semi-infinite long SSH lattice with right boundry, the zero-energy edge
state can be derived as

$|R\rangle=|\cdots,0,\xi^{N-n}\psi_{b_{N}},0,\cdots,0,\xi\psi_{b_{N}},0,\psi_{b_{N}}\rangle,$
(A5)

which is exponentially localized in the right side of the lattice in
topological nontrivial phase and only occupies the b-type sites.

For an even-sized SSH lattice composed of finite sites, we can get its
eigenvalues and corresponding eigenstates by diagnolizing the real-space
Hamiltonian under base vectors $|L\rangle$ and $|R\rangle$

$H_{M}^{\prime}=\left(\begin{array}[]{cc}O_{L,L}&O_{L,R}\\\
O_{R,L}&O_{R,R}\end{array}\right),$ (A6)

where $O_{L,L}=\left\langle L\mid H_{M}\mid L\right\rangle=0$,
$O_{L,R}=\left\langle L\mid H_{M}\mid
R\right\rangle=\frac{-J_{2}\xi^{N}(\xi^{2}-1)}{\xi^{2N}-1}$,
$O_{R,L}=\left\langle R\mid H_{M}\mid L\right\rangle=O_{L,R}^{*}$, and
$O_{R,R}=\left\langle R\mid H_{M}\mid R\right\rangle=0$. The eigenvalues and
corresponding eigenstates are

$E_{0,\pm}=\pm\left|O_{L,R}\right|,$ (A7a)
$\left|\Psi_{0,\pm}\right\rangle=(|L\rangle\pm|R\rangle)/\sqrt{2},$ (A7b)

Obviously, for an even-sized SSH lattice composed of finite sites, energies of
the hybridized edge states in topological nontrivial phase do not equal zero,
but a pair of numbers opposite to ecah other due to chiral symmetry
($E_{0,\pm}\rightarrow 0$ in the thermodynamic limit of $N\to\infty$). The
wavefunctions of almost-zero-energy eigenstates are odd and even
superpositions of states localized exponentially on the left and right edges.

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