# Fast quantum transfer mediated by topological domain walls

Juan Zurita  Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco,
E-28049 Madrid, Spain  Departamento de FÃsica de Materiales, Universidad
Complutense de Madrid, E-28040 Madrid, Spain<EMAIL_ADDRESS>Charles E.
Creffield  Departamento de FÃsica de Materiales, Universidad Complutense de
Madrid, E-28040 Madrid, Spain  Gloria Platero  Instituto de Ciencia de
Materiales de Madrid (CSIC), Cantoblanco, E-28049 Madrid, Spain

###### Abstract

Transfer protocols in 1D topological models are usually implemented between
the left and right end modes of the system, and their speed often
exponentially reduces with transfer distance. In this work, we propose a way
to harness the localization properties of a multidomain Creutz ladder, a flat-
band topological model with two protected states per domain wall, to choose
the two modes along the ladder which will be swapped using a transfer
protocol. Only protected states are involved in the transfer, and so it is
robust against symmetry-preserving disorder. Remarkably, one protected state
per domain wall is left undisturbed, even if it is located between the two
swapped states. An effective 1D chain of protected states is then established,
where any pair of them can be swapped without affecting the others, providing
a versatile and resilient platform for quantum information purposes.
Additionally, we propose transfer protocols in multidomain Creutz ladders and
SSH chains which exponentially speed up the process with respect to their
single-domain counterparts, thus reducing the accumulation of errors and
drastically increasing their performance, even in the presence of symmetry-
breaking disorder.

##  I Introduction

We are currently experiencing the NISQ era of quantum computing (Preskill,
2018), where noisy, intermediate-scale quantum processors start to show the
first signs of quantum advantage. The path ahead towards large error-
correcting codes and, eventually, fault-tolerant quantum computation will most
likely require a wide array of tools and techniques to fight decoherence. One
of these techniques can be the use of symmetry-protected states in topological
insulators, which are specially robust against some types of noise, and are
already available in the laboratory.

These topological boundary states naturally decouple from the bulk of the
system, and thus form an invariant subspace of the Hamiltonian around zero
energy, which has support on the boundaries of the material. This allows us to
define quantum information protocols in 2D (Yao  _et al._ , 2013; Dlaska  _et
al._ , 2017; Lemonde  _et al._ , 2019; Bello  _et al._ , 2017) and 1D (Leijnse
and Flensberg, 2011; Lang and Büchler, 2017; Jin  _et al._ , 2017; Estarellas
_et al._ , 2017; Mei  _et al._ , 2018; Boross  _et al._ , 2019; Yuce, 2019;
D’Angelis  _et al._ , 2020; Tan  _et al._ , 2020; Comaron  _et al._ , 2020;
Cao  _et al._ , 2021; Chen  _et al._ , 2021; Palaiodimopoulos  _et al._ ,
2021; Han  _et al._ , 2021; Yuan  _et al._ , 2021) which transport a particle
from one region of the boundary to another, with little to no loss to the
bulk, even in noisy regimes. These _particle_ transfer protocols can then be
used in different ways to implement _state_ transfer protocols –for example,
using these particles (e.g. photons) as a communication line between distant
qubits (Lang and Büchler, 2017), or building a quantum computer with specific
couplings so that its qubit states are equivalent to the empty ($\ket{0}$) and
filled ($\ket{1}$) states of a topological Hamiltonian (Mei  _et al._ , 2018)–
or other remote quantum operations (Lang and Büchler, 2017; Boross  _et al._ ,
2019). In this work we are mainly concerned with the particle transfer states
in the topological systems we study, leaving their specific applications in
quantum information protocols for future works.

In particular, we investigate the possibilities of particle control provided
by magnetic interference in the Creutz ladder (CL) (Creutz, 1999), a quasi-1D
topological insulator which was recently realized in three different cold atom
systems (Song  _et al._ , 2018; Hyoun Kang  _et al._ , 2020; He  _et al._ ,
2021), and can also be implemented with state-of-the-art technology in
superconducting circuits (Alaeian  _et al._ , 2019; Hung  _et al._ , 2021) (in
which a CL plaquette has been implemented) and photonic lattices (Mukherjee
_et al._ , 2018). In a particular regime, an orthogonal basis of spatially
compact energy eigenstates can be found, something associated with the
complete flattening of both bands in momentum space. This phenomenon is called
Aharonov-Bohm caging (Vidal  _et al._ , 1998; Creffield and Platero, 2010;
Bello  _et al._ , 2016; Jünemann  _et al._ , 2017), and it is caused by the
destructive interference of paths due to the magnetic Peierls phases. Flat
band models are interesting playgrounds to study the effects of interactions
and disorder (Tovmasyan  _et al._ , 2013; Takayoshi  _et al._ , 2013; Derzhko
_et al._ , 2015; Roy  _et al._ , 2020; Nakai and Hotta, 2022), and, like
topological models, also show interesting properties for quantum information
purposes (Röntgen  _et al._ , 2019; Kuno  _et al._ , 2020). In a Creutz ladder
with multiple topological domains, where each domain wall can hold up to two
protected states, it can allow us to decouple some of the boundary states of
the system from all the others at will.

Remarkably, if this is done to a state in a two-state domain wall, its partner
can be used to leapfrog over it, thus allowing the transfer of a particle
through the wall while leaving the decoupled state undisturbed and protected
inside it. Even though the CL topological phase diagram is well-known (Li and
Chen, 2015; Zurita  _et al._ , 2019), the properties of its possible domain
walls have not been studied in the literature, to the best of our knowledge.

Additionally, we tackle a well-known issue in protocols that involve the
hybridization of exponentially decaying end modes, which is the –also
exponential– increase in the time of transfer as a function of system length.
A solution for this problem can be found in multidomain models, in which the
protected states in domain walls along the system can act as signal
amplifiers, exponentially decreasing the transfer time between distant states
even with the most simple control protocols. After studying the case of the
Creutz ladder, we also include a version of these exponentially accelerated
transfer protocols for the SSH chain, a 1D model which is much easier to
implement experimentally, and where each domain wall can only hold one
boundary state. Domain walls in the SSH chain and related models have been
studied since its inception (Su  _et al._ , 1980; Munoz  _et al._ , 2018;
Longhi, 2019; D’Angelis  _et al._ , 2020; Qi  _et al._ , 2021), and other
transfer protocols using them exist in the literature (Longhi, 2019), but our
proposal have not been yet explored, as far as we know.

In Section II, we present the multidomain Creutz ladder, study its protected
states and describe the possible transfer protocols that can be implemented in
it. Then, we explore the exponentially accelerated transfer protocols that can
be obtained using domain walls as signal amplifiers, both in the CL (Section
III) and in the SSH chain (Section IV), as well as their resistance against
disorder.

Figure 1: (a) Imbalanced Creutz ladder in the most general case. The energy
imbalance parameters $\epsilon_{j}$ and the magnetic Peierls phases $\phi_{j}$
determine the topology, and can be used to define different domains. All
hopping amplitudes have an additional minus sign. (b) Topological phase
diagram of the imbalanced Creutz ladder. It is divided into two topological
phases with winding numbers $\nu=\pm 1$, and a trivial phase with $\nu=0$. The
topology of the crystalline TI regions are not described by the standard
Altland-Zirnbauer symmetry classes (Altland and Zirnbauer, 1997). Those who
can are marked with “AZ”. The flat-band points are labeled “FB”. (c) Two-
domain Creutz ladder with no imbalance in its domain wall ($\epsilon_{6}=0$),
and with an energy imbalance parameter equal to $\epsilon$ in all other rungs.
This model has four topological states: the left and right end modes, and two
states at the domain wall. (d) Spatial distribution of the AB-caged
topological states in a two-domain Creutz ladder with $\epsilon=0$: left,
right and $\mathcal{S}$\- and $\mathcal{P}$-type states. All of them are
compact and pinned at zero energy. (e) Spatial distribution of the topological
states in the system in (c), for $\epsilon=J$. A pair of antibonding and
bonding states $\ket{\pm}$ appear with small, opposite energies, while the
other two states remain at zero energy.

##  II Multidomain Creutz ladder

###  II.1 Topological phases in the Creutz ladder

We consider an imbalanced Creutz ladder Hamiltonian:

$\displaystyle\mathcal{H}$
$\displaystyle=-\sum_{j=1}^{L-1}\sum_{\sigma=A,B}\left[J\xi_{j,\sigma}c_{j+1,\sigma}^{\dagger}c_{j,\sigma}+Jc_{j+1,\overline{\sigma}}^{\dagger}c_{j,\sigma}+h.c.\right]$
$\displaystyle+\sum_{j=1}^{L}\sum_{\sigma=A,B}s_{\sigma}\epsilon_{j}c_{j,\sigma}^{\dagger}c_{j,\sigma}$
(1)

where $j=1,\dots,L$ labels the different rungs, $\sigma=A,B$ designates the
two legs, with $\overline{A}=B$ and vice versa,
$\xi_{j,\text{$\sigma$}}=e^{is_{\sigma}\phi_{j}/2}$, with
$s_{\sigma}=\delta_{\sigma,A}-\delta_{\sigma,B}$, $\phi_{j}$ is the magnetic
flux in the $j$-th plaquette, $J$ is the horizontal and diagonal hopping
amplitude and $2\epsilon_{j}$ is the energy imbalance between the two legs in
the $j$-th rung [see Fig. 1 (a)]. Operators $c_{j,\sigma}^{\dagger}$ create a
particle, which can be bosonic or fermionic, in site $j,\sigma$.

In a ladder with $\epsilon_{j}=\epsilon$, $\phi_{j}=\phi$ for all values of
$j$, the topology of the system depends on $\phi$ and $\epsilon/J$, as can be
seen in Fig. 1 (b). The system has two distinct nontrivial phases when
$\epsilon<2J$, with winding numbers $\nu=\pm 1$, depending on the value of
$\phi$, and a single trivial phase ($\nu=0$) when $\epsilon>2J$ (Li and Chen,
2015; Sun and Lim, 2017; Jünemann  _et al._ , 2017). When in a topological
phase, topological zero modes appear in the ends of the ladder.

Furthermore, the wall between domains that belong to different phases, with
winding numbers $\nu_{1}$ and $\nu_{2}$, will hold $|\Delta\nu|$ zero modes,
where $\Delta\nu=\nu_{2}-\nu_{1}$. Due to the particular form of the chiral
symmetry, these states will have a positive chirality if $\Delta\nu$ is
_negative_ , and vice versa. Thus, a topological-to-trivial wall will support
one zero mode, while the wall separating phases with $\nu=\pm 1$ will support
two zero modes with the same chirality.

Figure 2: (a) $\mathcal{L}$-to-$\mathcal{R}$ transfer protocol in a two-domain
Creutz ladder with $\ell=4$, induced by an energy imbalance of
$\pm\epsilon_{\mathrm{tr}}$. One of the topological states in the domain wall,
$\ket{\mathcal{S}}$, remains pinned at zero energy due to AB caging, while the
other one, $\ket{\mathcal{P}}$, hybridizes with the left and right states to
allow the transfer. (b) $\mathcal{L}$-to-$\mathcal{S}$ transfer in a two-
domain ladder. The initial state, not pictured, is the same as in (a). The
second domain has to be taken to the trivial phase using a control parameter
of $\epsilon_{\mathrm{bar}}>2J$ before inducing the transfer process. (c)
$\mathcal{L}$-to-$\mathcal{R}$ transfer in a three-domain ladder, starting in
state $(\ket{\mathcal{L}}+\ket{\mathcal{S}_{1}})/\sqrt{2}$. The final state is
$(\ket{\mathcal{S}_{1}}-\ket{\mathcal{R}})/\sqrt{2}$. As can be seen, the
transfer does not affect the intermediate $\mathcal{S}$-type states.

###  II.2 Topological domains and walls

We consider a Creutz ladder with $N$ topological domains, with their winding
number taking alternate values of $\nu=\pm 1$. We set $J=1$. Each domain will
consist of two end rungs and a number $\ell\geq 2$ of inner rungs. The length
of the ladder is then $L=N(\ell+1)+1$. Each end rung, except the left- and
rightmost ones, is shared between two domains, and constitutes a domain wall.
This model is obtained with the following parameter scheme:

$\phi_{j}=\begin{cases}\pi&\textrm{if }1\leq j\mod 2(\ell+1)\leq\ell+1\\\
-\text{$\pi$}&\textrm{otherwise }\end{cases}$ (2)

$\epsilon_{j}=\begin{cases}\epsilon_{w}^{(D[j])}&\textrm{if
}j\mod\ell+1=1\textrm{ (domain walls)}\\\
\epsilon_{b}^{(D[j])}&\textrm{otherwise (bulk sites)},\end{cases}$ (3)

where $D[j]=\lceil(j-1)/(\ell+1)\rceil$ is the domain number to which rung $j$
belongs, with $\lceil\cdot\rceil$ being the ceiling function. For this
purpose, we label each wall just like the domain to its left. The leftmost
rung of the ladder is labeled as belonging to domain $D[j=1]=0$.

Figure 3: Numerical simulations of the processes in Fig. 2, and the control
parameter pulses used. (a) Left-to-right transfer in a two-domain ladder. (b)
Left-to-center transfer in a two-domain ladder. The transfer-inducing
parameter $\epsilon_{\textrm{tr}}$ and the barrier $\epsilon_{\textrm{bar}}$
are shown on different scales. (c) Left-to-right transfer in a three-domain
ladder, acting on the initial state
$(\ket{\mathcal{L}}+\ket{\mathcal{S}_{1}})/\sqrt{2}$. The component at
$\mathcal{\ket{\mathcal{L}}}$ gets transferred, while the component at
$\ket{\mathcal{S}_{1}}$ remains unperturbed. The control parameter in the
central domain takes a maximum value of $0.952$, see main text. The rung
occupation number is represented as a function of time $t$ for each rung $j$.
Note that different colormaps are used for subfigures (a,b) and subfigure (c).
Preparation times of $t_{\textrm{prep}}=t{}_{\textrm{prep}}^{\prime}=30$ are
used. All times are expressed in units of $\hbar/J$.

An example with $\ell=4$ and two domains can be seen in Fig 1 (c). A given
domain $D_{0}$ with a flux of $\phi=\pm\pi$ will have a winding number of
$\nu=\pm 1$, respectively, as long as its energy imbalance satisfies
$\epsilon_{b}^{(D_{0})}<2J$. If $\epsilon_{b}^{(D_{0})}>2J$, then $\nu=0$ and
the domain is trivial.

If we set $\epsilon_{j}=0\,\,\forall j$ (balanced case) with the flux values
defined above, the model will have left ($\ket{\mathcal{L}})$ and right
($\ket{\mathcal{R}}$) topological end states, as well as two topological
states, $\ket{\mathcal{S}_{k}}$ and $\ket{\mathcal{P}_{k}}$, in each domain
wall $k=1,\ldots,N-1$. Given that the system presents AB caging (Jünemann  _et
al._ , 2017; Zurita  _et al._ , 2019), all of them will be compact, i.e.
localized in only a few nearby sites. An $N$-domain CL defined in this way has
$2N$ compact topological states pinned at exactly zero energy: $(N-1)$
$\mathcal{S}$\- and $\mathcal{P}$-type states, and the left and right end
modes. All of them are localized in two sites, except for the
$\mathcal{P}$-type states, which are localized in four. Their form is:

$\displaystyle\ket{\mathcal{L}}$
$\displaystyle=\frac{\ket{1,A}-i\ket{1,B}}{\sqrt{2}}$ (4)
$\displaystyle\ket{\mathcal{R}}$
$\displaystyle=\frac{\ket{L,A}+(-1)^{N+1}i\ket{L,B}}{\sqrt{2}}$ (5)
$\displaystyle\ket{\mathcal{S}_{k}}$
$\displaystyle=\frac{\ket{j_{k},A}+(-1)^{k+1}i\ket{j_{k},B}}{\sqrt{2}}$ (6)
$\displaystyle\ket{\mathcal{P}_{k}}$
$\displaystyle=\frac{1}{2}[\ket{j_{k}-1,A}-\ket{j_{k}+1,A}+$
$\displaystyle(-1)^{k+1}i(\ket{j_{k}-1,B}-\ket{j_{k}+1,B})],$ (7)

where $k=1,\ldots,N-1$, and $j_{k}=k(\ell+1)+1$ is the rung in which the
$k$-th wall is located. We use $\mathcal{S}$ and $\mathcal{P}$ as nomenclature
because their form and spatial symmetry are analogous to the $s$ and $p$
orbitals in an atom (or to the two lowest levels in a quantum well in
general). The four states in a two-domain ladder are represented in Fig. 1
(d), using their rung occupation number:

$\langle n_{j}\rangle=\langle n_{j,A}\rangle+\langle n_{j,B}\rangle.$ (8)

For reasons that will become apparent later, we will group left, right and
$\mathcal{S}$-type states under the name of “computational states”.

We now switch on the energy imbalance for all sites except those in the domain
walls without reaching the topological phase transition, that is,
$\epsilon_{w}^{(0)}=\epsilon_{w}^{(N)}=\epsilon_{b}^{(D)}=\epsilon_{0}<2J\,\,\forall
D$, $\epsilon_{w}^{(D)}=0\,\,\forall D\in[1,N-1]$. Then, the left, right and
$\mathcal{P}$-type topological states will acquire a finite overlap with each
other and hybridize 111Actually, their overlap (e.g.
$\braket{\mathcal{L}}{\mathcal{P}}$) is always zero due to their different
chiralities, but the effective hopping amplitude between them
($\bra{\mathcal{L}}\mathcal{H}\ket{\mathcal{P}}$) is not. We employ this
slight abuse of language throughout the paper for convenience. Their
analytical expression before hybridization is:

$\displaystyle\ket{\mathcal{L}}$
$\displaystyle=-i\mathcal{N_{L}}\sum_{j=1}^{\ell+1}\left(\frac{2iJ}{\epsilon}\right)^{-j}\ket{j}\otimes\left(\begin{array}[]{c}1\\\
-i\end{array}\right)$ (11) $\displaystyle\ket{\mathcal{R}}$
$\displaystyle=(-1)^{N+1}i\mathcal{N_{R}}\sum_{j=L-\ell}^{L}\left((-1)^{N}\frac{2iJ}{\epsilon}\right)^{j-L-1}\ket{j}\otimes\left(\begin{array}[]{c}1\\\
(-1)^{N+1}i\end{array}\right)$ (14) $\displaystyle\ket{\mathcal{P}_{k}}$
$\displaystyle=(-1)^{k+1}i\mathcal{N}_{\mathcal{P}_{k}}\left[\sum_{j=(k-1)(\ell+1)+2}^{k(\ell+1)}\left((-1)^{k}\frac{2iJ}{\epsilon}\right)^{j-k(\ell+1)-1}\ket{j}\otimes\left(\begin{array}[]{c}1\\\
(-1)^{k+1}i\end{array}\right)\right.$ (17)
$\displaystyle\left.-\sum_{j=k(\ell+1)+2}^{(k+1)(\ell+1)}\left((-1)^{k}\frac{2iJ}{\epsilon}\right)^{k(\ell+1)-j+1}\ket{j}\otimes\left(\begin{array}[]{c}1\\\
(-1)^{k+1}i\end{array}\right)\right],$ (20)

where $\mathcal{N}_{\mathcal{L},\mathcal{R},\mathcal{P}}$ are normalization
constants, and can be approximated with a very small margin of error for
$\ell\gtrsim 3$ as:

$\mathcal{N_{L}}=\mathcal{N_{R}}\approx\sqrt{\frac{4J^{2}/\epsilon^{2}-1}{2}}$
(21)
$\mathcal{N_{P}\approx}\frac{1}{\sqrt{2}}\left(\frac{1}{4J^{2}/\epsilon_{l}^{2}-1}+\frac{1}{4J^{2}/\epsilon_{r}^{2}-1}\right)^{-1/2},$
(22)

where $2\epsilon_{l,r}$ are the values of the energy imbalance in the domains
to the left and right of the relevant domain wall. The gauge was chosen to
coincide with Eqs. (4-7).

Given that all domain walls are still balanced (i.e. both their on-site
energies remain at zero), all $\mathcal{S}$-type states will remain localized
on the two sites, due to magnetic interference. This is pictured in Fig. 1 (e)
for a two-domain ladder, where bonding and antibonding states appear, with the
form $\ket{\pm}=(\ket{\mathcal{L}}-\ket{\mathcal{R}})/2\pm
i^{\ell}\ket{\mathcal{P}}/\sqrt{2}$, as well as a dark state at zero energy,
$\ket{0_{\mathcal{LR}}}=(\ket{\mathcal{L}}+\ket{\mathcal{R}})/\sqrt{2}.$

Each state is confined to the domains adjacent to its domain wall, or to the
first (last) domain in case of the left (right) end states. All states have
the same exponential profile, which is analogous to the Majorana bound states
in the Kitaev chain, as derived in (Alase  _et al._ , 2017; Cobanera  _et al._
, 2018) for the balanced CL. The states deviate slightly from these
expressions for very short domain lengths and values of $\epsilon$ close to
the topological phase transition, due to finite size effects
($\delta_{\mathcal{L}}=\norm{\ket{\mathcal{L}}_{\mathrm{num}}-\ket{\mathcal{L}}_{\mathrm{analyt}}}=0.16$
for $\ell=2$ and $\epsilon=1.5J$), but are remarkably accurate otherwise
($\delta_{\mathcal{L}}=0.016$ for $\ell=4$ and $\epsilon=J$).

###  II.3 Effective Hamiltonian

Using these analytical expressions, we can obtain an effective Hamiltonian for
the topological states in an $N$-domain imbalanced CL with $\epsilon=0$ in its
walls:

$\mathcal{H}_{\mathrm{eff}}\\!=\\!v_{1}\outerproduct{\mathcal{P}_{1}}{\mathcal{L}}\\!+\\!v_{N}\outerproduct{\mathcal{R}}{\mathcal{\mathcal{\mathcal{P}}}_{N-1}}\\!+\\!\\!\sum_{k=2}^{N-1}\\!\\!v_{k}\outerproduct{\mathcal{\mathcal{\mathcal{P}}}_{k}}{\mathcal{\mathcal{\mathcal{P}}}_{k-1}}\\!+\\!h.c.,$
(23)

where $v_{1}=\bra{\mathcal{\mathcal{P}}_{1}}\mathcal{H}\ket{\mathcal{L}}$,
$v_{k}=\bra{\mathcal{P}_{k}}\mathcal{H}\ket{\mathcal{P}_{k-1}}$ for
$k=2,\ldots,N-1$, and
$v_{N}=\bra{\mathcal{R}}\mathcal{H}\ket{\mathcal{P}_{N-1}}$.

It takes the form of a 1D chain of length $N+1$, with only nearest-neighbour
hopping amplitudes. Each of the $\mathcal{S}$-type states is completely
decoupled, and so they do not appear in the Hamiltonian.

Using expressions (11-20), we obtain the following expression for the hopping
amplitudes:

$v_{k}=2(-1)^{d+p_{k-1}}\epsilon\mathcal{N}_{k}\mathcal{N}_{k-1}\left[(-1)^{k}\frac{2iJ}{\epsilon}\right]^{-d-2},$
(24)

where $\mathcal{N}_{k},\mathcal{N}_{k-1}$ are the normalization constant for
the involved states and $d$ is the distance between the maxima between
topological states numbers $k-1$ and $k$ , with $\ket{\mathcal{L}}$ being the
zeroth state and $\ket{\mathcal{R}}$ being the $N$-th. The distance is
$d=\ell-1$ between two $\mathcal{P}$ states, $d=\ell$ between an end state and
a $\mathcal{P}$ state, and $d=\ell+1$ between the two end states in a single-
domain ladder. $p_{k-1}=0$ if the $(k-1)$-th state is a $\mathcal{P}$ state,
and $p_{k-1}=1$ otherwise.

Using this effective Hamiltonian, it is easy to understand the dynamics of the
topological states. If the system is initialized in state $\ket{\mathcal{L}}$,
time evolution will eventually take it to state $\ket{\mathcal{R}}$, using
each $\mathcal{P}$-type state of the ladder as a way to leapfrog its domain
wall without affecting its $\mathcal{S}$-type state. The values of the energy
imbalance in each domain can be tuned in order to implement a transfer
protocol between the two end states.

We have used this effective Hamiltonian to estimate transfer times and
parameter values, but all numerical results have been obtained solving the
full model.

Figure 4: (a) Topological states involved in an LR transfer in a four-domain
Creutz ladder. (b) Associated effective model, corresponding to a 1D chain.
(c) Occupation of the topological states during an LR transfer in a two-domain
ladder with $\ell=4$ ($L=11$), where the only $\mathcal{P}$ state is
represented in yellow. Numerical results for the full Creutz ladder
Hamiltonian are pictured in continuous line, while the analytical prediction
by the effective Hamiltonian is plotted in dotted lines. This process is also
pictured in Fig. 3 (a). (d) Occupation of the topological states in an LR
transfer in the four-domain ladder pictured in (a), with $\ell=4$ ($L=21$).
(e) Effective model for the cold atoms system in (Song  _et al._ , 2018),
corresponding to a single-domain CL. (f) Setup for a two-domain CL. The
corresponding Raman potentials are pictured in purple. All hopping amplitudes
have an additional minus sign.

###  II.4 Controlled transfer protocol

In order to start and stop the transfer when desired without disrupting the
dynamics, the control parameter of the system, the energy imbalance, must be
turned on and off in a smooth way. To this end, we employ a control pulse with
three parts: first, we increase the control parameter in a sinusoidal manner;
then, we keep the parameter constant for some time, and finally, the parameter
returns to zero, also in a sinusoidal way. In domain $D$, this pulse has the
form:

$\epsilon_{\mathrm{tr}}^{(D)}(t)=\begin{cases}\epsilon_{\mathrm{tr}}^{(D)}\sin^{2}(\Omega
t)&\textrm{for }0<t<t_{\mathrm{prep}}\\\
\epsilon_{\mathrm{tr}}^{(D)}&\textrm{for
}t_{\mathrm{prep}}<t<t_{\mathrm{tr}}-t_{\mathrm{prep}}\\\
\epsilon_{\mathrm{tr}}^{(D)}\sin^{2}[\Omega(t-t_{\mathrm{tr}})]&\textrm{for
}t_{\mathrm{tr}}-t_{\mathrm{prep}}<t<t_{\mathrm{tr}},\end{cases}$ (25)

where $t_{tr}$ is the total transfer time. We allow the maximum value of the
pulse to be different for each domain, but always consider mirror-symmetric
distributions (i.e.
$\epsilon_{\mathrm{tr}}^{(D)}=\epsilon_{\mathrm{tr}}^{(N-D+2)}$). All times
are expressed in units of $\hbar/J$, and we take $\hbar=1$. The preparation
time $t_{\mathrm{prep}}=\pi/(2\Omega)$ must be sufficiently large (numerical
results suggest $t_{\textrm{prep}}\apprge 20/J$) so that the Hamiltonian
changes adiabatically and the particle remains in the subspace of
computational states, with the same components as in the initial
configuration. We consider this type of pulse, already used in the literature
(Longhi  _et al._ , 2019), because it provides smooth initial and final
preparation stages, but also includes a period where the control parameter is
constant, therefore simplifying its implementation.

The fidelity threshold required for the final state in all transfer
simulations of the paper is taken to be $f_{0}=0.995$, well above the current
estimations of quantum error correction thresholds, which are around $0.99$
(Knill, 2005; Wang  _et al._ , 2011; Fowler  _et al._ , 2012; Egan  _et al._ ,
2020).

We can see a scheme of this process in Fig. 2 (a,c) for two and three-domain
ladders, respectively. In Fig. 2 (c), state
$(\ket{\mathcal{L}}+\ket{\mathcal{S}_{1}})/\sqrt{2}$ (which can itself be
prepared using transfer operations) evolves to
$(-\ket{\mathcal{R}}+\ket{\mathcal{S}_{1}})/\sqrt{2}$, demonstrating that the
$\mathcal{S}$-type states located between the origin and destination of the
transfer are unaffected by it. Numerical simulations of these processes are
shown in Fig. 3 (a,c) for $\ell=4$,$J=1$, $t_{\textrm{prep}}=30$. Case (a) has
$\epsilon_{\mathrm{tr}}^{(1)}=\epsilon_{\mathrm{tr}}^{(2)}=1$.

Using the effective Hamiltonian it was found that, in general, the transfers
with $N>2$ have to be implemented using a different value of
$\epsilon_{\mathrm{tr}}^{(D)}=\epsilon_{\mathrm{tr}}^{(N-D+2)}$ for each pair
of domains related by mirror symmetry, in order to achieve a fidelity greater
than $f_{0}$. For example, the three-domain protocol in Fig. 2 (c) uses
$\epsilon_{\mathrm{tr}}^{(1)}=\epsilon_{\mathrm{tr}}^{(3)}=1$ and
$\epsilon_{\mathrm{tr}}^{(2)}=0.952$. The different values of
$\epsilon_{\mathrm{tr}}^{(D)}$ for all protocols, as well as further
explanation on this point, are provided in Appendix B. The use of different
control pulses in different domains can provide an exponential speed-up in
cases with large $L$, as we explore in Section III.

More complex protocols acting on the topological states can also be
considered, and will be explored in future works.

Finally, to demonstrate the accuracy of the effective Hamiltonian, transfers
between the left and right end modes (which we will call _left-to-right_ or
_LR transfers_) were simulated in two- and four-domain ladders with $\ell=4$,
and then compared to the results predicted by the effective model. The
topological state occupation $\langle n_{k}\rangle$ is shown in Fig. 4 at each
time $t$, with $\langle n_{0}\rangle=|\braket{\mathcal{L}}{\psi(t)}$$|^{2}$,
$\langle n_{k}\rangle=|\braket{\mathcal{P}_{k}}{\psi(t)}|^{2}$ for $1<k<N-1$,
and $\langle n_{N}\rangle=|\braket{\mathcal{R}}{\psi(t)}|^{2}$. As can be
seen, both results are almost identical.

In this work, we choose an energy imbalance as a control parameter, but
analogous protocols can be implemented introducing a vertical hopping
amplitude between the legs instead, with a Hamiltonian term of the form
$\mathcal{H}_{m}=-\sum_{j=1}^{L}\left(mc_{j,A}^{\dagger}c_{j,B}+h.c.\right)$.
In most experimental implementations, this alternative is more difficult to
implement, and its protection against disorder is equivalent to the imbalanced
case, but we include a discussion on this version of the models in Appendix C
for completeness, including their different symmetry classes.

Figure 5: Transfer time between left and right states in a Creutz ladder as a
function of distance. (a) The three cases considered are (a.1) a single domain
of increasing length, in blue, (a.2) two domains of increasing length, with a
single amplifier between them, in red, and (a.3) an increasing number of
domains of length $\ell=4$, in yellow. (b) Transfer time $t_{\textrm{tr}}$ in
the three cases as a function of the total length of the ladder $L$, color
coded as in (a). A logarithmic scale is used for the vertical axis. The first
and second yellow points coincide with a blue point and a red point,
respectively. The simulations have a maximum control parameter of $=1$, a
preparation time of $t_{\textrm{prep}}=30$, and a time step of $\Delta t=0.1$.
Analytical results for the transfer times [Eqs. (33,34)] are included in
continuous red and blue lines. A linear fit is plotted for the yellow points.
In the inset, the same data is plotted using a linear vertical scale, in order
to appreciate the different trends of the data.

###  II.5 Transfers involving $\mathcal{S}$-type states

The same kind of transfer can also be implemented with the $\mathcal{S}$-type
states, with some additional steps. To swap the components of, for instance,
states $\ket{\mathcal{L}}$ and $\ket{\mathcal{S}_{k}}$, the first step is to
turn the $k$-th domain wall into a topological-to-trivial wall. This is
achieved by raising the value of $\epsilon$ in the domain immediately to its
right ($D=k+1$), to a sufficiently high value so that the domain becomes
trivial, $\epsilon_{k+1}^{(b)}=\epsilon_{\mathrm{bar}}>2J$, taking a time
$t_{\mathrm{prep}}^{\prime}$. This makes state $\ket{\mathcal{P}_{k}}$
disappear without affecting $\ket{\mathcal{S}_{k}}$, and acts like a barrier
for the spread of the computational states. A large value of the barrier,
$\epsilon_{\textrm{bar}}\gtrsim 20\epsilon_{\mathrm{tr}}$, has been observed
to be more effective in this task.

Then, just like in the LR transfer, the energy imbalance in all intermediate
bulk sites has to be switched on (now including domain wall $k$), while
leaving the intermediate domain walls at zero energy. The $\mathcal{S}$ state
will then take an exponential profile into the domain to its right (or left,
if the barrier was set up to its right):

$\displaystyle\ket{\mathcal{S}_{k}}_{\mathrm{left}}$
$\displaystyle=(-1)^{k+1}i\mathcal{N}_{\mathcal{S}_{k}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{j=(k-1)(\ell+1)+2}^{k(\ell+1)}\\!\\!\\!\\!\left((-1)^{k}\frac{2iJ}{\epsilon}\right)^{j-k(\ell+1)-1}\\!\\!\\!\\!\\!\ket{j}\\!\otimes\\!\\!\left(\begin{array}[]{c}1\\\
(-1)^{k+1}i\end{array}\right)$ (28)
$\displaystyle\ket{\mathcal{S}_{k}}_{\mathrm{right}}$
$\displaystyle=(-1)^{k+1}i\mathcal{N}_{\mathcal{S}_{k}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{j=k(\ell+1)+2}^{(k+1)(\ell+1)}\\!\\!\\!\\!\left((-1)^{k}\frac{2iJ}{\epsilon}\right)^{k(\ell+1)-j+1}\\!\\!\\!\\!\\!\ket{j}\\!\otimes\\!\\!\left(\begin{array}[]{c}1\\\
(-1)^{k+1}i\end{array}\right),$ (31)

where the subindex indicates the side into which the state extends.

The process is schematized for a two-domain ladder in Fig. 2 (b), and
simulated in Fig. 3 (b) for
$\ell=4,J=1,\epsilon_{\mathrm{tr}}=1,\epsilon_{\textrm{bar}}=20J,t_{\textrm{prep}}=t_{\textrm{prep}}^{\prime}=30$.
Instead of inducing trivial domains, an alternative that may be easier to
implement in some platforms would be to completely disconnect the rest of the
system, by switching off the appropriate hopping amplitudes.

Using the described protocols, the occupation of any two computational states
in a multidomain CL (left, right or $\mathcal{S}$-type) can be swapped. This
makes it a suitable model for the implementation of quantum information tasks
such as remote quantum gates, braiding or entanglement generation in photonic
lattices or superconducting circuits, which will be the subject of future
work. Given that our model is a quasi-1D platform with full connectivity
between its nodes, it can be used as a basic element in a more complex 2D or
3D structure, in order to achieve a quantum processor with a really high
connectivity.

###  II.6 Acquired phase

A complex phase is acquired in general by the transferred components of the
wavefunction. This phase factor can be easily obtained using the effective
model, and depends on the domain length $\ell$, the direction of transfer and
the chirality of the states involved, and is given by:

$\zeta(\ell,n_{w},x,\varsigma)=\begin{cases}(-1)^{n_{w}/2+\delta_{x,\varsigma}}[(-1)^{\delta_{x,\varsigma}}i]^{\ell}&\textrm{for
even }n_{w}\\\ (-1)^{(n_{w}-1)/2}&\textrm{for odd }n_{w},\end{cases}$ (32)

where $n_{w}$ is the number of domain walls between the transferred states,
$x=\pm 1$ is the chirality of the leftmost transferred state (e.g., $x=-1$ if
state $\ket{\mathcal{L}}$ is involved), and $\varsigma=1\,(-1)$ indicates the
direction of transfer, from left to right (from right to left) 222All transfer
protocols detailed in our work are actually bidirectional, that is, they can
be used to swap the two components of the involved states. In a bidirectional
transfer, the component travelling from left to right will acquire a phase of
$\zeta(\ell,n_{w},x,1)$, while the component traveling from right to left will
get $\zeta(\ell,n_{w},x,-1)$.. The two latter quantities are then compared in
a Kronecker delta $\delta_{x,\varsigma}$.

This formula does not depend on $\ell$ for odd $n_{w}$, due to the way the end
modes hybridize. The expression assumes the gauge chosen in Eqs. (4-7), and
its form is related to the presence or absence of reflection and inversion
symmetries in each case. The acquired phase, in any case, can only be a
multiple of $\pi/2$, and it is remarkably robust against disorder (even
symmetry-breaking disorder), as is usually the case in topological protocols.
This is further discussed in Apendix D.

###  II.7 Experimental proposal

As mentioned in the introduction, the imbalanced Creutz ladder can be
implemented using state-of-the-art technology in ultracold atoms (Jünemann
_et al._ , 2017; Song  _et al._ , 2018; Hyoun Kang  _et al._ , 2020; He  _et
al._ , 2021), superconducting circuits (Alaeian  _et al._ , 2019; Hung  _et
al._ , 2021) and photonic lattices (Mukherjee  _et al._ , 2018), in both
waveguide and cavity arrays. For our protocols, it is crucial that the energy
imbalances can be tuned individually, something that can be easily done in
most implementations. Additionally, the Peierls phases from the synthetic
magnetic field must be able to change sign from one cell to the next in order
to implement a domain wall with a width of a single rung. This is attainable
in the photonic and circuit QED implementations, given that the Peierls phases
are obtained with an on-site energy driving, which can be modified to
separately choose the phase of each bond. On the other hand, the ultracold
atoms setup in (He  _et al._ , 2021) might be able to engineer a multidomain
ladder if a more complex pattern of standing waves is used, where the
resulting Peierls phase depends on position.

However, the tight-binding model implemented with fermionic atoms in (Song
_et al._ , 2018), with spin acting as a synthetic dimension to create the two
rungs, is especially suited to implement multidomain Creutz ladders. The
tight-binding model that the authors present in said work is related to the
usual Creutz ladder model by the gauge transformation
$c_{j,\sigma}\to\exp[i\pi s_{\sigma}(3/2-j)/2]c_{j,\sigma}$,with
$s_{\sigma}=\delta_{\sigma,A}-\delta_{\sigma,B}$, which only affects the
relative phases between wavefunction components, but not the relevant physical
results. This can be easily seen by examining the magnetic flux for each
closed loop in both lattices, and realizing they are identical.

The authors build this model using a staggered Raman coupling with positive
and negative terms, which induces a pseudo-spin-orbit term that creates the
diagonal links [see Fig. 4 (e)]. Under this gauge transformation, a system
where all even (odd) unit cells have a negative Raman coupling term
corresponds to the $\nu=1\,(-1)$ phase of the Creutz ladder. Thus, the Raman
potential can be modified to create different domains by setting up regions
with different alternating patterns of signs 333This is reminiscent of the way
topological domains are created in the SSH chain, see Section IV for more
details., as shown in Fig. 4 (f). The appearance of flat bands requires the
horizontal and diagonal hopping amplitudes to be equal in magnitude. A
promising aspect of this setup is that the use of a synthetic dimension will
cause noise-induced fluctuations in the different links of each unit cell to
be highly correlated, which is the kind of disorder which does not break the
protecting chiral symmetry, as we explain in Section III.2 and Appendix C.

##  III Fast long-range transfer

###  III.1 Topological domain walls as amplifiers

In all protocols based on the hybridization of end modes in 1D topological
insulators, the transfer time increases exponentially with distance for a
fixed control parameter value 444As discussed in (Lang and Büchler, 2017), a
value of the control parameter closer to the topological phase transition can
be used to speed up the protocol, but this also reduces the gap, making it
more susceptible to noise, and requires a really precise control in order not
to cross the phase transition point accidentally when speeding up the
protocols to the point needed in large systems.. This is because the
exponential localization of the modes causes their overlap –and bonding
energy– to shrink exponentially as the distance between them increases. This
causes the transfer dynamics to be extremely slow in large systems.

However, there is a workaround that can help us with this issue. Using the
effective Hamiltonian, we obtain the left-to-right transfer times for the
single- and two-domain CL:

$t_{tr}^{(N=1)}=\frac{\pi\epsilon}{2(4J^{2}-\epsilon^{2})}\left(\frac{2J}{\epsilon}\right)^{\ell+3}=\frac{\pi\epsilon}{2(4J^{2}-\epsilon^{2})}\left(\frac{2J}{\epsilon}\right)^{L+1}$
(33)
$t_{tr}^{(N=2)}=\frac{\pi\epsilon}{4J^{2}-\epsilon^{2}}\left(\frac{2J}{\epsilon}\right)^{\ell+2}=\frac{\pi\epsilon}{4J^{2}-\epsilon^{2}}\left(\frac{2J}{\epsilon}\right)^{(L+1)/2}.$
(34)

This expression does not take into account the preparation time, which soon
becomes negligible as the transfer times increase. Even though both times grow
exponentially with the transfer distance $L$, the presence of an intermediate
$\mathcal{P}$ state in the two-domain ladder dramatically speeds up the
transfer, halving the coefficient in the exponent. The $\mathcal{P}$ state
works as a signal booster or amplifier that increases the effective hopping
amplitude between the left and right end states. The exponential dependence
turns out to depend only on domain length $\ell$, so we can lose the
exponential behaviour altogether by using a constant value of $\ell$ for
ladders of growing length, simply by increasing the number of domains.

We demonstrate this in Fig. 5, in which we plot the transfer time as a
function of distance for the single- and two-domain cases, as well as for a
ladder of increasing $N$ but fixed $\ell$. We include both numerical and
analytical results for one and two domains, which are remarkably accurate for
large systems. We set $\epsilon_{\textrm{tr}}^{(1)}=J,t_{\textrm{prep}}=30$,
and a time step of $\Delta t=0.1$ for all simulations. A domain length $\ell$
similar to the localization length of the topological states during the
transfer, $\lambda(\epsilon_{\textrm{tr}})=1/\log(2J/\epsilon_{\textrm{tr}})$
[obtained from Eqs. (11-20)], must be chosen, in order for the amplification
to be effective. We use $\ell=4$ for the multidomain simulations with
$\epsilon_{\textrm{tr}}^{(1)}=J$, given that the localization length is then
$\lambda=3.32$.

The transfer times with constant $\ell$ can be fitted to a linear function
$t_{\textrm{tr}}=t_{0}+A_{0}L$, with $t_{0}=81.66$ and $A_{0}=2.55$. The first
point was not included in the fit. The optimal maximum values for the control
parameter in simulations with $N>2$ were obtained numerically using the
effective Hamiltonian, and are included in Table 1, inside Appendix B.

The concern may arise that this could also make the protocols more fragile
against noise. However, as we show in the following section, the topological
protection of these protocols show the expected plateau at $f=1$ for low
disorder levels when the corresponding chiral symmetry is preserved, and when
it is not, their shortened times actually make them much more robust than the
single-domain case.

The notion of topological amplification in non-Hermitian models (i.e. using
topological states to exponentially amplify a signal) has been recently
explored in the literature (Porras and Fernández-Lorenzo, 2019; Wanjura  _et
al._ , 2020; Ramos  _et al._ , 2021; Wanjura  _et al._ , 2021; Ramos  _et al._
, 2022; Gómez-León  _et al._ , 2022). In some aspects, the protocols we
propose in our work are similar in spirit to this idea: both cases involve
amplifing a signal (in our case, the wavefunction of the states involved in
the transfer) by coupling it to the low-amplitude end of an exponentially
localized topological state. However, the nature of the amplification and the
topological protection in these works are different, given their non-Hermitian
Hamiltonians with intrinsic symmetries, robust to all kinds of disorder while
valid.

###  III.2 Robustness against disorder

Figure 6: Fidelity of the transfer protocols against quenched disorder. We
compare trivial transfer protocols in a 1D chain and in a Creutz ladder to the
single- and four-domain topological CL protocols, for a fixed length $L$. We
study the cases $L=13,21$. We consider both off-diagonal disorder which only
depends on $j$ (a,b) and general uncorrelated disorder, both diagonal and off-
diagonal (c,d), see main text. The effect of topological protection can be
seen in (a,b), where the topological protocols show a plateau for low values
of disorder. Note the different horizontal scale for (a,b) and (c,d). In the
case with symmetry-breaking disorder, the multidomain protocol performs much
better than the single-domain one, due to its shorter transfer time. It is
comparable with the fidelity of the trivial chain protocol, which has a much
smaller amount of degrees of freedom that can be affected by disorder. Sigmoid
functions fitted to the data are provided as a guide for the eye. Transfer
times for $L=13$ are $t_{\textrm{tr}}=332.3,3567.7,9013.8,63.0$ for the
trivial chain, the trivial CL and the single- and four-domain CL,
respectively. In the same order, the times for $L=21$ are
$t_{\textrm{tr}}=800.5,46739.8,2288679.9,135.5$.

In this section, we study the effects of disorder on the LR transfer protocols
defined above using their fidelity, which is defined as
$f=\left|\braket{\mathcal{R}}{\psi(t_{tr})}\right|^{2}$, where
$\ket{\psi(t_{tr})}$ is the state obtained at the end of the protocol. We
consider the standard, single-domain case, and also the case with four
domains. In order to identify the effects of topological protection, we also
consider two trivial transfer protocols, one using a 1D chain of sites with
hopping amplitude $J$, and the other using a Creutz ladder with no magnetic
flux ($\phi=0$). These protocols involve tuning the chemical potential of the
end sites of the models, starting at a low value $-\mu_{0}$ to create two
potential wells on either side, and then adiabatically increasing its value up
to zero during a preparation time $t_{\textrm{prep}}$. The potential wells are
then re-established adiabatically, by lowering the potentials down to
$-\mu_{0}$ again. We consider a fixed value of $\mu_{0}=10J=10\text{
and}t_{\textrm{prep}}=t_{\textrm{tr}}/2$, and then find the optimal transfer
time in the ideal system that satisfies $f>0.985$ (for the trivial chain) or
$f>0.95$ (for the Creutz ladder).

We consider quenched disorder in the timescale of the protocols (i.e. constant
in time for each realization). It is important to note that we do not
calibrate the transfer time for each realization, given that this would often
not be feasible in a real system. We study two disorder regimes: off-diagonal
disorder, which preserves the chiral symmetry of the CL, and a case with both
diagonal and off-diagonal disorder, which breaks it. In the former case, we
consider fluctuations which only depend on the longitudinal coordinate of the
ladder $j$, in order to preserve the chiral symmetry of the ladder
$\mathcal{X}_{C}$ (see Appendix C):

$\begin{cases}-J_{j,\sigma}^{(h)}&=-|J+\delta JR_{j}|e^{i\phi/2}\\\
-J_{j,\sigma}^{(d)}&=-|J+\delta JR_{j}|,\end{cases}$ (35)

where $\delta J$ is the level of off-diagonal disorder, $-J_{j,\sigma}^{(h)}$
and $-J_{j,\sigma}^{(d)}$ are the horizon tal and diagonal hopping terms
connecting site $j,\sigma$ to the sites on its right, and $R_{j}$ are random
numbers in the interval $[-0.5,0.5]$.

Figure 7: Transfer time between left and right states in an SSH chain as a
function of distance. (a) We consider (a.1) a single domain of increasing
length, in blue, (a.2) two domains of increasing length, in red, and (a.3) an
increasing number of domains of length $\ell=4$, in yellow. (b) Transfer time
$t_{\textrm{tr}}$ in the three cases as a function of the total length of the
chain $L$, color coded as in (a). A logarithmic scale is used for the vertical
axis. The first and second yellow points coincide with a blue point and a red
point, respectively. The simulations have a maximum control parameter of
$v_{\textrm{tr}}=0.5$, a preparation time of $t_{\textrm{prep}}=15$, and a
time step of $\Delta t=0.1$. Analytical results for the transfer times [Eqs.
(40,41)] are included in continuous red and blue lines. A linear fit is
plotted for the yellow points. In the inset, the same data is plotted using a
linear scale, in order to appreciate the different trends of the data.

For the case with general disorder, we consider the following fluctuations:

$\mu_{j,\sigma}=s_{\sigma}\epsilon+\delta\mu R_{j,\sigma}^{(\mu)}$ (36)
$\begin{cases}-J_{j,\sigma}^{(h)}&=-|J+\delta
JR_{j,\sigma}^{(h)}|e^{i\phi/2}\\\ -J_{j,\sigma}^{(d)}&=-|J+\delta
JR_{j,\sigma}^{(d)}|\end{cases},$ (37)

where $\delta\mu$ and $\delta J$ represent the level of diagonal and off-
diagonal disorder respectively, $\mu_{j,\sigma}$ is the total on-site
potential on site $j,\sigma$, and all different $R$ variables are independent
random numbers in the interval $[-0.5,0.5]$, and are now uncorrelated for all
different sites and bonds.

As we explain in Appendix C, fluctuations in the control parameters
$\epsilon_{j}$ do not break the chiral symmetry if they are equal for both
sites in each rung, something crucial for the robustness of the protocol.

We display the results in Fig. 6 for systems of length $L=13,21$. As can be
seen in Fig. 6 (a,b), the topological protocols show a plateau at low disorder
strength as long as the chiral symmetry is preserved. In the case with general
disorder [Fig. 6 (c,d)], the four-domain case performs considerably better
than the single-domain case, due to its decreased transfer time. Its fidelity
is then similar to the trivial chain case, even though the latter has a much
smaller number of degrees of freedom that can fluctuate. The complexity of the
model and the increased transfer time also explain the worse performance of
the trivial CL when compared to the chain. We include a detailed description
of the different factors that we observed that can affect the robustness of
the studied protocols in Appendix D.

The acquired phase in the transfer is, however, much more robust in the
topological case than in the trivial chain. This can be useful for quantum
information applications of these protocols, for example in photon-mediated
remote gates (Lang and Büchler, 2017) or superconducting qubit implementations
(Mei  _et al._ , 2018). We elaborate on this point in Appendix D, and showcase
its usefulness in a simple case involving the transfer of two superposed
states.

##  IV Amplifiers in the SSH chain

Figure 8: (a) Topological states involved in an LR transfer in a four-domain
SSH chain. (b) Associated 1D effective model. (c) Occupation of the
topological states during an LR transfer in a two-domain SSH chain with domain
length $\ell=4$ ($L=11$), where the only $\mathcal{S}$ state is represented in
yellow. Numerical results for the full SSH chain Hamiltonian are pictured in
continuous lines, while the analytical prediction by the effective Hamiltonian
is plotted in dotted lines. (d) Occupation of the topological states in an LR
transfer in the four-domain SSH chain pictured in (a), with $\ell=4$ ($L=21$).

The question arises whether these transfer protocols can work in simpler
topological models, like for example the SSH chain. Although this model only
possesses a trivial and a topological phase, different topologically
nontrivial domains can be established due to its particular chiral basis,
where the end of a topological system can be made trivial by removing a site
or vice versa. This allows us to create topological domain walls in SSH chains
by simply removing a site (or, equivalently, setting up two contiguous spatial
regions with the two different dimerizations). This is a well-known even-odd
effect that has been studied since the conception of the SSH model (Su  _et
al._ , 1980).

The Hamiltonian that describes an SSH chain with $N$ such domains is:

$\displaystyle\mathcal{H}_{\textrm{SSH}}$
$\displaystyle\\!=\\!-\\!\sum_{k=1}^{N}\\!\left\\{\\!\delta_{1,k\\!\\!\\!\\!\\!\mod\\!2}\\!\\!\sum_{x=x_{k}^{i}}^{x_{k}^{f}}\\!\\!\left(v_{k}c_{x,b}^{\dagger}c_{x,a}\\!+\\!wc_{x+1,a}^{\dagger}\\!c_{x,b}+\\!h.c.\right)\right.$
$\displaystyle\left.+\delta_{0,k\\!\\!\\!\\!\\!\mod\\!2}\\!\\!\sum_{x=x_{k}^{i}}^{x_{k}^{f}}\\!\\!\left(v_{k}c_{x-1,b}^{\dagger}c_{x,a}\\!+\\!wc_{x,a}^{\dagger}c_{x,b}\\!+\\!h.c.\right)\right\\},$
(38)

where $x_{k}^{i}=\left\lceil((k-1)(\ell+1)/2\right\rceil+1$ (resp.
$x_{k}^{f}=\left\lceil k(\ell+1)/2\right\rceil$) is the first (resp. last)
unit cell that the $k$-th domain has support on, $\ell$ is the number of inner
sites in a domain, $w$ takes a fixed value and $v_{k}$ are the control
parameters for each domain, and $\delta_{x,y}$ are Kronecker deltas. The first
term is nonzero for odd domains, and the second one, for even ones. Operator
$c_{x,\alpha}^{(\dagger)}$ destroys (creates) a particle in unit cell
$x=1,\ldots,\left\lceil L/2\right\rceil$ and sublattice $\alpha=a,b$.
$L=N(\ell+1)+1$ is the total length of the chain. We will restrict our study
to the case with even $\ell$, given that then, all boundary states are
localized in a single site in the compact limit. We also consider $w=1$.

Each of the $N-1$ domain walls holds a single state $\ket{\mathcal{S}_{k}}$,
which makes a total of $N+2$ protected boundary states, counting the left and
right end states. By manipulating the control parameters $v_{k}$ in the same
way as the energy imbalance in the Creutz ladder, the transfer protocols we
propose can be established between the left and right end modes. Now, the
compact states limit corresponds to the completely dimerized chain, in which
the first and last sites are completely decoupled from the rest. The acquired
phases are now:

$\zeta(\ell,n_{w})=\begin{cases}(-1)^{(n_{w}+\ell)/2}&\textrm{for even
}n_{w}\\\ (-1)^{(n_{w}+1)/2}i&\textrm{for odd }n_{w},\end{cases}$ (39)

where $n_{w}$ is the number of domain walls between the transferred states.

The absence of _$\mathcal{P}$_ -type states implies that processes like the
one depicted in Fig. 3 (c), where an intermediate state is unperturbed by a
transfer happening through it, are impossible. However, the domain wall states
can still be used as amplifiers between the left and the right end states,
also achieving an exponential speedup with respect to the single-domain case,
as shown in Fig. 7 for $v_{\textrm{tr}}=0.5$. All protocol parameters are
shown in Appendix B.

The transfer times for single- and two-domain models can be analytically
obtained with the appropriate effective Hamiltonian, which is also derived in
(Munoz  _et al._ , 2018). Its details are included in Appendix E. These
transfer times are:

$t_{tr}^{(N=1)}=\frac{\pi
v}{2(w^{2}-v^{2})}\left(\frac{w}{v}\right)^{\ell/2+2}=\frac{\pi
v}{2(w^{2}-v^{2})}\left(\frac{w}{v}\right)^{L/2+1}$ (40)

$\displaystyle t_{tr}^{(N=2)}$
$\displaystyle=\frac{\pi\sqrt{w^{2}+v^{2}}}{\sqrt{2}(w^{2}-v^{2})}\left(\frac{w}{v}\right)^{\ell/2+1}$
$\displaystyle=\frac{\pi\sqrt{w^{2}+v^{2}}}{\sqrt{2}(w^{2}-v^{2})}\left(\frac{w}{v}\right)^{(L+1)/4}$
(41)

In Fig. 8, we show a four-domain SSH chain with $\ell=4$ (a) and the
associated effective model (b). We also show the evolution of the boundary
states occupation, both exactly and using the effective model, in an LR
transfer in a two-domain (c) and in a four-domain (d) SSH chain, both with
$\ell=4,v_{\textrm{tr}}=0.5$.

Finally, we studied the behaviour against off-diagonal and general (both off-
diagonal and diagonal) disorder. The results can be seen in Fig. 9 for systems
of length 13 and 21 (or, in the case of the single-domain SSH chain, which
always has an even number of sites, $L=12$ and 20). In the SSH chain, the
chiral symmetry is preserved for all possible off-diagonal disorder terms, and
so the topological protocols show a plateau in their fidelity results for
disorder levels up to more than $0.2J$. In the case of general disorder, the
four-domain chain also shows an almost perfect performance up to at least
$0.1J$, showing a clear advantage over the single-domain and the trivial one,
due to its much shorter transfer time.

Figure 9: Fidelity of the SSH chain transfer protocols against quenched
disorder. We compare a transfer protocol in a trivial chain to the single- and
four-domain topological SSH chain protocols, for a fixed length $L$. We study
the cases $L=13,21$, except in the single-domain SSH chain, where the total
length has to be even, so we choose $L=12,20$. We consider off-diagonal
disorder (a,b) and both diagonal and off-diagonal disorder at the same time
(c,d), see main text. The effect of topological protection can be seen in
(a,b), where the topological protocols show a plateau for low values of
disorder. Note the different horizontal scale for (a,b) and (c,d). In the case
with symmetry-breaking disorder, the multidomain protocol performs much better
than the single-domain one, due to its shorter transfer time. Unlike the
Creutz ladder version, it also always outperforms the trivial chain protocol,
given that it has the same amount of degrees of freedom. Sigmoid functions
fitted to the data are provided as a guide for the eye. Transfer times for
$L=13,12$ are $t_{\textrm{tr}}=332.3,156.5,55.9$ for the trivial chain and the
single- and four-domain SSH chain, respectively. In the same order, the times
for $L=21,20$ are $t_{\textrm{tr}}=800.5,2175.4,55.9$.

##  V Conclusions

In this work, we analyze the control possibilities that magnetic interference
provides in the multidomain Creutz ladder, and propose a controlled particle
transfer protocol between any two of the topological interphases in the model.
Additionally, we also demonstrate that an exponential speed-up with respect to
single-domain protocols can be achieved by employing the protected states in
domain walls as signal amplifiers, both in the SSH chain and in the Creutz
ladder. Future prospects for our work include the implementation of more
complex protocols acting on the boundary states, and the study of the
different quantum information applications that such our protocols have, such
as braiding, entanglement generation or remote quantum gates.

###### Acknowledgements.

J.Z. would like to thank Álvaro Gómez León, David Fernández and Jordi Picó
Cortés for fruitful discussions. C.E.C. was supported by the Universidad
Complutense de Madrid through Grant No. FEI-EU-19-12. G.P. and J.Z. were
supported by Spain’s MINECO through Grant No. PID2020-117787GB-I00 and by CSIC
Research Platform PTI-001. J.Z. recognizes the FPU program FPU19/03575.

##  Appendix A Topological state preparation

The left, right and $\mathcal{S}$-type topological states of the rungless
Creutz ladder, which are localized in two sites, can be prepared starting from
a particle confined to a single site, while retaining topological protection.
To do this, the hopping amplitudes of the given site with its neighbours must
start off at zero, and have to be turned on adiabatically until they reach the
value of the rest of the hopping amplitudes, with the appropriate complex
phases. The symmetry protection is retained during this process, because the
state corresponds to a topological state of a rhomboid CSSH ladder of varying
$J/J^{\prime}$ (Zurita  _et al._ , 2021). Although the full system is not
equivalent to the CSSH ladder during the preparation, it is locally equivalent
to it around the computational states. Given the localized nature of all
eigenstates during the protocol due to AB caging, this local equivalence is
enough to make the computational state completely analogous to its CSSH ladder
counterpart, including the topological protection, so the preparation protocol
will be symmetry-protected against off-diagonal disorder.

Figure 10: Preparation of the computational states $\ket{\mathcal{L}}$ (left
of the figure) and $\ket{\mathcal{S}_{1}}$ (right) in the Creutz ladder. (a)
The initial state has a particle localized in a single site in leg $A$. (b)
The hopping amplitudes connected to that site (dashed lines) must start at a
value of zero, and then be adiabatically switched on, with the proper complex
phases. This protocol is topologically protected by chiral symmetry, given
that each intermediate state corresponds to a topological state in a CSSH
ladder (Zurita  _et al._ , 2021), to which the system is locally equivalent.
(c) The final state is the corresponding computational state in the Creutz
ladder. Magnetic interference will cause the site in leg $B$ to acquire the
necessary phase, represented by color. Red is used for phase $0$, green for
phase $\pi/2$ and purple for phase $-\pi/2$.

##  Appendix B Transfer protocols inside the protected manifold

As explained above, the protected boundary states at the ends and walls of the
models form an effective 1D chain. The left-to-right transfer is reduced to
the problem of how to transfer a particle from the first to the last site in
this effective 1D chain. Multiple avenues could be explored here, like for
example imposing an additional dimerization between the domain walls, in the
spirit of (Munoz  _et al._ , 2018) to try and obtain an additional degree of
protection, or considering the –also especially resilient– adiabatic passage
protocol recently proposed in a two-domain SSH chain (Longhi, 2019). However,
in this work we constrain our study to the simplest version of these
protocols, given that it provides an exponential speed-up over the single-
domain case: the modulation of the effective hopping amplitudes over time.

Among all possible ways to modulate them, we have chosen to follow the pulse
form detailed in Eq. (25) for the control parameter $c$ (which can be
$c=\epsilon,m$ or $v$, depending on the model) simultaneously in all domains,
but allowing for a different value of $c_{\textrm{tr}}$ for each domain. A
straightforward numerical simulation can easily provide the appropriate values
of these control parameters in order to obtain a fidelity of $f>0.995$ in the
shortest time possible, that is, in the first maximum of the last site
occupation. We have considered a value of $c_{\textrm{tr}}^{(1)}$ in the first
and last domain, and then a different value $c_{\textrm{tr}}^{(2)}$ for the
second and penultimate, and so on.

This, in turn, creates different effective hopping amplitudes in the effective
Hamiltonian. We label here their values in the middle of the protocol as
$v_{\textrm{1}}$ for the first and last domains, $v_{\textrm{2}}$for the
second and penultimate, and so forth.

We include a detailed list of all protocols with $N>2$ that were used in this
work, together with all their parameters, in Table 1.

Model | Ref. | $N$ | $\ell$ | $L$ | $c_{\textrm{tr}}^{(1)}$ | $c_{\textrm{tr}}^{(2)}$ | $c_{\textrm{tr}}^{(3)}$ | $t_{\textrm{tr}}$ | $t_{\textrm{tr}}^{\prime}$
---|---|---|---|---|---|---|---|---|---
ICL | Fig. 5 | 3 | 4 | 16 | 1 | 0.952 | – | 123.4 | 30
ICL | Fig. 5 | 4 | 4 | 21 | 1 | 0.969 | – | 135.7 | 30
ICL | Fig. 5 | 5 | 4 | 26 | 1 | 0.973 | – | 148.9 | 30
ICL | Fig. 5 | 6 | 4 | 31 | 1 | 0.975 | 0.979 | 162.2 | 30
ICL | Fig. 6 | 4 | 2 | 13 | 1 | 0.906 | – | 63.1 | 30
SSH | Fig. 7 | 3 | 4 | 16 | 0.5 | 0.543 | – | 50.6 | 15
SSH | Fig. 7 | 4 | 4 | 21 | 0.5 | 0.560 | – | 55.9 | 15
SSH | Fig. 7 | 5 | 4 | 26 | 0.5 | 0.561 | 0.566 | 62.2 | 15
SSH | Fig. 7 | 6 | 4 | 31 | 0.5 | 0.563 | 0.576 | 67.5 | 15
SSH | Fig. 9 | 4 | 2 | 13 | 0.5 | 0.560 | – | 35.0 | 15
Table 1: Multidomain transfer protocol parameters. We indicate the model in
which they are implemented (ICL = imbalanced Creutz ladder, SSH = SSH chain),
one of the Figs. which reference them, and the values of all parameters. The
control parameter ($v$ in SSH, $\epsilon$ in ICL) is represented by $c$.

##  Appendix C Runged vs. imbalanced Creutz ladder: symmetries and states

In this Appendix, we discuss the differences and similarities between the
imbalanced ($\epsilon\neq 0,m=0$) and the runged ($m\neq 0,\epsilon=0$) Creutz
ladders.

The rungless, balanced regime ($m=\epsilon=0$) of the CL belongs to the BDI
class as long as $\phi\neq 0\mod 2\pi$, with a hidden chiral symmetry that can
be expressed as
$\mathcal{X}_{S}^{(4)}=\textrm{diag}(\mathbbm{1}_{2},-\mathbb{\mathbbm{1}}_{2})$
if we choose a four-site unit cell (Zurita  _et al._ , 2021). The runged case
also belongs to the BDI class if $\phi=\pi\mod 2\pi$, with different
symmetries than in the previous case. The chiral symmetry is then
$\mathcal{X}_{C}=i\sigma_{y}$. The imbalanced CL belongs to the AIII class if
$\phi=\pi\mod 2\pi$, with the same chiral symmetry $\mathcal{X}_{C}$. When
either of the control parameters ($m$ or $\epsilon$) is nonzero and $\phi\neq
0,\pi\mod 2\pi$, the system belongs to one of the nontopological symmetry
classes in 1D. Topological edge states are still present in the model, but
they are related only to crystalline symmetries instead (Li and Chen, 2015).

Figure 11: Phase diagrams showing the symmetry classes and winding numbers of
the two Creutz ladder variants. (a) Imbalanced Creutz ladder phase diagram,
with $m=0$. (b) Runged Creutz ladder phase diagram, with $\epsilon=0$. (c)
Color code for the symmetry classes and winding number values ($\nu$).
Topological phases explained by the Altland-Zirnbauer classification (Altland
and Zirnbauer, 1997) are indicated by a white outline.

Topological states are protected by these chiral symmetries, but not for all
types of disorder. Diagonal (on-site) disorder always breaks the symmetries of
the models, while each of the symmetries protects against different kinds of
off-diagonal disorder, as shown in Table 2. The system is only topologically
protected if the off-diagonal disorder does not depend on the internal
coordinate $\sigma$. For this reason, implementations that use a synthetic
dimension for the two legs will most likely be advantageous over those using
two real dimensions, given that the inter-cell parameters will be more
correlated in the former case.

Crucially, fluctuations in any of the control parameters ($m$ or $\epsilon$)
preserve the chiral symmetry $\mathcal{X}_{C}$, and thus the topological
protection.

| $\delta J_{h/d}(j,\sigma)$ | $\delta J(j)$ | $\delta\phi(j,\sigma)$ | $\delta c(j)$ | $\delta\mu^{*}$
---|---|---|---|---|---
$\quad\mathcal{X}_{C}=\sigma_{y}\quad$ [$\phi=\pm\pi$] |  | $\checkmark$ |  | $\checkmark$ |
$\mathcal{X}_{S}^{(4)}=\textrm{diag}(\mathbbm{1}_{2},-\mathbb{\mathbbm{1}}_{2})$ [$m=\epsilon=0$] | $\checkmark$ | $\checkmark$ | $\checkmark$ |  |
Table 2: Types of disorder and the chiral symmetries in the Creutz ladder that
are preserved ($\checkmark$) or broken by them. The regime in which the
symmetry is present is shown in square brackets. The dependence of the
disorder terms is indicated in parentheses. $\delta J_{h/d}(j,\sigma)$ allows
for fluctuations of different value in each of the four inter-cell links,
while $\delta J(j)$ means that we apply the same fluctuation to all four
inter-cell hopping terms. $\delta c(j)$ represents a rung-dependent disorder
in one of the control parameters: the vertical links ($c=m$) or the energy
imbalance $(c=\epsilon$), $\delta\phi(j,\sigma)$ indicates a bond-dependent
fluctuation of the Peierls phases, and $\delta\mu^{*}$ stands for a general
non-homogeneous fluctuation in the chemical potentials. For a detailed
description of all symmetries, see (Zurita  _et al._ , 2021).

The runged multidomain CL with rungless domain walls has the following
topological states:

$\displaystyle\ket{\mathcal{L}}$
$\displaystyle=\mathcal{-N_{L}}\sum_{j=1}^{\ell+1}\left(-\frac{2J}{m}\right)^{-j}\ket{j}\otimes\left(\begin{array}[]{c}1\\\
-i\end{array}\right)$ (44) $\displaystyle\ket{\mathcal{R}}$
$\displaystyle=-\mathcal{N_{R}}\sum_{j=L-\ell}^{L}\left(-\frac{2J}{m}\right)^{j-L-1}\ket{j}\otimes\left(\begin{array}[]{c}1\\\
(-1)^{N+1}i\end{array}\right)$ (47) $\displaystyle\ket{\mathcal{P}_{k}}$
$\displaystyle=-\mathcal{N}_{\mathcal{P}_{k}}\sum_{j=(k-1)(\ell+1)+2}^{k(\ell+1)}\left(-\frac{2J}{m}\right)^{j-k(\ell+1)-1}\ket{j}\otimes\left(\begin{array}[]{c}1\\\
(-1)^{k+1}i\end{array}\right)$ (50)
$\displaystyle+\mathcal{N}_{\mathcal{P}_{k}}\sum_{j=k(\ell+1)+2}^{(k+1)(\ell+1)}\left(-\frac{2J}{m}\right)\ket{j}\otimes\left(\begin{array}[]{c}1\\\
(-1)^{k+1}i\end{array}\right),$ (53)

and its effective Hamiltonian has the same form as the imbalanced case, with
the following hopping amplitudes:

$v_{k}=2(-1)^{k+p_{k-1}}im\mathcal{N}_{k}\mathcal{N}_{k-1}\left(-\frac{2J}{m}\right){}^{-d-2},$
(54)

where $d$ is the distance between the maxima of the involved states, and
$p_{k-1}=0$ if the $(k-1)$-th state is a _$\mathcal{P}$ _state, and
$p_{k-1}=1$ otherwise.

In a transfer involving an $\mathcal{S}$ state that extends to the left or
right, this state will take the form:

$\ket{\mathcal{S}_{k}}_{\mathrm{left}}=-\mathcal{N}_{\mathcal{S}_{k}}\sum_{j=(k-1)(\ell+1)+2}^{k(\ell+1)}\left(-\frac{2J}{m}\right)^{j-k(\ell+1)-1}\ket{j}\otimes\left(\begin{array}[]{c}1\\\
(-1)^{k+1}i\end{array}\right)$ (55)
$\ket{\mathcal{S}_{k}}_{\mathrm{right}}=-\mathcal{N}_{\mathcal{S}_{k}}\sum_{j=k(\ell+1)+2}^{(k+1)(\ell+1)}\left(-\frac{2J}{m}\right)^{k(\ell+1)-j+1}\ket{j}\otimes\left(\begin{array}[]{c}1\\\
(-1)^{k+1}i\end{array}\right).$ (56)

The acquired phases in a transfer jumping over $n_{w}$ walls with a domain
length of $\ell$ are different from the imbalanced case, and can only be $0$
or $\pi$, making it easier to compensate if needed for quantum information
applications:

$\zeta(\ell,n_{w},x,\varsigma)=\begin{cases}(-1)^{\ell+n_{w}/2+\delta_{-x,\varsigma}}&\textrm{for
even }n_{w}\\\ (-1)^{(n_{w}-1)/2}&\textrm{for odd }n_{w},\end{cases}$ (57)

where $x=\pm 1$ is the chirality of the leftmost transferred state,
$\varsigma$ is the direction of transfer ($\pm 1$ for left-to-right/right-to-
left).

Apart from the phases, the rest of the dynamics of the model are identical to
those in the imbalanced CL. In particular, all transfer times are the same,
described by Eqs. (33,34), with $m$ playing the part of $\epsilon$. This was
confirmed both analytically and numerically.

##  Appendix D Robustness against disorder: factors and phases

In this Appendix, we elaborate on the results obtained for the disordered
protocols. We discuss the factors that can affect the performance of the
protocol, the highly robust behaviour of the acquired phase, and its effect on
the transfer of a superposition of states.

There are three main factors which determine the robustness against disorder
of a given protocol:

  * •

The topology of the system. When at least one protecting symmetry is
preserved, the fidelity of the protocol is better in the topological case, all
else being equal. Additionally, the phase of the wavefunction components is
preserved especially well, even if the protecting symmetries are broken. This
is relevant in the CL, due to the chiralities being defined by the relative
phase between the two legs, and also for more complex protocols, where
relative phases between different computational states carry information.

  * •

The number of degrees of freedom. More complicated models like the CL, with a
larger amount of moving parts, can induce more errors in the protocol for the
same level of disorder. However, the relative increase in errors will be
platform-dependent, given that some parameters of the Hamiltonian (and their
errors) can be correlated in some experimental realizations but not in others.

  * •

The speed of the protocol. Due to the accumulation of errors, protocols that
take more time usually have a lower fidelity than their faster counterparts,
all else being equal.

As mentioned in the main text, the value of the acquired phase in the
transfer, $\zeta$, is exceptionally robust in the presence of disorder. This
is because it is a geometric phase, not a dynamical phase, and so small
changes in the transfer time do not affect its value. We investigate this by
representing the circular standard deviation 555The circular standard
deviation of a set of data $\\{\zeta_{j}\\}_{j=1}^{M}$ can be found as
$\sigma(\zeta)=\sqrt{2\left(1-\left|\sum_{j=1}^{M}e^{i\zeta_{j}}\right|\right)}$.
It is needed to take into account the periodicity of the phases, and it ranges
between 0 and $\sqrt{2}$.  of the acquired phase values over 1000
realizations.

In the case of symmetry-preserving disorder, the acquired phases are almost
completely unperturbed for disorder strengths of more than 20% of the energy
scale of the model in both the SSH chain and CL protocols, with only the
single-domain CL protocol showing the effects of disorder near $\delta J\simeq
0.15$. This protection does not seem to carry over to the general disorder
case, but we see a better performance in the faster, four-domain SSH and
Creutz models than in a single-domain or trivial protocol, even in the case
where the fidelity in the CL did not outperform the trivial chain, compare
Fig. 12 (d) with Fig. 6 (d). In general, the stability of the acquired phase
seems to outlast the fidelity plateau as disorder increases.

This behaviour is useful in situations where the predictability of the phase
is desired, like quantum information implementations in which the topological
model is coupled to external qubits (e.g., if used as a photonic communication
line to implement remote quantum gates, like in (Lang and Büchler, 2017)), or
more complex transfer protocols where relative phases between boundary states
also have to be preserved.

To show a simple example of the latter, we consider the transfer of a
superposition of states from the two leftmost to the two rightmost
computational states in a six-domain Creutz ladder with $L=31$:

$\displaystyle\ket{+_{\textrm{left}}^{\textrm{CL}}}$
$\displaystyle=(\ket{\mathcal{L}}+\ket{\mathcal{S}_{1}})/\sqrt{2}\to$ (58)
$\displaystyle\to\ket{\psi_{f}^{\textrm{CL}}}$
$\displaystyle=\ket{-_{\textrm{right}}^{\textrm{CL}}}=\zeta_{1}\ket{\mathcal{S}_{5}}+\zeta_{2}\ket{\mathcal{R}}.$
(59)

This is achieved with two successive transfer protocols [see Fig. 13 (b)]. The
acquired phase factors are $\zeta_{1}=-1,\zeta_{2}=1$. We use
$t_{\textrm{prep}}=t_{\textrm{prep}}^{\prime}=30$,
$\epsilon_{\textrm{tr}}^{(1)}=J=1$,
$\epsilon_{\textrm{tr}}^{(2)}=\epsilon_{\textrm{tr}}^{(3)}=0.97$, and a total
transfer time of $T_{\textrm{tr}}=391.4$ for the whole protocol. The fidelity
of the transferred state for each system at the final time $t_{f}$,
$F=|\left\langle\psi_{f}^{(\textrm{ideal})}|\psi(t_{f})\right\rangle|^{2}$, is
$F=0.996$ for the pristine system.

As a topologically trivial protocol to compare against, we use a transmission
line consisting of a 1D chain of sites with a hopping amplitude of $J=1$, of
the same length as each of the transfers in the CL, and with two sites at the
left ($\ket{1,a/b}$) and right ($\ket{L,a/b}$) ends [see Fig. 13 (a)]. The
initial state is
$\ket{+_{\textrm{left}}^{\textrm{triv}}}=(\ket{1,a}+\ket{1,b})/\sqrt{2}$, and
two successive transfers are implemented using the same chain, until the final
state,
$\ket{\psi_{f}^{\textrm{triv}}}=\ket{+_{\textrm{right}}^{\textrm{triv}}}=(\ket{L,a}+\ket{L,b})/\sqrt{2}$
(plus a certain global phase), is achieved. An initial chemical potential of
$-\mu_{0}=-10J$ is set on the four end sites, and the optimal transfer time
for the full protocol was found to be $T_{tr}=2405.6$ (with $F=0.986$).

We now obtain the fidelity average over 1000 realizations for both symmetry-
preserving disorder, defined in Eq. (35), and general disorder, see Eqs.
(36,37). The results are shown in Fig. 14. The topological protocol shows a
plateau up to disorders of $0.15J$ in the former case, and still outperforms
the trivial protocol in the latter case, for low levels of disorder. The
trivial protocol falls quickly to $F=0.5$ in both cases as disorder is turned
on, given that the relative phase of the final superposition is almost random
even for small disorder values, while the phases in the topological protocol
are more reliable. In the case of general disorder, this is mainly due to the
much shorter times of the CL protocol.

Figure 12: Standard deviation of the acquired phases over 1000 realizations of
six different protocols, for symmetry-preserving (a,b) and general (c,d)
disorder. The lengths of the models are $L=13\text{(a,c) and }21$ (b,d)
($L=12,20$ for the single-domain SSH chain). The four topological protocols
show almost perfect results for symmetry-preserving disorder up to $0.2J$. In
the case of general disorder, the four-domain protocols show a clear advantage
over the rest, especially for $L=21$, due to their shorter transfer times.
Continuous lines fitted to the points are added as a guide for the eye.

Figure 13: Transfer of a particle in a superposition of two states. The
amplitude in each of the two states is transferred separately. (a) Transfer
protocol in a trivial chain with $L=26$ for a particle superposed between two
sites. The transfer is induced by using wells of chemical potential at both
edges. (b) Transfer protocol in a six-domain, $L=31$ Creutz ladder for the
superposition of two computational states. Each of the computational states is
translated $26$ rungs to the right. The first transferred state acquires an
additional $\pi$ phase in the CL protocol.

Figure 14: Average value of the fidelity of the transferred superposed state
over 1000 realizations in the presence of (a) symmetry-preserving [see Eq.
(35)] and (b) general disorder [Eqs. (36,37)], using a trivial chain (black)
and a six-domain topological CL (red). Note the different horizontal scale
between the figures. We include its standard deviation as error bars. The CL
protocol outperforms the trivial one by a wide margin in case (a), but it is
also better at low levels of disorder in case (b). This is due to the
robustness of the relative phase in the topological protocol, thanks to its
shorter transfer time. In the trivial case, the relative phase is soon lost,
making the fidelity quickly fall to $\sim 0.5$.

##  Appendix E The multidomain SSH chain effective Hamiltonian

The boundary states in the multidomain SSH chain have the form:

$\displaystyle\ket{\mathcal{L}}$
$\displaystyle=\mathcal{-N_{L}^{\prime}}\sum_{x=1}^{\ell+1}\left(-\frac{w}{v}\right)^{-x}\ket{x,a}$
(60) $\displaystyle\ket{\mathcal{R}}$
$\displaystyle=\mathcal{-N_{R}^{\prime}}\sum_{x=L-\ell}^{L}\left(-\frac{w}{v}\right)^{x-\left\lceil
L/2\right\rceil-1}\ket{x,\Delta_{N}}$ (61)
$\displaystyle\ket{\mathcal{S}_{k}}$
$\displaystyle=\mathcal{N}_{\mathcal{S}_{k}}^{\prime}\left[\ket{x_{0}^{(k)},\Delta_{k}}+\sum_{x=x_{0}^{(k)}-\ell/2}^{x_{0}^{(k)}-1}\left(-\frac{w}{v}\right)^{x-x_{0}^{(k)}}\ket{x,\Delta_{k}}\right.$
$\displaystyle\left.+\sum_{x=x_{0}^{(k)}+1}^{x_{0}^{(k)}+\ell/2}\left(-\frac{w}{v}\right)^{x_{0}^{(k)}-x}\ket{x,\Delta_{k}}\right],$
(62)

where $x_{0}^{(k)}=\left\lceil[k(\ell+1)+1]/2\right\rceil$ is the unit cell
where the $k$-th domain wall is, and $\Delta_{k}=a\,(b)$if $k$ is even (odd).
The effective Hamiltonian for the boundary states in the SSH chain has the
same form as Eq. (23), but substituting all $\mathcal{P}$ states –which do not
exist here– by _$\mathcal{S}$_ states. The effective hopping amplitudes now
have the form:

$\displaystyle t_{1}$
$\displaystyle=v\mathcal{N_{L}^{\prime}}\mathcal{N}_{\mathcal{S}_{1}}^{\prime}\left(-\frac{w}{v}\right)^{-\ell/2-1}$
(63) $\displaystyle t_{k}$
$\displaystyle=-v\mathcal{N}_{\mathcal{S}_{k}}^{\prime}\mathcal{N}_{\mathcal{S}_{k-1}}^{\prime}\left(-\frac{w}{v}\right)^{-\ell/2},\,\,\,\,\,k=2,\ldots,N-1$
(64) $\displaystyle t_{N}$
$\displaystyle=v\mathcal{N_{R}^{\prime}}\mathcal{N}_{\mathcal{S}_{N-1}}^{\prime}\left(-\frac{w}{v}\right)^{-\ell/2-1},$
(65)

where the normalization constants can be approximated as:

$\displaystyle\mathcal{N}_{\mathcal{L}}^{\mathcal{\prime}}$
$\displaystyle=\mathcal{N_{R}^{\prime}}=\sqrt{w^{2}/v^{2}-1}$ (66)
$\displaystyle\mathcal{N}_{\mathcal{S}_{k}}^{\prime}$
$\displaystyle=\sqrt{\frac{w^{2}-v^{2}}{w^{2}+v^{2}}}\,\,\,\,\,\,\,\forall k.$
(67)

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