# Optimizing edge state transfer in a Su-Schrieffer-Heeger chain via hybrid
digital-analog strategies

Sebastián V. Romero  0000-0002-4675-4452 TECNALIA, Basque Research and
Technology Alliance (BRTA), 48160 Derio, Spain  Department of Physical
Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080
Bilbao, Spain  Xi Chen  0000-0003-4221-4288 Department of Physical Chemistry,
University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain
EHU Quantum Center, University of the Basque Country UPV/EHU, Apartado 644,
48080 Bilbao, Spain  Gloria Platero  0000-0001-8610-0675 Instituto de Ciencia
de Materiales de Madrid (CSIC), Cantoblanco, E-28049 Madrid, Spain  Yue Ban
0000-0003-1764-4470<EMAIL_ADDRESS>TECNALIA, Basque Research and Technology
Alliance (BRTA), 48160 Derio, Spain

(August 28, 2024)

###### Abstract

Su-Schrieffer-Heeger (SSH) chain, serving as a paradigmatic model for
comprehending topological phases and their associated edge states, plays a
pivotal role in advancing our understanding of quantum materials and quantum
information processing and technology. In this paper, we introduce a hybrid
analog-digital protocol designed for the non-adiabatic yet high-fidelity
transfer of edge states in a SSH chain, featuring two sublattices A and B. The
core of our approach lies in harnessing the approximate time-dependent
counter-diabatic (CD) interaction, derived from adiabatic gauge potentials.
However, to enhance transfer fidelity, particularly in long-distance chains,
higher-order nested commutators become crucial. To simplify experimental
implementation and navigate computational complexities, we identify the next-
to-nearest-neighbour (NNN) hopping terms between sublattice A as dominant CD
drivings, and further optimize them by using variational quantum circuits.
Through digital quantum simulation, our protocol showcases the capability to
achieve rapid and robust solutions, even in the presence of disorder. This
analog-digital transfer protocol, an extension of quantum control methodology,
establishes a robust framework for edge state transfer. Importantly, the
optimal CD drivings identified can be seamlessly implemented across various
quantum registers, highlighting the versatility of our approach.

##  I Introduction

Building on Feynman’s seminal 1982 conjecture [1], quantum simulation has
emerged as a valuable toolkit, allowing the study of complex behaviours in
condensed matter systems by leveraging easily controllable quantum systems
[2]. In the realm of quantum computing platforms, topological quantum systems
have garnered substantial attention [3] due to their inherent robustness
arising from topological protection mechanisms. Among them, the Su-Schrieffer-
Heeger (SSH) model [4, 5, 6, 7], serving as a tight-binding model for the
simplest 1D lattice, encompasses topologically safeguarded edge states. This
model has sparked exploration of topological properties across diverse quantum
registers, known for their high tunability, including superconducting Xmon
qubit chains [8], a 1D bichromatic lattice [9], arrays of optically trapped
Rydberg atoms [10], as well as semiconductor quantum dots [11].

Digital quantum simulation (DQS) offers a remarkable avenue for investigating
a diverse range of intricate models, without the need for direct laboratory
engineering. Versatile digital simulation methodologies have been harnessed to
explore various domains, including topological phases [12, 13, 14] and quench
dynamics [15]. Unlike analog quantum simulation, which is confined to specific
quantum system types, DQS stands out for its execution via sequences of
quantum logic gates. This can be achieved using present-day quantum simulators
and holds promise for future fault-tolerant quantum computers [16]. To foster
the development of DQS, hybrid quantum-classical optimization algorithms like
variational quantum algorithms [17] grounded in gradient descent [18], or
variational quantum eigensolvers [19], play pivotal roles. In particular,
quantum-control-inspired algorithms have demonstrated themselves invaluable
approached for addressing complex quantum control problems by leveraging
classical optimizers [20, 21, 22, 18].

Moreover, facilitating the long-distance transfer of quantum states holds
paramount importance in the context of large-scale quantum information
processing endeavours. In this context, the utilization of edge states within
the SSH model provides a compelling avenue, with these edge states serving as
a quantum channel, that enables the implementation of steadfast quantum state
transfer between distant qubits. The realization of robust edge state transfer
is achievable by adiabatic passage [23, 24], drawing inspiration from the
original concept of Thouless pumping [25]. Recent experimental achievements
have further confirmed the adiabatic robust transfer of atomic momentum states
across synthetic lattices of Bose-Einstein condensed states [26]. While
adiabatic passage exhibits commendable resistance against uncorrelated
disorder, its vulnerability to environmental noise remains a challenge. To
circumvent this limitation, non-adiabatic transfer protocols [27, 28] have
emerged as effective strategies. Operating within shorter time frames, these
protocols mitigate the adverse impact of decoherence. However, it is important
to note that Thouless pumping is not generically robust to nonadiabatic
effects, as revealed in nonquantized charge transport scenarios [29] within a
Floquet framework. Consequently, the pursuit of transfer schemes that combine
both speed and robustness remains of paramount importance in advancing the
field of quantum state transfer methodologies.

In this paper, we present a hybrid analog-digital way to design the non-
adiabatic protocol for transferring edge states in a SSH model with odd-number
sites. Building on the concept of shortcuts to adiabaticity [30], our method
involves deriving the counter-diabatic (CD) terms through a variational
approach [31], also referred to as quantum transitionless driving [32]. This
implies the incorporation of approximate two-body and many-body interactions
[33], easily implemented in digitized adiabatic computation [34]. By
incorporating these CD terms, specifically the next-to-nearest neighbour (NNN)
hopping terms between sublattice A, into the original Hamiltonian, we achieve
an efficient representation of adiabatic dynamics. This ensures that the state
transfer consistently aligns with the zero-energy (instantaneous) eigenstate
of the original Hamiltonian. Furthermore, we leverage DQS within variational
quantum circuits to obtain optimal CD drivings during the transfer process.
Our hybrid analog-digital protocol remains remarkably resilient even when
subjected to disorder.

The subsequent sections of this article are organized as follows: in Section
II we introduce the SSH model for an odd number of sites and delve into the
characteristics of its edge state. Section III outlines the non-adiabatic
protocol for transferring edge states in the SSH model by using NC,
elucidating CD drivings located as NNN hopping terms between sublattice A. In
Section IV, a detailed description of the quantum circuit implementation of
the time evolution is provided where we find the optimal CD driving digitally.
In Section V, we discuss the feasibility of various experimental platforms and
robustness against disorder. Finally, we conclude in Section VI. This work not
only provides an efficient protocol for the edge state transfer which is
applicable in different quantum platforms, but also opens a door to the
synergy of quantum control and digital quantum simulation.

##  II SSH model: Hamiltonian and Preliminaries

The Su-Schrieffer-Heeger (SSH) model, initially proposed in 1979 to depict
solitons in polyacetylene [4], describes a one-dimensional chain formed by two
sublattices with chiral symmetry. For the case of an odd number $2N-1$ of
sites, comprised of $N$ sites on sublattice A and $N-1$ sites on sublattice B,
the Hamiltonian is given by

$H_{0}(t)=t_{2}(t)\sum_{j=1}^{N-1}c^{\dagger}_{2j}c_{2j-1}+t_{1}(t)\sum_{j=1}^{N-1}c^{\dagger}_{2j+1}c_{2j}+\text{h.c.},$
(1)

where $c^{\dagger}_{i}$ ($c_{i}$) creates (annihilates) a fermion in the $i$th
site of the chain. Sites with odd (even) indices belong to sublattice A (B).
The alternative nearest-neighbouring hopping terms $t_{2}(t)$ and $t_{1}(t)$
represent the intra- and inter-unit-cell interaction, respectively. This
Hamiltonian, with $2N-1$ sites, exhibits chiral symmetry [35], resulting in a
symmetric spectrum with $2N-2$ non-zero-energy levels and one zero-energy
level. Particularly, the exact zero-energy eigenstate

$\ket{\Phi_{0}(t)}=\frac{1}{\mathcal{N}}\left[\sum_{i=1}^{N}a_{i}\ket{A_{i}}+\sum_{i=1}^{N-1}b_{i}\ket{B_{i}}\right],$
(2)

has the components with site occupation amplitudes,

$\displaystyle b_{i}$ $\displaystyle=0,$ $\displaystyle\forall i\in[1,N-1],$
(3) $\displaystyle a_{i}$
$\displaystyle=\left[-\frac{t_{2}(t)}{t_{1}(t)}\right]^{i-1},$
$\displaystyle\forall i\in[1,N],$

where $\mathcal{N}^{2}=\sum\nolimits_{i=0}^{N-1}[t_{2}(t)/t_{1}(t)]^{2i}$ is
the normalization factor. Here, Eq. (3) implies that the zero-energy
eigenstate only has non-zero probabilities on sublattice A sites, which
satisfies chiral symmetry.

[width=]./images/ssh.pdf

Figure 1: Schematic configuration of a SSH chain. The edge state can be non-
adiabatically transferred from the left side to the right one by adding the
hopping between the neighbouring sites of sublattice A. These additional
hoppings can be implemented by producing a gauge potential via applying
geometric phases [36], artificial magnetic fields [37] or angular momentum
states [38].

Making use of the state $\ket{\Psi(t)}$ evolving along with
$\ket{\Phi_{0}(t)}$, one can adiabatically transfer the state from the left at
the initial time $t=0$ to the right bound at the final time $t=T$ (see Fig. 1)
when the boundary conditions $t_{1}(0)/t_{2}(0)\gg 1$, $t_{1}(T)/t_{2}(T)\ll
1$, and the general adiabatic condition $T\gg\pi/t_{0}$ are satisfied [23].
Without the loss of generality, we set $t_{1}(t)=\lambda(t)$ as our control
function and $t_{2}(t)=\Lambda-\lambda(t)$, with a constant $\Lambda$. For
instance, applying the hoppings [27]

$t_{1,2}(t)=t_{0}(1\pm\cos\Omega t),$ (4)

where $\Omega=0.01t_{0}$, $\Lambda=2t_{0}$, $\dot{\lambda}(t)=-\Omega
t_{0}\sin\Omega t$ and for simplicity we set $\lambda\equiv\lambda(t)$ in the
text. Defining the transfer fidelity

$F(t)=|\braket{\Phi_{0}(t)}{\Psi(t)}|^{2},$ (5)

which is used to characterize the extent that the state evolves along the
instantaneous eigenstate in the interval $t\in[0,T]$, we can obtain the
fidelity robustness $F(T)>99.9\%$ at $T\equiv\pi/\Omega=100\pi/t_{0}$,
fulfilling the adiabatic condition motioned above.

Indeed, it is worth noting that previous efforts have primarily focused on
expediting the slow adiabatic protocol for edge state transfer by using
inverse engineering methods [27, 39]. While inverse engineering provides the
advantage of flexibility in designing the time-dependent couplings
$t_{1,2}(t)$ (if it is allowed experimentally), it does run the risk of
exciting other intermediate states, rather than the zero-energy eigenstate. In
contrast, CD driving offers the distinct advantage of ensuring adiabatic
following of the original Hamiltonian [32, 31, 33, 34], but within a shorter
time frame. In what follows, we shall focus on the optimal CD terms, providing
a promising pathway to achieve more robust and high-fidelity state transfers,
even in the presence of noise and other experimental imperfections.

##  III Non-adiabatic edge state transfer

###  III.1 Approximate CD and NC

In this section, we provide a speed-up protocol inspired by the CD driving
where the state evolution is always along $\ket{\Phi_{0}(t)}$, by cancelling
the nonadiabatic transitions from all instantaneous eigenstates. This leads to
the total Hamiltonian $H(t)=H_{0}(t)+H_{\text{cd}}(t)$. However, instead of
finding the exact CD terms which requires the spectral knowledge, we apply the
adiabatic gauge potentials [40] to achieve the approximate CD terms

$H^{(l)}_{\text{cd}}=\dot{\lambda}\mathcal{A}_{\lambda}^{(l)}$ (6)

in the pre-selected form with the coefficients determined by minimizing NC
[31, 33] and

$\mathcal{A}_{\lambda}^{(l)}=i\sum_{k=1}^{l}\alpha^{(l)}_{k}\underbrace{[H_{0},[H_{0},\dots,[H_{0}}_{2k-1},\partial_{\lambda}H_{0}]]],$
(7)

where $\alpha_{k}^{(l)}\equiv\alpha_{k}^{(l)}(t)$, with $k\in\\{1,\dots,l\\}$
and $l$ the order of the expansion, are time-dependent coefficients. In the
limit of $l\rightarrow\infty$, Eq. (7) becomes the exact gauge potential.
Instead, given by a larger value of $l$, the approximate CD terms approach the
exact CD driving more closely. The coefficients $\alpha_{k}^{(l)}$ are
determined by minimizing the action $S_{l}=\mathrm{tr}[G_{l}^{2}]$ with
$G_{l}=\partial_{\lambda}H_{0}-i[H_{0},\mathcal{A}_{\lambda}^{(l)}]$, see
details in Appendix A. The first-order NC,
$\mathcal{A}_{\lambda}^{(1)}=i\alpha_{1}^{(1)}[H_{0},\partial_{\lambda}H_{0}]$,
is given by

$\mathcal{A}_{\lambda}^{(1)}=i\kappa^{(1)}\left[\sum_{j=1}^{N-2}c^{\dagger}_{2j+2}c_{2j}-\sum_{j=1}^{N-1}c^{\dagger}_{2j+1}c_{2j-1}\right]+\text{h.c.},$
(8)

where $\kappa^{(1)}=-\Lambda\dot{\lambda}\alpha_{1}^{(1)}$, and
$\alpha_{1}^{(1)}$ is obtained by minimizing $S_{1}=\mathrm{tr}[G_{1}^{2}]$.
In this scenario, the CD drivings are situated as the NNN hopping between odd-
odd and even-even sites with the same absolute strength $|\kappa^{(1)}|$. More
detailed analytical solution on the first-order NC for a general SSH chain
with $2N-1$ sites can be found in Appendix A.

As the non-adiabatic transfer protocol needs the NNN hoppings, as shown in Eq.
(8), chiral symmetry [41, 27] is broken. However, aiming at transferring the
edge state along the zero-energy eigenstate [Eq. (3)], the probabilities in
the even sites are zero leading to zero hopping between even and even sites.
Consequently, the first-order NC terms are simplified to be only located
between the neighbouring sublattice A.

Subsequently, the second-order NC term goes as

$\mathcal{A}_{\lambda}^{(2)}=i\left(\alpha_{1}^{(2)}[H_{0},\partial_{\lambda}H_{0}]+\alpha_{2}^{(2)}[H_{0},[H_{0},[H_{0},\partial_{\lambda}H_{0}]]]\right),$
(9)

where

$\begin{split}[H_{0},[H_{0},[H_{0},\partial_{\lambda}H_{0}]]]&=\Lambda(\Lambda-\lambda)^{2}\left[4\sum_{j=1}^{N-2}(c^{\dagger}_{2j+1}c_{2j-1}-c^{\dagger}_{2j+2}c_{2j})+c^{\dagger}_{2N-1}c_{2N-3}\right]\\\
&+\Lambda\lambda^{2}\left[\sum_{j=1}^{N-2}(3c^{\dagger}_{2j+3}c_{2j+1}+c^{\dagger}_{2j+1}c_{2j-1}-4c^{\dagger}_{2j+2}c_{2j})+c^{\dagger}_{2N-1}c_{2N-3}\right]\\\
&+4\Lambda\lambda(\Lambda-\lambda)\left[\sum_{j=1}^{N-3}(c^{\dagger}_{2j+3}c_{2j-1}-c^{\dagger}_{2j+4}c_{2j})+c^{\dagger}_{2N-1}c_{2N-5}\right]-\text{h.c.}\end{split}$
(10)

The variational parameters $\alpha_{1}^{(2)}$, $\alpha_{2}^{(2)}$ are obtained
by minimizing $S_{2}=\mathrm{tr}[G_{2}^{2}]$ where $G_{2}$ is both
$\alpha_{1}^{(2)}$\- and $\alpha_{2}^{(2)}$-dependent. As a result, the CD
driving resulted from the second-order NC
$H_{\text{cd}}^{(2)}(t)=\dot{\lambda}\mathcal{A}_{\lambda}^{(2)}$ involves
NNN, fourth-neighbouring terms between sublattice A and B. Similar to the
analysis on the first-order NC, the hoppings between the NNN and the fourth-
neighbouring sublattice B can be neglected for the second-order NC due to the
transfer trajectory along the zero-energy eigenstate [Eq. (3)].

Indeed, obtaining higher orders of NC terms require significantly more effort.
As detailed in Appendix B, the gauge potential expansion
$\mathcal{A}_{\lambda}^{(l)}$ introduces hoppings from the NNNs to the $2l$th-
neighbours. Therefore, computing this quantity for higher orders involves more
interactions constrained in the same sublattice. In general, for a system with
$2N-1$ sites, at least $\mathcal{A}_{\lambda}^{(N-1)}$ should be computed to
account all possible interactions. From $\mathcal{A}_{\lambda}^{(N)}$, higher
orders will start accumulating new contributions to the hoppings already
considered with the lower order NC. Consequently, all possible long-range
hoppings in same sublattices are needed to accomplish exact gauge potential,
which is challenging and impractical for realistic setups. Given these
challenges, exploring simpler strategies or approximations may be necessary
for practical implementation. It is a common practice to strike a balance
between accuracy and computational complexity in quantum simulations,
particularly when dealing with higher-order terms.

###  III.2 Numerical results with simplified CD drivings

./images/alpha_kappa.pdf

Figure 2: Using different schemes with the Hamiltonians $H_{0}$,
$H_{0}+H_{\text{cd}}^{(1)}$, $H_{0}+H_{\text{cd}}^{(2)}$ to transfer the edge
state of a SSH chain with $5$ sites, we illustrate (a) fidelity robustness
$F(T)$ in the dependence of the operation time $T$ ranging from adiabatic
($\Omega=0.01t_{0}$) to non-adiabatic regime (up to $\Omega=10t_{0}$); (b)
Transfer fidelity $F(t)$ during the transfer, where $T=\pi$, $\Omega=t_{0}$.
(c) Variational parameters $\alpha^{(1)}_{1}$, $\alpha^{(2)}_{1}$ and
$\alpha^{(2)}_{2}$ derived by minimizing the action
$S_{l}=\mathrm{tr}[G_{l}^{2}]$ ($l=1,2$). (d) sublattice A hoppings for the
edge state transfer: different strengths $\kappa^{(2)}_{31}$,
$\kappa^{(2)}_{53}$ and $\kappa^{(2)}_{51}$ [Eq. (13)], equal strengths
$\kappa^{(2)}=-\Lambda\dot{\lambda}\alpha^{(2)}_{1}$ [Eq. (14)] from the
second-order NC in comparison with
$\kappa^{(1)}=-\Lambda\dot{\lambda}\alpha^{(1)}_{1}$ derived from the first-
order NC and $\Lambda=2t_{0}$ with $t_{0}=1$.

Next, we exemplify our protocol using a simple chain with $5$ sites (denoted
by $N=3$). The CD driving terms derived from the first-order and the second-
order NC terms are obtained as:

$\displaystyle H^{(1)}_{\text{cd}}$
$\displaystyle=i\Lambda\dot{\lambda}\alpha_{1}^{(1)}(c_{3}^{\dagger}c_{1}+c_{5}^{\dagger}c_{3}-c_{4}^{\dagger}c_{2})+\text{h.c.}$
(11)
$\displaystyle\begin{split}H^{(2)}_{\text{cd}}&=i\Lambda\dot{\lambda}[\alpha_{1}^{(2)}+\alpha_{2}^{(2)}(t_{1}^{2}+4t_{2}^{2})]c_{3}^{\dagger}c_{1}\\\
&+i\Lambda\dot{\lambda}[\alpha_{1}^{(2)}+\alpha_{2}^{(2)}(4t_{1}^{2}+t_{2}^{2})]c_{5}^{\dagger}c_{3}\\\
&-i\Lambda\dot{\lambda}[\alpha_{1}^{(2)}+4\alpha_{2}^{(2)}(t_{1}^{2}+t_{2}^{2})]c_{4}^{\dagger}c_{2}\\\
&+4i\Lambda\dot{\lambda}\alpha_{2}^{(2)}t_{1}t_{2}c_{5}^{\dagger}c_{1}+\text{h.c.},\end{split}$
(12)

where the second-order NC introduces the NNN hoppings, resulting from the
competition between $\alpha_{1}^{(2)}$ and $\alpha_{2}^{(2)}$, and the
emergence of fourth-neighbouring hopping. By employing the hopping terms in
the form of Eq. (4), with $\Omega=0.01t_{0}$, the adiabatic transfer for this
$5$-site chain is achieved at $T=\pi/\Omega=100\pi$, with $t_{0}=1$
throughout. However, shortening the operation time into the non-adiabatic
regime induces the unwanted excitation. In Fig. 2(a), transfer protocols for
$5$ sites including using $H_{0}$, $H_{0}+H_{\text{cd}}^{(1)}$,
$H_{0}+H_{\text{cd}}^{(2)}$, present the fidelity robustness $F(T)$ as the
function of the operation time $T$ (equivalently in $\Omega$). Adding
$H_{\text{cd}}^{(2)}$ to $H_{0}$ exhibits high-fidelity transfer even in the
non-adiabatic regime. In particular, we choose the specific case of
$\Omega=t_{0}$ so that $T=\pi$ where the adiabatic condition is not satisfied.
As shown in Fig. 2(b), the protocol with $H_{0}+H_{\text{cd}}^{(2)}$ keeps
high fidelity during the transfer. Fig. 2(c) shows the variational parameters
$\alpha_{1}^{(1)}$, $\alpha_{1,2}^{(2)}$, while their derivation can be found
in Appendix A. The CD drivings from the second-order NC vary in different
locations, manifested as the competition of terms of $\alpha_{1}^{(2)}$ and
$\alpha_{2}^{(2)}$, indicated in Eq. (11).

The CD drivings from the second-order NC can be further simplified with the
following two procedures: (i) Neglecting NNN hopping terms between sublattice
B due to the transfer trajectory along the zero-energy state; (ii) Performing
CD drivings only for NNN hopping terms between sublattice A while neglecting
longer-range hopping terms. As a first step, the hoppings between $2$ and $4$
sites (couplings on sublattice B) in Eq. (12) are neglected, resulting in a
simplified form,

$\tilde{H}^{(2)}_{\text{cd}}=-i\kappa^{(2)}_{31}c_{3}^{\dagger}c_{1}-i\kappa^{(2)}_{53}c_{5}^{\dagger}c_{3}-i\kappa^{(2)}_{51}c_{5}^{\dagger}c_{1}+\text{h.c.}$
(13)

with symmetric CD driving strengths being
$\kappa^{(2)}_{31}=-\Lambda\dot{\lambda}[\alpha_{1}^{(2)}+\alpha_{2}^{(2)}(t_{1}^{2}+4t_{2}^{2})]$,
$\kappa^{(2)}_{53}=-\Lambda\dot{\lambda}[\alpha_{1}^{(2)}+\alpha_{2}^{(2)}(4t_{1}^{2}+t_{2}^{2})]$
and $\kappa^{(2)}_{51}=-4\Lambda\dot{\lambda}\alpha_{2}^{(2)}t_{1}t_{2}$. The
simplified scheme $\tilde{H}^{(2)}_{\text{cd}}$ [Eq. (13)], with the CD
drivings shown in Fig. 2(d), yields the exact same transfer fidelity as the
original $H^{(2)}_{\text{cd}}$ that includes all terms [Eq. (11)], shown in
Fig. 2(b) (solid red). Secondly, continuing to simplify CD driving strengths
by neglecting $\alpha_{2}^{(2)}$ terms leads to

$\kappa^{(2)}=\kappa^{\prime(2)}_{31}=\kappa^{\prime(2)}_{53}=-\Lambda\dot{\lambda}\alpha_{1}^{(2)},$
(14)

as shown in Fig. 2(d). Equal CD drivings in different locations, can also
fulfil the transfer in a $5$-site chain with high fidelity very similar to
(indistinguishable from) Fig. 2(b) (solid red). This indicates that for a
$5$-site chain, applying the hopping terms only between neighbouring
sublattice A from the second-order NC as the CD drivings maintains the
transfer with high fidelity, although the chiral symmetry of the SSH chain is
broken [41, 27].

For longer chains, the contribution from higher-order NC is necessary. The
exact CD driving
$H_{\text{cd}}^{(l)}(t)=\dot{\lambda}\mathcal{A}_{\lambda}^{(l)}$ should be
derived rigorously from $\mathcal{A}_{\lambda}^{(l)}$ with the limit
$l\rightarrow\infty$ to retrieve the actual gauge potential. However, to this
end, one needs to derive
$\alpha_{1}^{(l)},\alpha_{2}^{(l)},\dots,\alpha_{l}^{(l)}$ with its
corresponding large computational cost. To further simplify the protocol for a
$2N-1$ chain, we aim to find

$H_{\text{cd}}=-i\sum_{j=1}^{N-1}\kappa_{2j+1,2j-1}c^{\dagger}_{2j+1}c_{2j-1}+\text{h.c.},$
(15)

and seek for optimal ways to uncover $\kappa_{2j+1,2j-1}$ so that CD drivings
are only attributed from NNN hoppings in sublattice A.

##  IV Digital simulation and Optimization

Given the complexity of calculating high-order NC terms, particularly the
variational parameter $\alpha^{(l)}_{1}$, that digital quantum simulation
emerges as an efficient way to learn the prefactor for the NC, i.e. the CD
drivings. In Section IV.1, a general scheme for digital simulation of a $2N-1$
SSH chain in quantum circuits is listed where the general Hamiltonian is
digitally encoded and digitized. Aiming to achieve high fidelity transfer, we
separately test two different cost functions by making use of the transfer
fidelity and a Hellinger distance-based method, deriving the optimal CD
drivings concurrently. In Section IV.2, we exemplify the edge state transfer
in $5$\- to $15$-site chains by using qiskit [42] in a quantum circuit with
ibmq_qasm_simulator [43].

###  IV.1 Encoding and classical optimizers

The edge state transfer in a SSH chain of $2N-1$ sites requires at least
$n=\lceil\log_{2}(2N-1)\rceil$ qubits to encode the process by using the
Hamiltonian $H_{0}$ [Eq. (1)]. Such Hamiltonian has to be padded as

$H_{0}^{(c)}(t)=\begin{bmatrix}H_{0}(t)&0\\\ 0&0\\\ \end{bmatrix},$ (16)

with $H_{0}^{(c)}(t)\in\mathcal{M}_{2^{n}}(\mathbb{R})$.

Our goal is to transfer the edge state non-adiabatically by introducing CD
drivings, represented by NNN interactions acting exclusively on sublattice A

$\displaystyle
H_{\text{cd}}^{(c)}=-i\sum_{j=1}^{N-1}\kappa^{(c)}_{2j+1,2j-1}c_{2j+1}^{\dagger}c_{2j-1}+\text{h.c.}$
(17)

Here, the superscript $(c)$ denotes that the optimal but unknown prefactor
$\kappa^{(c)}$ is derived from the circuit, approaching $\kappa^{(l)}$
$(l\rightarrow\infty)$. The subscript indicates the location of the hopping
term. It is worth noting that the forms of these drivings are based on the
first-order NC as shown in Eq. (8). To digitally implement the time evolution,
both Hamiltonians must be decomposed into the Pauli basis. The decomposition
can be performed using methods presented in previous works, such as [44, 18,
45]. In Appendix C, general decompositions of Hamiltonians are attached,
considering all possible NNN interactions between sublattice A.

In the circuit, the state is evolved as $\ket{\Psi(t)}=U(t,0)\ket{\Psi(0)}$
with $t\in[0,T]$. The initial state is $\ket{\Psi(0)}=\ket{\Phi_{0}(0)}$ (the
probability is $1$ at the first site and $0$ otherwise), which can be encoded
in the computational basis as $\ket{\Psi(0)}=\ket{0\dots 0}=\ket{0}$. We
employ product formulas for exponentials of commutators [46], digitally
exploring them for 1D fermion chains with nearest- and next-nearest-neighbour
hopping terms. In our work, the circuit can be constructed with Pauli gates to
digitize the time evolution of Hamiltonian $H_{0}^{(c)}+H_{\text{cd}}^{(c)}$
with the unitary operator,

$U(T,0)\approx\prod_{k=1}^{r}\exp\left[{-iH_{0}^{(c)}(k\Delta t)\Delta
t}\right]\exp\left[{-iH_{\text{cd}}^{(c)}(k\Delta t)\Delta t}\right].$ (18)

This is achieved by computing the Trotter-Suzuki decomposition [47, 48] up to
the first order, with $r\coloneqq T/\Delta t$ steps, where $\Delta t$ is the
interval of each Trotter step. We discretize $\kappa^{(c)}_{2j+1,2j-1}$ as a
set of $r+1$ parameters

$\vec{\kappa}^{(c)}_{2j+1,2j-1}(t)\coloneqq[\kappa^{(c)}_{2j+1,2j-1}(0),\kappa^{(c)}_{2j+1,2j-1}(\Delta
t),\dots,\kappa^{(c)}_{2j+1,2j-1}(T)],\leavevmode\nobreak\
\leavevmode\nobreak\ $ (19)

with boundary conditions
$\kappa^{(c)}_{2j+1,2j-1}(0)=\kappa^{(c)}_{2j+1,2j-1}(T)=0$ so that $r-1$
parameters will be optimized. The digital form of the Hamiltonian
$H_{0}^{(c)}+H_{\text{cd}}^{(c)}$, i.e. Eq. (18), is implemented in the
quantum circuit as depicted in Fig. 3. Our goal is to generate the target
state $\ket{\Psi(T)}$ as close as possible to $\ket{\Phi_{0}(T)}=\ket{2N-2}$
(the probability is $1$ at site $2N-1$ and $0$ otherwise), up to the set of
optimal parameters $\vec{\kappa}^{(c)}$.

[width=.95]./images/qc.pdf

Figure 3: Circuit implementation of the state evolution by using the unitary
operator $U(T,0)$ [Eq.(18)] trotterizing the Hamiltonian which is the sum of
$H_{0}^{(c)}$ (yellow) and $H_{\text{cd}}^{(c)}$ (purple).

In general, various classical optimizers, such as adaptive moment estimation
algorithm (ADAM), constrained optimization by linear approximations (COBYLA),
are available in the selection pool, with their pros and cons. Simultaneous
Perturbation Stochastic Approximation (SPSA), a gradient descent method for
optimizing systems with multiple unknown parameters, is a suitable simulation
optimization method with a relatively small number of measurements of the
objective function, appropriate to be applied in the SSH model in this
context. In order to ease the optimization routine utilizing _warm-starting_ ,
we initialize the parameters at each time step with an initial guess
proportional to $\kappa^{(1)}$. Additionally, adaptive bounds are set for the
$r-1$ parameters to optimize CD drivings.

To guarantee the state evolution along the zero-energy eigenstate, we apply
two cost functions respectively: one designed from the transfer fidelity and
the other one retrieved from the measurement in quantum circuits. The first
cost function $C(\vec{\kappa}^{(c)}(t))=F(T)$ [Eq. (5)], where
$\vec{\kappa}^{(c)}$ collects all the NNN couplings $\kappa^{(c)}_{2j+1,2j-1}$
$\forall j\in[1,N-1]$, conceived from the control problem itself, maximizes
the fidelity at the final time $T$ between the zero-energy eigenstate and the
state derived in the circuit. These overlaps can be computed in quantum
circuits thanks to SWAP tests [49, 50], where as much as qubits to encode both
overlapping states are needed, among other techniques [51, 52]. To use reduced
number of qubits and to get optimization from utilizing a finite number of
measurement in quantum circuits, we apply Hellinger distance [53] (also called
as Jeffreys distance [54]) which quantifies the similarity between two given
probability distributions $P$ and $Q$ over a measure space $\mathcal{X}$, with
corresponding probability density functions $p(x)$ and $q(x)$ where
$x\in\mathcal{X}$. Its integral form is defined as

$H^{2}(P,Q)=1-\int_{\mathcal{X}}\sqrt{p(x)q(x)}\text{d}x,$ (20)

where $0\leq H^{2}(P,Q)\leq 1$ using Cauchy-Schwarz inequality. For discrete
distributions of $m$ values, let $P=\\{p_{i}\\}_{i=0}^{m-1}$ and
$Q=\\{q_{i}\\}_{i=0}^{m-1}$ so that Eq. (20) becomes

$1-H^{2}(P,Q)=\sum_{i=0}^{m-1}\sqrt{p_{i}q_{i}}.$ (21)

Among all the $2^{n}$ possible outcomes in the circuit (ranging from $\ket{0}$
to $\ket{2^{n}-1}$), we expect to maximize the probability of measuring as
circuit state $\ket{2N-2}$. The Hellinger distance
$H^{2}(P(\vec{\kappa}^{(c)}(t)),Q)$ is used as the cost function, where
$P(\vec{\kappa}^{(c)}(t))$ is the probability distribution after
$N_{\text{shot}}$ measurements. The target probability distribution $Q$ is
defined as $q_{2N-2}=1$ and $0$ otherwise. To get $P(\vec{\kappa}^{(c)}(t))$
as close as possible to $Q$, i.e. to maximize the probability of transferring
the edge state, we define the cost function as

$\min_{\vec{\kappa}^{(c)}}H^{2}(P(\vec{\kappa}^{(c)}),Q)\equiv\max_{\vec{\kappa}^{(c)}}p_{2N-2},$
(22)

which acts analogously to the transfer fidelity [Eq. (5)] in the large
$N_{\text{shot}}$ limit. Therefore, seen from Eq. (22), minimizing the
Hellinger distance is translated into maximizing the population on the site
$a_{N}$ at $t=T$, which is indeed the main goal of the optimization process.

###  IV.2 Numerical results and analysis

In this subsection, we give a detailed example for the transfer over SSH
chains with different numbers of sites, going from $5$ ($N=3$) to $15$ sites
($N=8$), thus requiring at most $n=4$ qubits for the digitalization in a
quantum circuit approaching $\ket{\Phi_{0}(T)}=\ket{2N-2}$ starting from
$\ket{\Phi_{0}(0)}=\ket{0}$. Hereinafter, working in the non-adiabatic regime
$t_{0}=\Omega=1$, thus $T=\pi$, is set. The Pauli basis decomposition of Eqs.
(16) and (17) can be computed following Appendix C. To exemplify the
technique, the decomposition for a $5$-site chain becomes

$\displaystyle H_{0}^{(c)}(t)$
$\displaystyle=\frac{\Lambda-\lambda(t)}{2}(I+Z)IX+\frac{\lambda(t)}{4}[(I+Z)(XX+YY)+X(XX-
YY)+Y(XY+YX)],$ (23) $\displaystyle H_{\text{cd}}^{(c)}(t)$
$\displaystyle=\frac{1}{4}[\kappa^{(c)}_{31}(t)(I+Z)Y(I+Z)+\kappa^{(c)}_{53}(t)(YX-
XY)(I+Z)].$ (24)

Regarding the NNN hoppings $\vec{\kappa}^{(c)}$, two approaches are going to
be tested: time-symmetric hoppings due to symmetry and equal NNN hoppings
inspired from the 1NC matrix form. For the first one, the NNN hoppings obey
$\kappa^{(c)}_{2i+1,2i-1}(t)=\kappa^{(c)}_{2(N-i)+1,2(N-i)-1}(T-t)$ with
$i\in[1,\lfloor N/2\rfloor]$ so that $\lfloor N/2\rfloor(r-1)$ parameters are
required. For the latter one, we factor out the hopping strengths of Eq. (17),
thus the NNN hoppings become site-independent, i.e.
$\kappa^{(c)}_{2j+1,2j-1}(t)\mapsto\kappa^{(c)}(t)$. Consequently, $r-1$
parameters are needed to be optimized. The latter simplified protocol is
easier for the implementation.

We trotterize the time evolution operator [Eq. (18)] into $r=22$ steps and run
the variational quantum circuit with $1000$, $2000$, $3000$, $5000$, $6000$
and $10000$ optimizer iterations for the chains with $5$ to $15$ sites,
respectively. Such a realization is repeated with $10$ times in order to see
the variance (thus the stability) of our method. In order to guarantee
$\mathcal{O}(\left\lVert H_{0}\right\rVert)\sim\mathcal{O}(\left\lVert
H_{\text{cd}}\right\rVert)$ for feasibility purposes, i.e. CD drivings are in
the same order of $t_{0}$, we take the value of $\kappa^{(1)}$ times a
constant $\sigma_{\kappa}$ as the initial guess, while the optimal parameters
are set to lay between a constant $\kappa_{\text{bound}}$ in the same order of
magnitude of the initial guess minima and zero. For our study, we take
$\sigma_{\kappa}$ a random value within $[4,5]$, $[6,7]$, $[7,8]$, $[9,10]$,
$[10,11]$ and $[11,13]$ as well as
$\kappa_{\text{bound}}\in[-2.5,-3,-3.5,-4,-4.5,-5]$ for $5$ to $15$ sites,
respectively, to show the stability of the results.

images/kappa.pdf

(a)

images/costs.pdf

(b)

Figure 4: From top to bottom: results for chains of $5$, $7$, $9$, $11$, $13$
and $15$ sites ((a)-(f), respectively) using both equal (orange squares) and
different NNN hoppings (rest) on sublattice A after $10$ runs. Left: NNN
couplings for each chain are represented in the case of $\Omega=t_{0}=1$ with
the labelled average values and their corresponding standard deviation. Right:
Evolution of the cost function $F(t)$ for the chains with $5$, $7$, $9$, $11$,
$13$ and $15$ sites against the rescaled number of iterations with
$k\in[1,2,3,5,6,10]$ in each case, respectively. Shaded areas show the region
of the returned minimum and maximum values.

./images/kappa_circ_hellinger.pdf

Figure 5: Using Hellinger distance as the cost function [Eq. (22)], we find
optimal CD drivings for a $5$-site chain from running quantum circuits $10$
times where $\Omega=t_{0}=1$, when considering equal NNN hopping terms on
sublattice A. (a) $\kappa^{(c)}_{\text{H}}$. (b) $p_{4}$ in terms of
iterations. (c) $p_{4}$ as a function of the number of measurements
$N_{\text{shot}}$, and simultaneously check (d) $F(T)$ in the presence of the
corresponding $N_{\text{shot}}$ by introducing the optimal drivings obtained
from Hellinger distance. Each point derived at a certain $N_{\text{shot}}$ in
(c-d) is from running $1000$ times iteration in the circuit. In (a), (c), (d),
the labelled average values and standard deviation are shown, while the area
of minimum and maximum returned values are plotted in (b).

Two distinct cost functions are used here. The first one is to use the
transfer fidelity as cost function [Eq. (5)]. The optimal ${\kappa}^{(c)}(t)$
are found in Fig. 4(a1-f1), while the evolution of the final fidelity $F(T)$
with respect to the number of iterations considering only NNN hoppings in
sublattice A is shown in Fig. 4(a2-f2). Using only NNN hoppings as CD drivings
can transfer the edge state with $\mathcal{O}(\left\lVert
H_{0}\right\rVert)\sim\mathcal{O}(\left\lVert H_{\text{cd}}\right\rVert)$ with
fidelity $F(T)\sim 90\%$ in a $15$-site chain. If we relax the
$\mathcal{O}(\left\lVert H_{0}\right\rVert)\sim\mathcal{O}(\left\lVert
H_{\text{cd}}\right\rVert)$ constraint by increasing $\sigma_{\kappa}$,
$|\kappa_{\text{bound}}|$ values, stronger NNN hoppings will be found. In this
way, it is possible to accomplish higher fidelities but at the cost of
deriving much stronger NNN hoppings in contrast to $t_{1,2}$. It is also
possible to replicate CD drivings similar to the ones derived from inverse
engineering [27] if one uses loose constraints for the strength in the
circuits. However, the CD driving strengths have to be limited depending on
the physical setup in order to avoid strong heating.

The other separate study is to use the same procedure and hyper-parameters in
the quantum circuit but apply the cost function Eq. (22) from Hellinger
distance with $N_{\text{shot}}=1024$. Resembling the study of Fig. 4, we
obtain similar results shown in Fig. 5(a-b), which proves the feasibility of
using measurement-based quantum circuits. To estimate the number of
measurements needed to acquire a good and stable performance of the edge state
transfer, in Fig. 5(c-d) we illustrate the relation between $p_{4}$ and the
number of measurements $N_{\text{shot}}$, where each $p_{4}$ value is obtained
by $1000$-times iteration in the circuit. Meanwhile, we also check the
transfer fidelity $F(T)$ in the presence of the corresponding
$N_{\text{shot}}$ by using the optimal drivings obtained from Hellinger
distance. Running the variational circuit $10$ times, one can get the average
fidelity above $90\%$ from $N_{\text{shot}}=1$, and stable average fidelity
above $99\%$ with $N_{\text{shot}}>200$.

##  V Discussions

###  V.1 Physical feasibility

Firstly, demonstrating long-range edge state transfer is feasible across
various experimental platforms. In the realm of superconducting qubits [8],
the engineered hopping strengths, depicted as $t_{1,2}(t)=g_{0}\mp
g_{1}\cos\Omega t$ similar to our model in Eq. (4), provide a promising
avenue. Experimental parameters such as
$g_{1}/2\pi=$250\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$, $g_{0}=g_{1}$, and
$T=\pi/\Omega=$0.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$$ in the
adiabatic regime ($\Omega=0.01g_{1}$) were considered. Notably, shortening the
operation time to $T=$2\text{\,}\mathrm{n}\mathrm{s}$$ ($\Omega=g_{1}$) while
keeping $g_{1}$ constant necessitates the addition of CD hoppings between
neighbouring sites on sublattice A with a magnitude comparable to $g_{1}$.
Tuning this CD driving can be achieved smoothly by varying the current passing
through the coupler linking corresponding Xmon qubits [55, 56], where
imaginary couplings are indeed realizable in superconducting qubit setups
[57]. Experimental validation of this CD driving has been performed by
generating gauge-invariant phases in three-level superconducting quantum
circuits [58, 59].

images/disorder.pdf

Figure 6: Robustness of the fidelity $F(T)$ for a SSH chain of $5$ sites using
both different and equal NNN hoppings (see Fig. 4) as well as in the adiabatic
regime $\Omega=0.01t_{0}$ ($t_{0}=1$) against (a) relative strength of the
staggered Rice-Mele contribution in Eq. (26) in terms of $\delta/t_{0}$
setting $\delta_{j}=\tau_{j}=0$ for $\Delta(t)=\delta$ (solid) and
$\Delta(t)=\delta\sin\Omega t$ (dashed), where fidelities of both
parameterizations almost overlap; (b) diagonal disorder contribution in Eq.
(26) in terms of $\sigma_{\delta}/t_{0}$ setting $\Delta(t)=\tau_{j}=0$, (c)
hopping disorder contribution of Eq. (27) in terms of $\sigma_{\tau}/t_{0}$
setting $\Delta(t)=\delta_{j}=0$. For the latter two (b-c), results are
averaged over $1000$ runs.

On the other hand, the configuration involving sites $A_{i}$ and $B_{i}$ in
each unit cell, along with $A_{i+1}$—the neighbouring site in sublattice A,
constitutes the spatial adiabatic passage block of a three-level system. This
configuration is relevant to systems like cold atoms trapped in optical
lattices [60, 61, 62] and electrons in quantum dots [63, 39, 64]. Achieving
non-adiabatic transfer in a longer chain, as exemplified by our SSH model,
becomes increasingly necessary and significant for long-range information
transfer. Experimental demonstrations in dynamically modulated SSH models have
shown adiabatic edge-to-edge transport of atomic momentum states in synthetic
lattices of Bose-Einstein condensates [26] on the timescale of hundreds of
microseconds. In contrast, our proposed protocol offers robust transfer on a
nanosecond timescale before decoherence sets in. Moreover, digital quantum
simulation provides discretized solutions, which are expected to be more
easily implemented in experiments.

###  V.2 Robustness against the disorder

In the context of adiabatic transfer in a SSH model, external sources such as
diagonal disorder acting over on-site energies can break chiral symmetry,
while hopping disorder preserves chiral symmetry [41]. In our non-adiabatic
protocol, chiral symmetry is intentionally broken by adding CD drivings
between atoms from the same sublattice [41, 65]. To assess the robustness of
our non-adiabatic protocol against deviations from the reference Hamiltonian
(1), we examine the probability of transferring the edge state using a
modified Hamiltonian:

$H^{\prime}(t)\coloneqq
H_{0}(t)+H_{\text{cd}}(t)+H_{\text{diag}}(t)+H_{\text{off-diag}},$ (25)

where

$H_{\text{diag}}(t)=\sum_{j=1}^{2N-1}[(-1)^{j-1}\Delta(t)+\delta_{j}]c^{\dagger}_{j}c_{j},$
(26)

adds a staggered-diagonal potential with
$\delta_{j}\in[-\sigma_{\delta},\sigma_{\delta}]$ a random value encoding a
diagonal disorder to the staggered potential contribution. For the Rice-Mele
model [35], $\Delta(t)$ can be taken as constant with $\Delta(t)=\delta$ or
$\Delta(t)=\delta\sin\Omega t$, with $\delta$ a real constant [23]. Following
similar methods as proposed in [8, 41], we consider an additional off-diagonal
contribution given by

$H_{\text{off-
diag}}=\sum_{j=1}^{2N-2}\tau_{j}c^{\dagger}_{j+1}c_{j}+\text{h.c.},$ (27)

with $\tau_{j}\in[-\sigma_{\tau},\sigma_{\tau}]$ a random value encoding a
disorder on nearest-neighbour hoppings.

Taking the average CD driving derived from the circuit as shown in Fig. 4, we
solve the time-dependent Schrödinger equation [Eq. (25)] under the
consideration of different scenarios with the aforementioned extra
contributions. When a staggered-diagonal term is introduced (Fig. 6(a)), the
fidelity degrades slower by using different hoppings rather than equal ones,
showing that using different hoppings provides more robustness for both
parameterizations $\Delta(t)=\delta$ and $\Delta(t)=\delta\sin\Omega t$. Shown
in Fig. 6, the staggered Rice-Mele contribution (a), diagonal disorder
contribution (b) and off-diagonal disorder contribution (c) separately present
high fidelities over long range until the ratio of their strengths over
$t_{0}$ is around $0.1$. All of the above results indicate that even though
chiral symmetry is broken [41], a high stability can be preserved in both
staggered Rice-Mele contributions and disorder with reasonable strengths.

###  V.3 Non-adiabatic transfer solely by CD driving

images/kappa_noh0.pdf

(a)

images/costs_noh0.pdf

(b)

Figure 7: Left: Derived different NNN hopping terms on sublattice A for a
chain of $15$ sites setting $H_{0}=0$. NNN couplings are represented in the
case of $t_{0}=1$ and $\Omega=1$ with the labelled average values and their
corresponding standard deviation after $10$ runs. Right: Evolution of the cost
function $F(t)$ against the number of iterations, where shaded areas show the
region of the returned minimum and maximum values.

images/disorder_noh0.pdf

Figure 8: Robustness of the fidelity $F(T)$ for a SSH chain of $15$ sites
using different NNN hoppings on sublattice A, setting $H_{0}=0$ (blue lines)
as well as the expected behaviour (black lines) in the adiabatic regime
$\Omega=0.01t_{0}$ ($t_{0}=1$) against (a) relative strength of the staggered
Rice-Mele contribution in Eq. (26) in terms of $\delta$ setting
$\delta_{j}=\tau_{j}=0$ for $\Delta(t)=\delta$ (solid) and
$\Delta(t)=\delta\sin\Omega t$ (dashed), where fidelities of both
parameterizations almost overlap; (b) diagonal disorder contribution in Eq.
(26) in terms of $\sigma_{\delta}$ setting $\Delta(t)=\tau_{j}=0$, (c) hopping
disorder contribution of Eq. (27) in terms of $\sigma_{\tau}$ setting
$\Delta(t)=\delta_{j}=0$. For the latter two, results are averaged over $1000$
runs.

With the goal of achieving a simpler and more reliable setup for edge state
transfer, we propose encoding the entire SSH system information in the CD
drivings by turning off the reference Hamiltonian ($H_{0}=0$). This
simplification is motivated by the negligible contribution of NNN hoppings on
sublattice B (even-even interactions), satisfying the conditions for
transferring the edge state. Moreover, setting $H_{0}=0$ significantly reduces
the depth of the variational quantum circuit needed to extract the desired CD
drivings, as depicted in Fig. 3. Simultaneously, this simplification
eliminates population excitation in sublattice B.

We conducted experiments with different NNN hoppings, as shown in Fig. 4,
while setting $H_{0}=0$ for chains with $5$, $7$, $9$, $11$, $13$ and $15$
sites. Each case was tested with 1000 iterations. This simplification led to a
shorter circuit depth, faster convergence, and even better fidelities compared
to including $H_{0}$ in the calculations. Fidelity consistently exceeded
$99\%$.

In Fig. 7, we demonstrate the high fidelity of a 15-site chain when $H_{0}=0$
and different NNN hoppings on sublattice A are used in $10$ runs. The
robustness of this approach against disorder is illustrated in Fig. 8, where
we compare it with the expected behaviour in the adiabatic regime
($\Omega=0.01t_{0}$). Notably, setting $H_{0}=0$ makes the method stable
against Rice-Mele contributions, and it exhibits slight performance
degradation for diagonal and hopping disorder but remains stable when disorder
strengths, $\sigma_{\delta}$ and $\sigma_{\tau}$, are less than $0.1$.

##  VI Conclusion

By synergistically integrating analog and digital strategies, we present a
non-adiabatic scheme for edge state transfer in SSH chains with different odd
number of sites. In particular, we perform several detailed studies in the
non-adiabatic regime $T=\pi$ ($\Omega=t_{0}$), thus transferring the edge
state $100$ times faster than the regime $T=100\pi$ ($\Omega=0.01t_{0}$) set
to meet adiabatic criteria [27]. Starting by computing the NCs from
approximate gauge potentials, we find the patterns between the chain length
and the location of different order of NC. Essentially, the CD drivings can be
interpreted as NNN hoppings between sublattice A, although the deviation up to
higher order requires a lot of computational cost. Resorting to digital
quantum simulation, we employ variational quantum circuits to extract optimal
CD drivings. The CD drivings from the first-order NC formula are set as the
initial values in the circuit with some deviation as degrees of freedom during
the optimization process. For this purpose, two different kinds of cost
functions, based on the transfer fidelity and Hellinger distance, are used,
ultimately yielding high fidelity. The analog analysis to derive the CD
drivings from NCs displays explicitly the precise location of additional
hoppings required to achieve non-adiabatic driving. Meanwhile, the digital
simulation to optimize CD drivings facilitates the routines to achieve the
additional hoppings. Both ways are mutually reinforcing and complementary,
enhancing the ability to effectively navigate the non-adiabatic state transfer
in the SSH chain.

We also discuss the practicability of this protocol in different physical
platforms such as superconducting qubits and cold atoms trapped in optical
lattices, especially the feasibility to generate CD drivings utilizing
cutting-edge techniques available in these contexts. In a comprehensive
assessment, we subject the protocol to rigorous testing, gauging its
resilience against an array of supplementary contributions commonly
encountered within laboratory settings, including: Rice-Mele detuning,
diagonal disorder and hopping disorder. The high fidelity proves the
robustness of our method, showing that our protocol is indeed feasible for
practical experimental implementation.

###### Acknowledgements.

We thank Koushik Paul and Pranav Chandarana for the fruitful discussions. We
acknowledge the use of IBM Quantum services for this work. The views expressed
are those of the authors, and do not reflect the official policy or position
of IBM or the IBM Quantum team. S.V.R. and Y.B. thank the support of the
Basque Government through HAZITEK program (Q4_Real project, ZE-2022/00033),
the Government of Biscay (Bizkaiko Foru Aldundia–Diputación Foral de Bizkaia)
through Lantik and its Industry Focused Quantum Ecosystem initiative which
provided access to the IBM quantum computers. X.C. appreciates the financial
supported from the Basque Government through Grant No.IT1470-22, the Project
Grant No.PID2021-126273NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by
“ERDFA way of making Europe” and “ERDF Investinyour Future,” Nanoscale NMR and
complex systems (Grant No.PID2021-126694NB-C21), EU FET Open Grant EPIQUS
(No.899368), and Ramón y Cajal program of the Spanish MINECO (Grant No.
RYC-2017-22482). G.P. was supported by Spain’s MINECO through Grant No.
PID2020-117787GB-I00 and by CSIC Research Platform PTI-001.

##  Appendix A Analytical solution to the first- and second-order NC for a SSH
chain with $2N-1$ sites

We derive step-by-step the approximate gauge potentials
$\mathcal{A}_{\lambda}^{(1)}$ starting from the base Hamiltonian (1) to
retrieve the analytical expressions of $\alpha_{1}^{(1)}(t)$ from (7) via
action minimization. To ease the calculations, remember that spinless creation
and annihilation operators obey the anticommutation rules
$\\{c_{i},c_{j}\\}=0$ and $\\{c^{\dagger}_{i},c_{j}\\}=\delta_{ij}$ with
$\delta_{ij}$ equal to 1 if $i=j$ and 0 otherwise. Therefore,
$[c^{\dagger}_{i}c_{j},c^{\dagger}_{k}c_{l}]=c^{\dagger}_{i}c_{l}\delta_{jk}-c^{\dagger}_{k}c_{j}\delta_{il}$.
We start from the reference Hamiltonian (1) with hopping parameters
$t_{1}(t)=\lambda(t)$ and $t_{2}(t)=\Lambda-\lambda(t)$, with $\Lambda$ some
constant given by the chosen parameterization of the hoppings. Before
computing the first-order NC (8), we need the following quantities:

$\displaystyle\frac{[H_{0},\partial_{\lambda}H_{0}]}{\Lambda}$
$\displaystyle=\left[\sum_{j=1}^{N-1}c^{\dagger}_{2j+1}c_{2j-1}-\sum_{j=1}^{N-2}c^{\dagger}_{2j+2}c_{2j}\right]-\text{h.c.},$
(28)
$\displaystyle\begin{split}\frac{[H_{0},[H_{0},\partial_{\lambda}H_{0}]]}{\Lambda}&=t_{2}\left[2\sum_{j=1}^{N-2}(c^{\dagger}_{2j+1}c_{2j}-c^{\dagger}_{2j+2}c_{2j})+c^{\dagger}_{2N-1}c_{2N-2}\right]\\\
&+t_{1}\left[\sum_{j=1}^{N-2}(2c^{\dagger}_{2j+3}c_{2j}-c^{\dagger}_{2j+2}c_{2j+1}-c^{\dagger}_{2j}c_{2j-1})-c^{\dagger}_{2N-2}c_{2N-3}\right]+\text{h.c.}\end{split}$
(29)

We can compute
$G_{1}=\partial_{\lambda}H_{0}-i[H_{0},\mathcal{A}_{\lambda}^{(1)}]$ for
$2N-1$ sites thanks to (29). Grouping terms, one can derive

$\begin{split}G_{1}&=\sum_{j=1}^{N-2}\left[(1+2\alpha^{(1)}_{1}\Lambda
t_{2})c^{\dagger}_{2j+1}c_{2j}-(1+2\alpha^{(1)}_{1}\Lambda
t_{1})c^{\dagger}_{2j+2}c_{2j+1}-2\alpha^{(1)}_{1}\Lambda
t_{2}c^{\dagger}_{2j+2}c_{2j-1}+2\alpha^{(1)}_{1}\Lambda
t_{1}c^{\dagger}_{2j+3}c_{2j}\right]\\\
&\qquad\qquad+(1+\alpha^{(1)}_{1}\Lambda
t_{2})c^{\dagger}_{2N-1}c_{2N-2}-(1+\alpha^{(1)}_{1}\Lambda
t_{1})c^{\dagger}_{2}c_{1}+\text{h.c.}\end{split}$ (30)

Following Wick’s theorem [31], the only surviving terms of the action
$S_{1}=\mathrm{tr}[G^{2}_{1}]$ will be proportional to the sum of squares of
each individual contribution associated coefficients. Therefore,

$\begin{split}\frac{S_{1}}{2}&=\sum_{i=1}^{2}\Big{(}(N-2)\left[(1+2\alpha^{(1)}_{1}\Lambda
t_{i})^{2}+(2\alpha^{(1)}_{1}\Lambda t_{i})^{2}\right]\\\
&\qquad+(1+\alpha^{(1)}_{1}\Lambda t_{i})^{2}\Big{)}.\end{split}$ (31)

Minimizing the action $S_{1}$ solving $\partial
S_{1}/\partial\alpha^{(1)}_{1}=0$ implies that

$\alpha_{1}^{(1)}=-\frac{2(N-2)+1}{[8(N-2)+1](t_{1}^{2}+t_{2}^{2})}=-\frac{\mathcal{C}(N)}{t^{2}_{1}+t^{2}_{2}},$
(32)

with $\mathcal{C}(N)=[2(N-2)+1]/[8(N-2)+1]$. In particular, for $5$ sites,
$\mathcal{C}(3)=1/3$.

For deriving the second-order NC we have to minimize the action
$S_{2}=\mathrm{tr}[G_{2}^{2}]$. Apart from Eqs. (28) and (29), the following
quantities are also needed:

$\displaystyle\begin{split}\frac{[H_{0},[H_{0},[H_{0},\partial_{\lambda}H_{0}]]]}{\Lambda}&=t^{2}_{2}\left[4\sum_{j=1}^{N-2}(c^{\dagger}_{2j+1}c_{2j-1}-c^{\dagger}_{2j+2}c_{2j})+c^{\dagger}_{2N-1}c_{2N-3}\right]\\\
&+t^{2}_{1}\left[\sum_{j=1}^{N-2}(3c^{\dagger}_{2j+3}c_{2j+1}+c^{\dagger}_{2j+1}c_{2j-1}-4c^{\dagger}_{2j+2}c_{2j})+c^{\dagger}_{2N-1}c_{2N-3}\right]\\\
&+4t_{1}t_{2}\left[\sum_{j=1}^{N-3}(c^{\dagger}_{2j+3}c_{2j-1}-c^{\dagger}_{2j+4}c_{2j})+c^{\dagger}_{2N-1}c_{2N-5}\right]-\text{h.c.},\end{split}$
(33)
$\displaystyle\begin{split}\frac{[H_{0},[H_{0},[H_{0},[H_{0},\partial_{\lambda}H_{0}]]]]}{\Lambda}&=t^{3}_{2}\left[8\sum_{j=1}^{N-2}(c^{\dagger}_{2j+1}c_{2j}-c^{\dagger}_{2j+2}c_{2j-1})+c^{\dagger}_{2N-1}c_{2N-2}\right]\\\
&+t^{3}_{1}\left[\sum_{j=1}^{N-2}(8c^{\dagger}_{2j+3}c_{2j}-c^{\dagger}_{2j}c_{2j-1}-7c^{\dagger}_{2j+2}c_{2j+1})-c^{\dagger}_{2N-2}c_{2N-3}\right]\\\
&+t_{1}t^{2}_{2}\left[\sum_{j=1}^{N-2}(9c^{\dagger}_{2j+3}c_{2j}-4c^{\dagger}_{2j}c_{2j-1}-4c^{\dagger}_{2j+2}c_{2j+1})-c^{\dagger}_{2N-2}c_{2N-3}\right]\\\
&+t^{2}_{1}t_{2}\left[\sum_{j=1}^{N-2}(5c^{\dagger}_{2j+1}c_{2j}-9c^{\dagger}_{2j+2}c_{2j-1})+4c^{\dagger}_{2N-1}c_{2N-2}\right]\\\
&+t_{1}t_{2}\sum_{j=1}^{N-3}[t_{2}(7c^{\dagger}_{2j+3}c_{2j}-8c^{\dagger}_{2j+4}c_{2j-1})+t_{1}(8c^{\dagger}_{2j+5}c_{2j}+3c^{\dagger}_{2j+3}c_{2j+2}-7c^{\dagger}_{2j+4}c_{2j+1})]+\text{h.c.}\end{split}$
(34)

Notice that some of the latter terms of (33) and (34) only appear if $N\geq
4$. Following the same strategy as in Eqs. (30) and (31),
$G_{2}=\partial_{\lambda}H_{0}-i[H_{0},\mathcal{A}_{\lambda}^{(2)}]$ and
$S_{2}=\mathrm{tr}[G_{2}^{2}]$ return

$\displaystyle\begin{split}G_{2}&=\sum_{j=1}^{N-3}\Big{[}2\Lambda\left(t_{1}[\alpha^{(2)}_{1}+4\alpha^{(2)}_{2}(t^{2}_{1}+2t^{2}_{2})]c^{\dagger}_{2j+3}c_{2j}-t_{2}[\alpha^{(2)}_{1}+4\alpha^{(2)}_{2}(2t^{2}_{1}+t^{2}_{2})]c^{\dagger}_{2j+4}c_{2j+1}+4\alpha^{(2)}_{2}t_{1}t_{2}[t_{1}c^{\dagger}_{2j+5}c_{2j}-t_{2}c^{\dagger}_{2j+4}c_{2j-1}]\right)\\\
&\qquad+\left(1+2\Lambda
t_{1}[\alpha^{(2)}_{1}+4\alpha^{(2)}_{2}(t^{2}_{1}+t^{2}_{2})]\right)c^{\dagger}_{2j+2}c_{2j+1}-\left(1+2\Lambda
t_{2}[\alpha^{(2)}_{1}+4\alpha^{(2)}_{2}(t^{2}_{1}+t^{2}_{2})]\right)c^{\dagger}_{2j+3}c_{2j+2}\Big{]}\\\
&-\Lambda
t_{2}[2\alpha^{(2)}_{1}+\alpha^{(2)}_{2}(9t^{2}_{1}+8t^{2}_{2})]c^{\dagger}_{4}c_{1}+\Lambda
t_{1}[2\alpha^{(2)}_{1}+\alpha^{(2)}_{2}(8t^{2}_{1}+9t^{2}_{2})]c^{\dagger}_{2N-1}c_{2N-4}\\\
&-\left(1+\Lambda
t_{1}[\alpha^{(2)}_{1}+\alpha^{(2)}_{2}(t^{2}_{1}+4t^{2}_{2})]\right)c^{\dagger}_{2}c_{1}+\left(1+\Lambda
t_{2}[2\alpha^{(2)}_{1}+\alpha^{(2)}_{2}(5t^{2}_{1}+8t^{2}_{2})]\right)c^{\dagger}_{3}c_{2}\\\
&-\left(1+\Lambda
t_{1}[2\alpha^{(2)}_{1}+\alpha^{(2)}_{2}(8t^{2}_{1}+5t^{2}_{2})]\right)c^{\dagger}_{2N-2}c_{2N-3}+\left(1+\Lambda
t_{2}[\alpha^{(2)}_{1}+\alpha^{(2)}_{2}(4t^{2}_{1}+t^{2}_{2})]\right)c^{\dagger}_{2N-1}c_{2N-2}+\text{h.c.},\end{split}$
(35)
$\displaystyle\begin{split}\frac{S_{2}}{2}&=(N-3)\Big{[}4\Lambda^{2}\left(t^{2}_{1}[\alpha^{(2)}_{1}+4\alpha^{(2)}_{2}(t^{2}_{1}+2t^{2}_{2})]^{2}+t^{2}_{2}[\alpha^{(2)}_{1}+4\alpha^{(2)}_{2}(2t^{2}_{1}+t^{2}_{2})]^{2}+[4\alpha^{(2)}_{2}t_{1}t_{2}]^{2}(t^{2}_{1}+t^{2}_{2})\right)\\\
&\qquad+\left(1+2\Lambda
t_{1}[\alpha^{(2)}_{1}+4\alpha^{(2)}_{2}(t^{2}_{1}+t^{2}_{2})]\right)^{2}+\left(1+2\Lambda
t_{2}[\alpha^{(2)}_{1}+4\alpha^{(2)}_{2}(t^{2}_{1}+t^{2}_{2})]\right)^{2}\Big{]}\\\
&+\Lambda^{2}\left(t^{2}_{2}[2\alpha^{(2)}_{1}+\alpha^{(2)}_{2}(9t^{2}_{1}+8t^{2}_{2})]^{2}+t^{2}_{1}[2\alpha^{(2)}_{1}+\alpha^{(2)}_{2}(8t^{2}_{1}+9t^{2}_{2})]^{2}\right)+\left(1+\Lambda
t_{1}[\alpha^{(2)}_{1}+\alpha^{(2)}_{2}(t^{2}_{1}+4t^{2}_{2})]\right)^{2}\\\
&+\left(1+\Lambda
t_{2}[2\alpha^{(2)}_{1}+\alpha^{(2)}_{2}(5t^{2}_{1}+8t^{2}_{2})]\right)^{2}+\left(1+\Lambda
t_{1}[2\alpha^{(2)}_{1}+\alpha^{(2)}_{2}(8t^{2}_{1}+5t^{2}_{2})]\right)^{2}+\left(1+\Lambda
t_{2}[\alpha^{(2)}_{1}+\alpha^{(2)}_{2}(4t^{2}_{1}+t^{2}_{2})]\right)^{2}.\end{split}$
(36)

Now we minimize the action $S_{2}$ by solving $\partial
S_{2}/\partial\alpha^{(2)}_{1,2}=0$, yielding

$\displaystyle\frac{\partial
S_{2}}{\partial\alpha^{(2)}_{1}}=0\implies\alpha^{(2)}_{1}$
$\displaystyle=-\frac{\alpha^{(2)}_{2}\left([32(N-2)+1](t^{4}_{1}+t^{4}_{2})+32[3(N-3)+2]t^{2}_{1}t^{2}_{2}\right)}{[8(N-2)+1](t^{2}_{1}+t^{2}_{2})}-\frac{\mathcal{C}(N)}{t^{2}_{1}+t^{2}_{2}},$
(37) $\displaystyle\frac{\partial
S_{2}}{\partial\alpha^{(2)}_{2}}=0\implies\alpha^{(2)}_{2}$
$\displaystyle=-\frac{[8(N-2)+1](t^{2}_{1}+t^{2}_{2})+\alpha^{(2)}_{1}\left([32(N-2)+1](t^{4}_{1}+t^{4}_{2})+32[3(N-3)+2]t^{2}_{1}t^{2}_{2}\right)}{[128(N-2)+1](t^{6}_{1}+t^{6}_{2})+6[128(N-3)+59]t^{2}_{1}t^{2}_{2}(t^{2}_{1}+t^{2}_{2})}.$
(38)

We can solve $\alpha^{(2)}_{1}$ and $\alpha^{(2)}_{2}$ by solving the system
of linear equations given by (37) and (38), returning

$\displaystyle\alpha^{(2)}_{1}$
$\displaystyle=-\frac{(t^{2}_{1}+t^{2}_{2})\left(90(N-2)(t^{4}_{1}+t^{4}_{2})+[2N(256N-1039)+1725]t^{2}_{1}t^{2}_{2}\right)}{72(N-2)(t^{8}_{1}+t^{8}_{2})+[8N(128N-581)+4851]t^{2}_{1}t^{2}_{2}(t^{4}_{1}+t^{4}_{2})+2[16N(32N-99)+193]t^{4}_{1}t^{4}_{2}},$
(39) $\displaystyle\alpha^{(2)}_{2}$
$\displaystyle=\frac{18(N-2)(t^{4}_{1}+t^{4}_{2})+2[32N(N-4)+111]t^{2}_{1}t^{2}_{2}}{72(N-2)(t^{8}_{1}+t^{8}_{2})+[8N(128N-581)+4851]t^{2}_{1}t^{2}_{2}(t^{4}_{1}+t^{4}_{2})+2[16N(32N-99)+193]t^{4}_{1}t^{4}_{2}}.$
(40)

In particular, for $5$ sites we have that

$\displaystyle\alpha^{(2)}_{1}$
$\displaystyle=-\frac{9(t^{2}_{1}+t^{2}_{2})\left[10(t^{4}_{1}+t^{4}_{2})+11t^{2}_{1}t^{2}_{2}\right]}{72(t^{8}_{1}+t^{8}_{2})+123t^{2}_{1}t^{2}_{2}(t^{4}_{1}+t^{4}_{2})+98t^{4}_{1}t^{4}_{2}},$
(41) $\displaystyle\alpha^{(2)}_{2}$
$\displaystyle=\frac{6\left[3(t^{4}_{1}+t^{4}_{2})+5t^{2}_{1}t^{2}_{2}\right]}{72(t^{8}_{1}+t^{8}_{2})+123t^{2}_{1}t^{2}_{2}(t^{4}_{1}+t^{4}_{2})+98t^{4}_{1}t^{4}_{2}}.$
(42)

In Fig. 9, time-dependent Schrödinger equation is solved for a SSH chain of
$5$ sites setting $\Omega\in[0.1,0.2,0.5,1]$ and $t_{0}=1$ without and adding
second order NC contribution, left and right columns respectively, showcasing
the robustness of the protocol against non-adiabatic effects.

images/probs_nocd.pdf

(a)

images/probs_cd.pdf

(b)

Figure 9: On-site probability time evolution starting from $\ket{\Phi_{0}(0)}$
by solving the time-dependent SE without and adding the second order NC
contribution ((a)-(d) and (e)-(h), respectively). From top to bottom on both
subfigures: $\Omega$ is equal to $0.1$, $0.2$, $0.5$ and $1$.

##  Appendix B Relation between the order of NC and their interactions
involved

As shown in (28) and (33), first order NC introduces interactions among NNNs
and second order NC introduces interactions between NNNs and fourth-nearest
neighbours. This behaviour extends for arbitrary order of NC.

For the $d$th order NC ($d<N-1$), interactions with the $2l$th nearest
neighbours ($l\in[1,d]$) will appear. Considering terms like
$c^{\dagger}_{2(m+l)-r}c_{2m-r}$ with $2(m+l)-r<2N-1$ and $r\in\\{0,1\\}$, the
$(d+1)$th order NC will be given by

$\begin{split}&[c^{\dagger}_{2j+p}c_{2j+p-1},[c^{\dagger}_{2k+q}c_{2k+q-1},c^{\dagger}_{2(m+l)-r}c_{2m-r}]]\\\
&\qquad=[c^{\dagger}_{2j+p}c_{2j+p-1},c^{\dagger}_{2(m+l)-r+1}c_{2m-r}\delta_{r+q,1}-c^{\dagger}_{2(m+l)-r}c_{2m-r-1}\delta_{rq}]\\\
&\qquad=c^{\dagger}_{2(m+l)-r}c_{2(m-1)-r}\delta_{r+p,1}\delta_{rq}+c^{\dagger}_{2(m+l+1)-r}c_{2m-r}\delta_{rp}\delta_{r+q,1}\\\
&\qquad-c^{\dagger}_{2(m+l)-r+1}c_{2m-r-1}(\delta_{r+p,1}\delta_{rq}+\delta_{rp}\delta_{r+q,1})\end{split}$
(43)

so, as expected, the relative distance between indices increased by two with
respect the original term $c^{\dagger}_{2(m+l)-r}c_{2m-r}$, which translates
that $2(l+1)$th nearest neighbours are now also considered.

As a conclusion, for a SSH model of $2N-1$ sites,
$\mathcal{A}_{\lambda}^{(N-1)}$ in (6) is the minimum order of the adiabatic
gauge potential expansion needed to involve all possible interactions along
the same sublattice.

##  Appendix C Pauli basis decomposition of SSH model

Here we present the general Pauli basis decomposition of the SSH model padded
Hamiltonian with $2N-1$ sites and its corresponding first-order NC, based on
the work presented in [45, 44]. Moreover, we do the same calculation for the
Rice-Mele model [35]. Notice that $n=\lceil\log_{2}(2N-1)\rceil$ qubits will
be at least needed to encode such Hamiltonians.

We start from the regular Hamiltonian (1) and split their terms in two
contributions: one proportional to $t_{2}$ ($H_{0}^{t_{2}}$) and another to
$t_{1}$ ($H_{0}^{t_{1}}$), such that
$H_{0}=t_{2}H_{0}^{t_{2}}+t_{1}H_{0}^{t_{1}}$.

Let $N-1=\sum_{i}c_{i}2^{i}$ and $N=\sum_{i}d_{i}2^{i}$ with
$c_{i},d_{i}\in\\{0,1\\}$ $\forall i$, and
$\mathcal{D}_{x}\coloneqq[1+(-1)^{x}Z]/2$. For $N$ odd, we have

$H_{0}^{t_{2}}(N)=\left[\sum_{i=0}^{n-3}c_{n-2-i}\left(\bigotimes_{j=0}^{i-1}\mathcal{D}_{c_{n-2-j}}\right)\mathcal{D}_{0}I^{\otimes
n-2-i}\right]X.$ (44)

For $N$ even, we have that

$\begin{split}H_{0}^{t_{2}}(N)&=H_{0}^{t_{2}}(N-1)+\left[\bigotimes_{i=0}^{n-3}\mathcal{D}_{c_{n-2-i}}\right]\mathcal{D}_{0}X.\end{split}$
(45)

Following [45], before starting with $H_{0}^{t_{1}}$ decomposition, let
$\sigma^{s}_{\\{0,1\\}}\coloneqq\\{X,Y\\}$ (skew-diagonal Pauli matrices) and
$\gamma_{m}\coloneqq(-1)^{(\lfloor n_{Y}/2\rfloor+\tilde{n}_{Y})\text{ mod
}2}$ with $\tilde{n}_{Y}$ the amount of $Y$ matrices in the $(m-1)$-rightmost
skew-diagonal matrices of the decomposition. Moreover, we define

$\mathcal{S}^{\text{\\{e,o\\}}}_{m}\coloneqq\sum_{\begin{subarray}{c}i_{1},\dots,i_{m}\\\
n_{Y}\text{
\\{even,odd\\}}\end{subarray}}\frac{\gamma_{m}}{2^{m-1}}\bigotimes_{k=1}^{m}\sigma^{s}_{i_{k}},$
(46)

which returns a summation of all possible Pauli strings of length $m$ only
involving $X$ or $Y$ matrices with $n_{Y}$ even or odd. Such terms returns
hermitian contributions for the tridiagonal matrix entries present in the SSH
model. Therefore, $H_{0}^{t_{1}}$ decomposition becomes

$\begin{split}H_{0}^{t_{1}}(N)&=\sum_{m=2}^{n-2}\left[\sum_{i=0}^{n-m-1}d_{n-2-i}\left(\bigotimes_{j=0}^{i-1}\mathcal{D}_{d_{n-2-j}}\right)\mathcal{D}_{0}I^{\otimes
n-m-i-1}\right]\mathcal{S}^{\text{e}}_{m}\\\
&+\frac{I+Z^{c_{n-3}+1}}{2}\mathcal{S}^{\text{e}}_{n-1}+\mathcal{S}^{\text{e}}_{n},\end{split}$
(47)

except for $N$ an exact power of $2$, where (47) simplifies to

$H_{0}^{t_{1}}(N)=\sum_{m=2}^{n}I^{\otimes n-m}\mathcal{S}^{\text{e}}_{m}.$
(48)

Parameterizing $t_{1,2}(t)$ as in (4), we have that
$\partial_{\lambda}t_{1,2}=\pm 1$, implying that
$\partial_{\lambda}H_{0}=H_{0}^{t_{1}}-H_{0}^{t_{2}}$. With this information,
we can decompose in the Pauli basis the matrix entries related to NNN
couplings, taking into account that it is hermitian with purely imaginary
entries, thus $n_{Y}$ must be odd for each Pauli string of the resulting
decomposition.

If we only consider sublattice A NNN interactions, its corresponding Pauli
basis decomposition becomes

$\mathcal{K}(N)=\sum_{m=1}^{n-3}\left[\sum_{i=0}^{n-m-1}d_{n-2-i}\left(\bigotimes_{j=0}^{i-1}\mathcal{D}_{d_{n-2-j}}\right)\mathcal{D}_{0}I^{\otimes
n-m-i-2}\right]\mathcal{S}^{\text{o}}_{m}\mathcal{D}_{0}+\left(\frac{I+Z^{c_{n-3}+1}}{2}\mathcal{S}^{\text{o}}_{n-2}+\mathcal{S}^{\text{o}}_{n-1}\right)\mathcal{D}_{0}$
(49)

except for $N$ an exact power of $2$, where (49) reduces to

$\mathcal{K}(N)=\sum_{m=1}^{n-1}I^{\otimes
n-m-1}\mathcal{S}^{\text{o}}_{m}\mathcal{D}_{0}.$ (50)

For completeness, Rice-Mele model can be decomposed as

$\begin{split}\frac{1}{\Delta}H_{\Delta}(N)&=\left[\sum_{i=0}^{n-2}c_{n-2-i}\left(\bigotimes_{i=0}^{i-1}\mathcal{D}_{c_{n-2-i}}\right)\mathcal{D}_{0}I^{\otimes
n-2-i}\right]Z\\\
&+\left[\bigotimes_{i=0}^{n-2}\mathcal{D}_{c_{n-2-i}}\right]\mathcal{D}_{0}.\end{split}$
(51)

##  Appendix D Another hopping parameterizations

We take polynomial hopping parameters from Ref. [27], where

$t_{1}(t)=t_{0}P(t/T),\quad t_{2}(t)=t_{0}[1-P(t/T)],$ (52)

with $P(x)=2x^{3}-3x^{2}+1$, satisfying that $P(0)=1$ and $P(1)=0$. If we take
$t_{1}(t)=\lambda(t)\implies\dot{\lambda}(t)=6t_{0}t(1-t/T)/T^{2}$ and
$t_{2}(t)=\Lambda-\lambda(t)$ with $\Lambda=t_{0}$, our previously derived
results for computing general first- and second-order nested commutators [Eqs.
(28), (33), (39), (40)] can be used, thus our technique can be implemented
analogously as shown for the hopping parameters given by Eq. (4) with
$\Lambda=t_{0}$ instead of $2t_{0}$.

Moreover, based on [27, 35], we can propose

$t_{1}(t)=t_{0}\cos\Omega t/2,\quad t_{2}(t)=t_{0}\sin\Omega t/2,$ (53)

with transfer time $T=\pi/\Omega$ and now setting
$t_{1}(t)=\lambda(t)\implies\dot{\lambda}(t)=-\Omega t_{2}(t)/2$ and
$t_{2}(t)=\sqrt{t^{2}_{0}-\lambda^{2}(t)}$ $\forall t\in[0,T]$.

In Fig. 10, the time evolution of all hopping parameterizations reviewed on
this document are plotted. Note that Eq. (4) and Eq. (52) almost overlap along
the transfer window $t\in[0,T]$.

images/hoppings.pdf

Figure 10: Hopping parameterizations attached with $t_{1}$ ($t_{2}$) shown in
solid (dashed) line.

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