# Multipartite entanglement distribution in a topological photonic network

Juan Zurita  Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco,
E-28049 Madrid, Spain  Departamento de Física de Materiales, Universidad
Complutense de Madrid, E-28040 Madrid, Spain<EMAIL_ADDRESS>Andrés Agustí
Casado  Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco,
E-28049 Madrid, Spain  Charles E. Creffield  Departamento de Física 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

In the ongoing effort towards a scalable quantum computer, multiple
technologies have been proposed. Some of them exploit topological materials to
process quantum information. In this work, we propose a lattice of photonic
cavities with alternating hoppings to create a modified multidomain SSH chain,
that is, a sequence of topological insulators made from chains of dimers. A
qubit is then coupled to each boundary. We show this system is well suited for
quantum information processing because topological transfer of photons through
this one-dimensional lattice can entangle any set of qubits on demand,
providing a scalable quantum platform. We verify this claim evaluating
entanglement measures and witnesses proving that bipartite and multipartite
entanglement is produced, even in the presence of some disorder.

##  I Introduction

Quantum mechanics allows for the creation of a computer able to solve problems
with a higher performance than classical ones Feynman (1982); Shor (1994);
Grover (1996). Plenty of research nowadays deals with this statement, from the
study of algorithms and metrics that prove quantum advantage Boixo  _et al._
(2018); Bravyi  _et al._ (2018) to the design of hardware primitives able to
create said computer Cirac and Zoller (1995); Zhong  _et al._ (2020). In this
work we focus on the latter, the physical elements of a quantum computer, and
analyse them with a metric that is a necessary condition to quantum advantage:
multipartite entanglement production Nielsen (2006). A list of proposals for
quantum computers includes superconducting circuits Blais  _et al._ (2021),
ion or cold atoms traps Pogorelov  _et al._ (2021), photonic systems, quantum
dots, and topological materials, to name but a few. It is unclear which
platform will have the most impact, but the history of classical computing
teaches us that multiple implementations coexisted and all are worth
researching. Here we focus on a topological photonic lattice coupled to color
center emitters, given the outstanding progress taking place on those
respective areas of research, that we briefly introduce now.

Figure 1: (a) A three-domain, 14-cavity Si photonic lattice. The four color
center qudits are represented as blue spheres. (b) Close-up of a color center
qubit in a cavity (highlighted).

Topological phases of matter are an interesting quantum platform because of
their robustness against some kinds of disorder. This protection helps with
the survival of quantum properties for longer times, the main limiting factor
in quantum computing applications. One of the applications topological matter
finds in quantum technologies is the use of 1D symmetry-protected edge states
to transfer information across the material Bello  _et al._ (2016, 2017); Lang
and Büchler (2017); Longhi  _et al._ (2019); Longhi (2019); Yuce (2019);
Pérez-González  _et al._ (2019); D’Angelis  _et al._ (2020). This transfer can
take place along several regions with different topological characteristics,
that is, along multiple domains. Recently, it was shown that a multidomain
approach drastically accelerates transfer Zurita  _et al._ (2023), which
greatly increases the technological potential of this phenomenon. In that
regard, photonic lattices are experiencing an outstanding growth in the number
and complexity of topological models they can implement Malkova  _et al._
(2009); Wang  _et al._ (2009); Hafezi  _et al._ (2013); Weimann  _et al._
(2017); St-Jean  _et al._ (2017); Zhao  _et al._ (2018); Yang  _et al._
(2018); Huda  _et al._ (2020); Zurita  _et al._ (2020); Gong  _et al._ (2021);
Pernet  _et al._ (2022); Rosiek  _et al._ (2023); Kim  _et al._ (2022) and use
in applications like lasing Bandres  _et al._ (2018); Contractor  _et al._
(2022), sensing Parto  _et al._ (2023) or, as mentioned, quantum information
processing Blanco-Redondo  _et al._ (2018); Hu  _et al._ (2020); Tschernig
_et al._ (2021); Dai  _et al._ (2022). In particular, multidomain SSH-like
models have been implemented experimentally in several platforms Blanco-
Redondo  _et al._ (2016); Zhao  _et al._ (2018); Huda  _et al._ (2020); Pernet
_et al._ (2022).

Another reason to turn our attention to photonics is their ability to couple
to emitters, forming highly scalable platforms that are created using advanced
fabrication techniques Khoury and Abbarchi (2022). Recent breakthroughs have
been achieved like the production of high-brightness G-centers and the control
of their position Hollenbach  _et al._ (2020, 2022), the coupling of G-centers
to a photonic cavity in a single-photon basis Baron  _et al._ (2022); Redjem
_et al._ (2023), or the creation of vacancies or W-, T- and G-centers coupled
to waveguides Tait  _et al._ (2020); MacQuarrie  _et al._ (2021); DeAbreu  _et
al._ (2022); Prabhu  _et al._ (2023); Lukin  _et al._ (2023).

The advancements cited so far have led to theoretical works about whether
these emitters can be used as qubits of a quantum computer, coupled to each
other through the edge states of a photonic topological system Lang and
Büchler (2017); Yan  _et al._ (2021); Hollenbach  _et al._ (2020); Guimbao
_et al._ (2021). In this work we address this question, that is, we study
whether coherent operation of multi-qubit states is possible. Furthermore, we
propose protocols for coherent operation that scale well with the number of
qubits. To show this we propose a nearly one-dimensional system and prove that
it produces entanglement between any pair or bigger subset of qubits in
experimentally feasible settings, proving that coherent processes mediated by
the edge states overcome dissipation and decoherence.

The structure of this work is as follows: in Section II, we introduce the
system, consisting of a quasi-1D topological photonic lattice, with qubits
placed at its interfaces. We also propose an experimental implementation for
it, using color center qubits in a silicon cavity lattice. Next, we
demonstrate in Section III how the topological transfer of a photon through
the lattice can be used to prepare a Bell state between arbitrary pairs of
these qubits. To quantify the performance of the protocol we study the
concurrence between the target qubits in the presence of different kinds of
disorder. We then use similar topological transfers in Section IV to prepare
the W and GHZ states between three qubits and evaluate the performance of the
protocol using genuine multipartite entanglement witnesses in the same
parameter regimes as with Bell states. Finally, we state our conclusions about
the usability and scalability of this system as a quantum computing platform,
together with possible future lines of research.

##  II Description of the system

We consider a cavity lattice made up of several topological regions, with a
number of qubits coupled to the boundaries between them. The full system will
be described by:

$\mathcal{H}_{T}=\mathcal{H}_{\gamma}+\mathcal{H}_{Q}+\mathcal{H}_{Q\gamma},$
(1)

where $\mathcal{H}_{\gamma}$ describes the topological photonic lattice,
$\mathcal{H}_{Q}$ describes the qubits and $\mathcal{H}_{Q\gamma}$ governs the
interactions between them.

###  II.1 Topological photonic lattice

We first focus on the topological lattice. The simplest topological insulator,
the SSH chain, consists of a chain of atoms with alternating tunnelling
amplitudes $v$ and $w$. Its end states are protected against disorder in said
amplitudes as long as the intracell links $v$ are weaker than the intercell
ones $w$. We consider a lattice of photonic cavities made up of several SSH-
like domains, where each new domain is coupled to the penultimate site in the
previous one, creating domain walls with additional sites (or stubs) coupled
to them [see Fig. 1 (a)]. We will refer to this lattice as the stub-SSH (SSSH)
model. As in the SSH chain, each unit cell has two sites, $A$ and $B$, that we
label with $\alpha$. They take on alternating roles in each domain. The
Hamiltonian describing the model is:

$\displaystyle\mathcal{H}_{\gamma}=-u_{0}(b_{1}^{\dagger}a_{1}+a_{1}^{\dagger}b_{1})-\sum_{k=1}^{N}\mathcal{H}_{\gamma}^{(k)},$
(2)

where $\mathcal{H}_{\gamma}^{(k)}$ is the Hamiltonian belonging to the $k$-th
domain. It takes a different form for odd and even values of $k$. For odd $k$,
it is given by:

$\displaystyle\mathcal{H}_{\gamma}^{(k)}=\sum_{j=1}^{\ell/2}wa_{j_{0}(k)+j}^{\dagger}b_{j_{0}(k)+j-1}+$
(3) $\displaystyle+\sum_{j=1}^{\ell/2-1}v_{k}$ $\displaystyle
b_{j_{0}(k)+j}^{\dagger}a_{j_{0}(k)+j}+u_{k}b_{j_{0}(k)+\ell/2}^{\dagger}a_{j_{0}(k)+\ell/2}+h.c.,$

while for even $k$, the $a$ and $b$ operators are swapped:

$\displaystyle\mathcal{H}_{\gamma}^{(k)}=\sum_{j=1}^{\ell/2}wb_{j_{0}(k)+j}^{\dagger}a_{j_{0}(k)+j-1}+$
(4) $\displaystyle+\sum_{j=1}^{\ell/2-1}v_{k}$ $\displaystyle
a_{j_{0}(k)+j}^{\dagger}b_{j_{0}(k)+j}+u_{k}a_{j_{0}(k)+\ell/2}^{\dagger}b_{j_{0}(k)+\ell/2}+h.c.$

In the expressions above, $j_{0}(k)=k\ell/2+1$ is the unit cell of the $k$-th
domain wall and operator $a_{j}^{\dagger}$ $(b_{j}^{\dagger})$ creates a
photon in the cavity in unit cell $j$ and sublattice $\alpha=A(B)$. The
$v_{k}$ and $w$ hopping amplitudes form the main SSH domains, while the
$u_{k}$ amplitudes connect the cavities at the ends and stubs to the rest. The
lattice is determined by its number of SSH-like domains $N$ and their domain
length $\ell$. An additional cavity is coupled to each domain wall, such that
each domain $k=1,\ldots,N$ has $\ell$ inner sites and is bounded by two
extremal sites. We consider even values of $\ell$. The total number of
cavities is $N_{cav}=N\ell+2$.

###  II.2 Coupled qubits

Qubits can be coupled to any number of extremal cavities, as shown in Fig. 1
(a,b), and we will label them starting at 0. They will act like photon
emitters with the frequency of the cavity. In one of the protocols, we will
extend the qubits to qutrits, so we keep the notation general. The coupling
between the qubits and the cavities housing them is, in the rotating wave
approximation:

$\mathcal{H}_{Q\gamma}=\sum_{k=0}^{p-1}g_{k}c_{j(k),\alpha(k)}|s_{u}\rangle\langle
s_{l}|+h.c.,$ (5)

where $j(k),\alpha(k)$ labels the cavity where the $k$-th qudit is located,
$c_{j(k),\alpha(k)}$ destroys a photon in that cavity, $p$ is the number of
qudits, and $|s_{l}\rangle$ and $|s_{u}\rangle$ are the lower and upper qudit
states connected by the cavity-mediated transition, with
$s_{l,u}=0,\ldots,d-1$, where $d$ is the dimension of the qudits used.

The odd domains in this model are SSH chains, while the even ones can be
obtained as a limiting case of the extended SSH model studied in Pérez-
González  _et al._ (2019). The use of domains with topological invariants of
$\pm 1$ (see Appendix A) provides domain walls with two states each, one of
them can be used to interface with a qubit while the other is used to transfer
information along the wall. This allows us to create a 1D network of qubits
with full connectivity. A system with $N$ domains then has $2N$ boundary
states, that are topologically protected against off-diagonal disorder as long
as $w$ is greater than all $v$-type (intra-domain) and $u$-type (stub) links,
as shown in Fig. 2 (a,b). Some of these states are mainly localized in a
single cavity and can thus be easily coupled to a qubit (the left and right
states $|\mathcal{L}\rangle$, $|\mathcal{R}\rangle$ and the
$|\mathcal{S}_{k}\rangle$ states inside the domain walls), while others
($|\mathcal{P}_{k}\rangle$) can be used to transfer information along a wall.
These last states disappear if we set $v_{k}>w$ in one of their contiguous
domains. We show this, as well as the extent of the topological protection, in
Appendix A.

All of the edge states that are not decoupled from the rest of the system
(e.g. by setting their $u_{k}=0$) will hybridize, forming bonding and
antibonding eigenstates. The topological protection ensures that they will
stay decoupled from all bulk states. The effective overlap between each pair
of states depends on the links between them, given that these dictate their
decay length.

Since the effective overlap between each pair of states depends on the links
between them, a transfer protocol between the two ends of all the hybridized
states can be designed by tuning some of the hopping amplitudes. Thus, by
tuning the weaker hopping amplitudes, marked in orange in Fig. 2 (a), the
dynamics in this subspace can be controlled. Using the right pulses, a photon
initially on the leftmost state of the set of hybridized states can always be
made to end up on the rightmost one, as explored in Zurita  _et al._ (2023).
We show the pulses we used in Appendix B. This transfer process, which we will
refer to as $T$, is always topologically protected and can be established
between any two domain boundaries.

These pulses are also used to completely localize the initial and final states
of the transfer in a single cavity, in order for the qubit-end mode coupling
to be perfect. If no parameters could be tuned at all, end-to-end transfers
would still be possible, although their fidelity would be smaller because the
end modes would not coincide exactly with the end cavity states.

Apart from connectivity, there is an additional advantage to using domains. As
discussed in the literature Lang and Büchler (2017); Zurita  _et al._ (2023),
a downside of bidirectional transfer protocols in 1D topological insulators is
the exponential scaling of their duration as a function of distance, which in
turn allows small symmetry-breaking perturbations to become catastrophic for
the performance of the protocol. This is caused by their exponential decay
into the bulk, which determines their effective overlap. This problem can be
solved by adding intermediate domain walls between the ends of the model, as
shown in Zurita  _et al._ (2023). This makes the transfer duration grow only
linearly with distance.

###  II.3 Experimental proposal

The cavity lattice can be built using a triangular lattice of holes in a slab
of silicon, where the absence of holes creates a cavity, as shown in Fig. 1
(a). The hopping amplitude between contiguous cavities can be modified by
changing the medium between cavities, by driving the cavity frequencies Liao
_et al._ (2010) or using ancilliary cavities (either containing impurities Qin
and Nori (2016) or with high frequencies Liao  _et al._ (2010); Mukherjee  _et
al._ (2018)), or even by implementing nanowire-induced movable cavities, as
shown experimentally in Birowosuto  _et al._ (2014). As we explain below, the
ability to tune these amplitudes is not necessary to implement the two-qubit
protocol, although it does increase the resulting fidelity.

The qubits can be impurities in the silicon like G- or T-centers, located
within the appropriate cavities, as shown in Fig. 1 (b). The qudit-cavity
coupling parameters, $g_{k}$, could be modified by tuning the energies of the
impurity with an electrical gate in or out of resonance with the cavity
Hollenbach  _et al._ (2020), by tuning the cavity frequency as mentioned
above, or otherwise by using an indirect transition with an auxilliary laser
Lang and Büchler (2017).

As mentioned in Hollenbach  _et al._ (2020); Redjem  _et al._ (2023),
controlling the exact placement and orientation of the impurities can pose a
challenge, although advances are being made with controllable growth
techniques Hollenbach  _et al._ (2020, 2022); Redjem  _et al._ (2023). A
solution could be to create the cavity lattice around the impurities after
they are placed Redjem  _et al._ (2023). The topological robustness against
off-diagonal disorder in our model greatly dampens the effects of the
fabrication defects that these systems may have, especially the distances
between cavities.

Figure 2: (a) Photonic cavity lattice with $N=4$ domains and $\ell=4$ cavities
per domain. Qubits (in green) are coupled to the cavities at the ends and
domain walls, with a coupling amplitude of $g_{k}$. Hopping amplitudes used as
control parameters are shown in orange: we use $u$ for extremal links, $v$ for
all the rest. (b) Topologically protected states in this system. Two chiral
states, $\mathcal{S}$ and $\mathcal{P}$, are located at each wall. States in
odd (even) boundaries are completely located in sublattice $B$ ($A$), and
shown in blue (yellow). (c) Transfer process between the left end and the
first wall. The parameter $v_{\mathrm{tr}}<w$ allows the transfer, while
$v_{\mathrm{bar}}\gg w$ stops it. (d) Transfer process between the two ends.
The $\mathcal{P}$ states are involved now. Two different pulses are used for
the extremal ($v_{\mathrm{tr}}^{(1)}$) and central ($v_{\mathrm{tr}}^{(2)}$)
domains to achieve a high-fidelity transfer.

##  III Bell state preparation

###  III.1 Preparation protocol

In this section, we use this photonic multidomain topological system to
prepare maximally entangled states on two qubits, i.e. Bell states. In order
to explain the protocol, we need to define some notation. Let $n_{d}$ be the
total number of domains and $\ell$ the number of sites per domain. As
explained above, qubits sit on the stubs between domains. Let $R$ and $L$
stand for the rightmost and leftmost qubits, which are the furthest apart and
so the hardest to entangle. Therefore, we describe the protocol for these
qubits only, given that the protocol for any other pair is analogous. The
ground and excited states of qubit $L$ are $\ket{0}_{L}$ and $\ket{1}_{L}$,
repectively, and we will refer to its coupling to the photonic cavity stub
here as $g_{L}$. Furthermore, since coherent single-photon emission has been
reported experimentally for color-center qubits Baron  _et al._ (2022); Redjem
_et al._ (2023), we consider $\pi$\- and $\pi/2$-pulses to be an accesible
operation between qubit $L$ and its cavity 111We use sine-shaped pulses for
all $g$ parameters, see Appendix B for details.. We label the latter unitary
as $(\Pi/2)_{01}^{(L)}$. We use similar notation for qubit $R$. The vacuum
state of the photonic lattice is $\ket{\varnothing}_{\gamma}$. The topological
transfer of a photonic excitation from the leftmost cavity to the rightmost
through the edge states is $T_{LR}$.

With this notation now fixed, we describe the protocol that prepares a Bell
state from a separable state as shown in Fig. 3 (a): (i) Prepare the initial
state $\ket{1}_{L}\otimes\ket{0}_{R}\otimes\ket{\varnothing}_{\gamma}$ (ii)
Execute a $(\Pi/2)_{01}^{(L)}$ so that the excited qubit emits half its
probability amplitude to the photonic lattice. Note it does so with a $-i$
relative phase (iii) Produce a full transfer $T_{LR}$ so the photonic
amplitude is now at the rightmost cavity. Note that the relative phase changes
again. (iv) Execute a second $\pi$-pulse on $g_{R}$, that is,
$\Pi_{01}^{(R)}$. The final qubit state will be

$\displaystyle\ket{\Phi}_{LR}=\frac{1}{\sqrt{2}}\left[\ket{10}+e^{i\theta}\ket{01}\right]$
(6)

where the relative phase is

$e^{i\theta}=[(-1)^{\ell/2+1}i]^{n_{d}},$ (7)

See Appendix C for a detailed derivation of this relative phase 222This
formula is related to those given in Zurita  _et al._ (2023) for the SSH
chain, where $\ell$ was defined differently and $n_{d}=n_{w}+1$..

###  III.2 Performance against disorder: concurrence

While the calculations reported so far demonstrate that this system generates
entanglement in the ideal case, we need to explore the impact of disorder if
we are to consider the system a scalable quantum platform. First of all, the
final state will no longer be Eq. 6. Moreover, the final state has a nonzero
probability amplitude of having a leftover photon in the photonic lattice,
leading to decoherence in the state of the qubits. In this case, entanglement
certification is not as simple as inspecting the state in Eq. 6, so instead we
turn to concurrence, an entanglement measure Hill and Wootters (1997), to
perform this task. We remind the reader that, if a two-qubit state has high
concurrence, any latter state with lower concurrence may be prepared from the
former with only local operations and classical communication (LOCC); in
particular, without using the photonic lattice. In this sense, Bell states
have maximal concurrence. We now turn our attention to the disorder model
used, and then show that the protocol can produce high-concurrence states
despite disorder.

We consider quasistatic disorder, that is, constant in time for each
simulation, on each matrix element of the tight-binding Hamiltonian:

$\mathcal{H}_{j,\alpha;j^{\prime},\alpha^{\prime}}=\mathcal{H}_{j,\alpha;j^{\prime},\alpha^{\prime}}^{(0)}+h_{j,\alpha;j^{\prime},\alpha^{\prime}}^{(\sigma)},$
(8)

where each $h_{j,\alpha;j^{\prime},\alpha^{\prime}}^{(\sigma)}$ is an
independent random variable sampled once per realization with a normal
probability distribution centered at zero and with a standard deviation of
$\sigma$, which corresponds to the intensity of disorder. We tune the diagonal
and off-diagonal levels of disorder independently to showcase the topological
protection of our system towards off-diagonal (OD) symmetry-preserving
disorder as well as to check the impact of general (G) symmetry-breaking
noise.

There is a second feature of the system with regards to noise protection. In
Zurita  _et al._ (2023) it was shown that the use of topological domains
drastically accelerates transfer protocols—thus reducing the effects of
disorder—and increases connectivity. To clarify this quantitatively, we
consider two different kinds of simulations: a single, long SSH chain between
the qubits with 14 cavities in a line, and the four-domain lattice shown in
Fig. 2 (a), with 15 cavities between the qubits. The transfer time goes from
$t_{\mathrm{tr}}=304.0/w$ for one domain (with a preparation time of
$t_{\mathrm{prep}}=25/w$) to $t_{\mathrm{tr}}=45.3/w$ for four domains (with
$t_{\mathrm{prep}}=20/w$). This speed-up, which is exponential in the length
of the chain, is crucial in reducing the effects of diagonal disorder on
entanglement production.

We show the results in Fig. 3 (b). In the case of off-diagonal disorder (OD),
topological protection helps keep the concurrence greater than $0.7$ even at
$\sigma=0.1$ (a disorder of $10\%$ with respect to $w$). The single- and four-
domain cases perform very similarly, with the single-domain case being
slightly better, given that more edge states participate in the transfer in
the four-domain case. However, when general disorder is considered, the four-
domain case performs much better than the single-domain one. With the
protecting symmetry broken, the deciding factor is now the speed of the
protocol, and the presence of domain walls allows it to be over in a much
shorter time. For more details on the kind and importance of the observed
errors see Appendix D.

We have considered no qubits in the domain walls for this protocol. If we
included them and wanted to create entanglement between two of them, only the
transfer protocol would have to be modified as detailed above.

Figure 3: (a) Preparation protocols for the $\Phi$-type Bell states. Qudit
states are represented as blue (0) or yellow (1). Photonic cavities are
represented as thin pink lines, with an additional shading if a photon is
present. Superpositions are represented with dotted lines. (b) Average
concurrence and its standard deviation over 1000 realizations of the Bell
states as a function of off-diagonal (OD) and general (G) disorder. We show
the single-domain case (blue) and the four-domain case (red).

To prepare the other kind of Bell state,
$|\Psi\rangle=(|00\rangle+\xi|11\rangle)/\sqrt{2}$, the same protocol can be
used, with an additional $X$ gate acting on the second qubit at the end. We
also explore another possibility, using qutrits, in Appendix E.

##  IV Multipartite entanglement generation

The full connectivity of the topological photonic lattice opens the door to
other protocols that, instead of entangling arbitrary pairs of qubits,
entangle triplets or larger subsets of qubits. This approach displays the
benefits of this nearly one-dimensional platform when it comes to scaling the
number of qubits. To demonstrate this, we design protocols that prepare
multipartite entangled states. Unlike in bipartite entanglement, there is no
unique notion of maximally entangled multi-qubit states that, once generated
with the photonic lattice, allows for the generation of any other state
without the use of the photonic lattice. Instead, there are multiple families
of states, each of them with a maximal state. Thus, we propose protocols for
the generation of these maximal states for different families Gour and Wallach
(2013). The simplest of these can be regarded as the W and GHZ states for
three qubits Dür  _et al._ (2000).

###  IV.1 W state preparation

We propose a protocol for the preparation of the W state that is similar to
the previous one preparing the $\Phi$ state in Eq. 6, but with extended
notation regarding the third qubit, which we label $C$ after its centered
position somewhere between the leftmost and rightmost qubits. The setup is
shown in Fig. 4 (a). With this extension in notation, the protocol is as
follows (i) Start in state
$\ket{1}_{L}\otimes\ket{0}_{C}\otimes\ket{0}_{R}\otimes\ket{\varnothing}_{\gamma}$.
(ii) Implement a $2\pi/3$-pulse on the left qubit-cavity coupling parameter
$g_{L}$, so that the cavity now has a probability of $2/3$ of having a photon.
(iii) Perform a topological transfer to the cavity $C$ where qubit $C$ is
found. (iv) Execute a $\pi/2$-pulse on qubit $C$, resulting in a qubit
excitation probability of one third. Its cavity now has a $1/3$ probability of
having a photon. (v) Implement a topological transfer from cavity $C$ to
cavity $R$. (vi) Perform a final $\pi$-pulse on qubit $R$. Then, the final
state of the qubits is:

$\displaystyle\ket{\textrm{W}}_{LCR}=\frac{1}{\sqrt{3}}\left[\ket{100}-i\ket{010}+\ket{001}\right]$
(9)

after taking into account all the relative phases produced by each step of the
protocol.

###  IV.2 GHZ state preparation

To prepare the GHZ state, we need qutrits instead of qubits. We now suppose
the gap between states $|1\rangle$ and $|2\rangle$, $\Delta\epsilon_{12}$, can
be tuned to the frequency of the cavity. State $|2\rangle$ of each qutrit is
used as an auxilliary state and is not present in the final result. The
protocol is as follows: (i) Start in state
$(|011\rangle+\eta|211\rangle)/\sqrt{2}$ (with $|\eta|=1$), which has no
entanglement, and an empty photonic lattice. (ii) Implement a $\pi$-pulse on
the left qutrit. (iii) Perform a full transfer along the first domain. (iv)
Implement a $\pi$-pulse on the second qutrit. (v) Apply an $X$ gate to that
qutrit333We implement operations $X$ and $\mathcal{A}$ using a sine-shaped
pulse on the appropriate elements of the qudit hamiltonian: the off-diagonal
elements that flip the qutrit states and the diagonal elements that fix the
ensuing phase shift. Due to the method used, a global phase of $-i$ is
acquired, in addition to the implemented operation (e.g. to do an $X$
operation on the left qubit we actually implement
$-i(X_{L}\otimes\mathbbm{1}_{R})$).. (vi) Repeat operations (ii-v) along the
second domain: do a $\pi$-pulse on the second qutrit, realize a transfer along
the second domain and an $X$ gate on the third qutrit. (vii) Finish the
protocol by applying operation $\mathcal{A}=|1\rangle\langle 2|+h.c.$ to the
last qutrit [see Fig. 4 (a.2)]. The final state is:

$i|\textrm{GHZ}\rangle=i(|000\rangle+\eta\zeta^{2}|111\rangle)/\sqrt{2}$ (10)

In our case, $\zeta=i$, so we choose $\eta=-1$ to get no relative phase.

###  IV.3 Performance against disorder: entanglement witnesses

Just as with the bipartite case above, the calculations provided so far show
that the system produces the maximally entangled states of the two different
families of tripartite entanglement in the absence of disorder. To extend this
claim to experimental settings, we remind the reader that certifying the
presence of entanglement becomes more sophisticated than simple inspection of
the state. Since disorder might leave some photonic probability amplitude in
the domain walls, as opposed to vacuum as in the pristine case, the three-
qubit reduced state is no longer pure. Then, we need to define what we mean by
tripartite entanglement. In the pure state setting this is straightforward: a
state is tripartitely entangled iff there is no separable subsystem; that is,
$\psi\neq\phi_{i}\otimes\phi_{j,k}$ for any $i,j,k$, where the subindices
range without repetition over the three subsystems. But in the mixed state
case special care must be taken into account to rule out possibilities like

$\displaystyle\rho_{inseparable}=P_{1}\rho_{1}\otimes\rho_{23}+P_{2}\rho_{2}\otimes\rho_{13}+P_{3}\rho_{3}\otimes\rho_{12}$
(11)

where the entanglement, if any, is of bipartite nature, but our information
about the system is not enough to know between which qubit pair, nor which is
the separable one. The subindices of each density matrix $\rho$ indicate the
subsystems they span. Whenever a density matrix cannot be expressed as in Eq.
11, or as incoherent mixtures of these matrices, we claim that its tripartite
entanglement is genuine. For the sake of brevity, we use witnesses instead of
entanglement measures to make assesments about the presence of entanglement as
a function of disorder. Witnesses are functions of the state that, in our
case, report the presence of genuine entanglement whenever they are positive.
If they are non-positive the state might be entangled, but these quantities do
not witness it in those particular settings, hence the name. In other words,
witnesses are typically built to be sensitive to a particular state and its
surroundings. This is the case for the two witnesses that we consider. The
first one is tailored to detect the W state in Eq. 9 and is

$\displaystyle\mathcal{W}_{\mathrm{W}}=-\frac{2}{3}+\textrm{tr}(\rho\Pi_{\textrm{W}}),$
(12)

where $\rho$ is the final density matrix for one realization of the protocol
and $\Pi_{\textrm{W}}$ is the projector on state $|\textrm{W}\rangle_{LCR}$.
Moreover, we consider another witness, tailored to the GHZ state with no
relative phase in Eq. 10:

$\displaystyle\mathcal{W}_{\textrm{GHZ}}=-\frac{3}{4}+\textrm{tr}(\rho\Pi_{\textrm{GHZ}}),$
(13)

where $\Pi_{\textrm{GHZ}}$ is the projector on that state. For the derivation
of these witnesses, see Acίn et al. Acín  _et al._ (2001).

Figure 4: (a) Preparation protocols for (a.1) W and (a.2) GHZ states for 3
qudits: $L,C,R$. Qudit states are represented as blue (0), yellow (1) or red
(2). Photonic cavities are represented as thin pink lines, with an additional
shading if a photon is present. Superpositions are represented with dotted
lines. (b) Average values over 1000 realizations of the witnesses in Eq. 12
and Eq. 13 as a function of off-diagonal (blue) and general (red) disorder.

The average values of these witnesses as a function of disorder are presented
in Fig. 4 (b). From the data we conclude that topological protection plays a
fundamental role in the generation of genuine entanglement, since the
witnesses remain positive for higher values of symmetry-preserving disorder
(blue) than when general, symmetry-breaking disorder is present (red).
Nevertheless, the requirements placed on the system’s disorder are higher with
multipartite than with bipartite entanglement, since the former is a more
valuable resource than the latter. In the bipartite case, some entanglement is
still present when the disorder takes values as high as $0.1w$, whereas here
we need a disorder smaller than $0.05w$ to detect multipartite entanglement.
However, we must point out that, while concurrence is a measure for two-qubit
entanglement, the witnesses used for three-qubit entanglement are not
measures. Therefore, it could be the case that multipartite entanglement is
still present for disorder higher than $0.05w$ and further developments in the
theory of genuine entanglement certification Friis  _et al._ (2018) could
detect them. We include the fidelity results for both protocols in Appendix D,
and discuss the most common errors observed that harm performance.

On a similar note, we remind the reader that this multipartite entanglement
can be produced on demand between any three qubits in the scheme of Fig. 2.
Additionally, the topological transfer considered here could entangle more
than three qubits in a genuine way, leading to further technological
applications while paving the way to become a quantum computing platform. The
modifications needed to prepare generalized W and GHZ states for any number of
qubits are straightforward, and we discuss them in Appendix F.

##  V Conclusions

In this work, we propose a realistic experimental setting in which fast
multidomain topological protocols can be used to quickly and reliably generate
multipartite entanglement. We consider an easily scalable photonic lattice in
silicon, with coupled qudits implemented natively with impurities, but our
protocols could also be considered in other systems like superconducting or
spin qubits, both of which are being investigated as platforms for photon-
mediated entanglement generation Yang  _et al._ (2016); Chan  _et al._ (2023).

We show how topological protection can help overcome some of the main issues
associated with these setups, and we benchmark the effectiveness of that
protection by studying the behaviour of entanglement measures and witnesses
for increasing disorder levels. The versatility of the proposed setup and its
effectiveness in entanglement distribution opens the door for quantum
information applications like the generation of highly-entangled states, such
as cluster states, to use as a resource for quantum computation. This
constitutes an intriguing avenue for future research. Finally, the
universality of operations over the protected computational states in the
photonic lattice can allow an array of possible quantum operations, which will
be explored further in future works.

###### Acknowledgements.

C.E.C. was supported by the Spanish MICINN through Grant No.
PID2022-139288NB-100. G.P., A.A.C. and J.Z. were supported by Spain’s MINECO
through Grant No. PID2020-117787GB-I00 and by CSIC Research Platform PTI-001.
We also acknowledge support from National Project QTP2021-03-002. J.Z.
recognizes the FPU program FPU19/03575.

##  Appendix A Topological protection

In order to prove the topological protection of the model analytically, we
first look at the case where all $v$ terms are nonzero but smaller than $w$.
As noted recently in Dias and Marques (2022), site indexing plays a vital role
in the topological description of 1D models. By choosing the appropriate
indexing [which we also use in Eq. (2)], we can conceptualize the SSSH lattice
as a series of topological domains with alternating winding numbers of
$\nu=\pm 1$. We show this in Fig. 7 (a) for a 5-domain model. The odd-numbered
domains are regular SSH chains with a winding number of $1$, while the even
domains have a winding number of $-1$. Given that each site connects to the
contiguous unit cell that is further from it, instead of closer like the SSH
chain, the bulk Hamiltonian $\mathcal{H}(k)$ is identical to the SSH chain but
with $k\to-k$. That reverses the path traced by $\mathcal{H}(k)$ in the space
of possible Hamiltonians, changing the sign of its winding number, as shown in
Fig. 7 (b). The model we use here is completely equivalent to the one in the
main text, as shown in Fig. 7 (c). The latter does not use long-range hopping
terms.

The difference in topological invariant between the two sides of a boundary
dictates the number and chirality of protected states. The ends of the system
will have one protected state ($\mathcal{L}$ and $\mathcal{R}$), and each wall
will have two ($\mathcal{S}$ and $\mathcal{P}$), see Fig. 2 (b). The
chiralities (i.e. the sublattice that supports the states) alternate between
successive walls. These states will be protected as long as the chiral
symmetry is preserved, and so they can be used to store information
($\mathcal{L},\mathcal{R}$ and $\mathcal{S}$) or help in the transfer
($\mathcal{P}$). This is the general case.

Another possible case, that we use to prepare the 3-qubit states, is that one
of the domains is made trivial due to a high value of $v=v_{\mathrm{bar}}\gg
w$. In that case, the interface between that domain and its topological
neighbour will only have one protected state ($\mathcal{S}$), see Fig. 2 (c).
The transfer process will not continue into the trivial domain, which acts
like a sort of barrier.

During a multidomain transfer, the intermediate qubit cavities are left
unconnected, as shown in Fig. 2 (d). This case can be seen as a multidomain
SSH chain, the topology of which has been discussed in the literature at
length Su  _et al._ (1980); Munoz  _et al._ (2018); Qi  _et al._ (2021);
Zurita  _et al._ (2023). The topological protection is also guaranteed under
this point of view.

Having proven the presence of topological protection using the bulk-boundary
correspondence, we now stop to closely examine the meaning of that protection
in each of our protocols. To that end, we study numerically the behaviour of
the spectrum of the photonic lattice with respect to the level and type of
disorder in the system.

In each realization, the presence of disorder will change the eigenvalues of
the system randomly, and this effect will increase with the disorder strength
$\sigma$. If a semi-infinite SSH chain was considered, the topological
protection would imply that the only end state, which would be its own chiral
partner, would be pinned at zero as long as the chiral symmetry is preserved.

In a long, single-domain SSH model, such as the 14-cavity dimerized chain used
in this work, the two end modes hybridize and acquire a small positive and
negative energy. The two energy eigenstates are now each other’s chiral
partner, and so they have opposite energies in the presence of chiral
symmetry. This situation can be seen in Fig. 5 (a).

This small energy, however, is the cause of the exponentially long durations
achieved with single-domain chains Zurita  _et al._ (2023). To fight this, we
increase the number of domains, making them closer to each other. As a
consequence, their hybridization energies are dramatically larger, as shown in
Fig. 5 (c) for the four-domain, 15-cavity SSH model corresponding to the
connected sites in Fig. 2 (d). They are still clearly separated from the bulk,
but their energy fluctuations are larger than in the single-domain case.

To study the amplitude of these fluctuations, we calculate the standard
deviation of the energy of each photonic lattice eigenstate,
$\sigma_{\epsilon}$. We group the states in different realizations by their
order in energy, and we can differentiate between topological and bulk states
by calculating the probability $p_{e}$ of finding the particle at the
topological end modes444We use the completely dimerized form of the
topological modes for simplicity to find $p_{e}$, but this is enough to
clearly differentiate between bulk and edge states..

We show the results in Fig. 6 for the single-domain, 14-site model (a,b) and
the four-domain, 15-site one (c,d), for both kinds of disorder and $10^{4}$
realizations. Each of the realizations has a fixed disorder profile, scaled to
match $\sigma$. As expected, all edge states show a smaller
$\sigma_{\epsilon}$ than the bulk states for off-diagonal disorder, although
the single-domain case fluctuates much less, as expected. In the general
disorder case, however, the single-domain model end modes inmediately lose all
protection, while the four-domain model states continue being the more
constant ones. This shows that, for a fixed length, the topological protection
of a multidomain model is weaker than the single-domain case but, in the more
realistic case of general disorder, can resist for larger disorder levels.

The other hallmark of topological protection, bulk-boundary decoupling, is
studied in depth in Appendix D, where we break down the results of all
disorder simulations.

Figure 5: Spectrum of the single- (a,b) and four-domain (c,d) models over a
single realization, for increasing values of both off-diagonal (a,c) and
general disorder (b,d). The color of each point represents an approximate
probability of finding it in a topological state (see main text), $p_{e}$.
Topological edge states are shown in red, and trivial states in blue. Figure
6: Standard deviation of each state in the spectrum of the single- (a,b) and
four-domain (c,d) models over $10^{4}$ realizations, for increasing values of
both off-diagonal (a,c) and general disorder (b,d). The color of each point
represents an approximate probability of finding it in a topological state
(see main text), $p_{e}$. Topological edge states are shown in red, and
trivial states in blue.Topological protection in the presence of undisturbed
chiral symmetry is more robust in the single-domain case, but it is also more
fragile when the symmetry is broken. Figure 7: (a) Possible 1D indexing of the
cavities, that allows the domains to be identified as having a winding number
of $\nu=\pm 1$. (b) Path taken in the space of possible Hamiltonians
$\mathcal{H}(k)=\vec{d}\cdot\vec{\sigma}$, where $\vec{\sigma}$ are the Pauli
matrices and $k\in BZ$, for systems $I$ and $II$ above. (c) Under deformation,
the lattice in (a) is equivalent to a three-domain lattice of the type shown
in Fig. 2 (a). Figure 8: Disaggregated probabilities for all disordered
protocols, i.e. the single- and four-domain $\Phi$ and $\Psi$ protocols, and
the W and GHZ ones; both with off-diagonal (OD) and general (G) disorder. The
fidelity $f$, the probability of finding no photon in the cavities
$p(\varnothing_{\gamma})$, and that of finding no photon in the bulk states
$p(\textrm{no bulk})$ are represented in blue, purple and brown lines,
respectively. The purple shaded region shows the probability of a failed final
state with no photon, caused by imperfect qudit-cavity coupling operations.
The orange shaded region shows the probability of a failed final state with a
photon in the topological states, while the white region above $p(\textrm{no
bulk})$ is the probability of leakage into the bulk. The latter is almost
zero, except in the protocols that use trivial domains as barriers (W and
GHZ).

##  Appendix B Details on the control pulses

The pulses used have three parts: an initial preparation time where the
control parameter is switched on slowly during an interval of
$t_{\mathrm{prep}}$, a time where the parameter is kept constant, and a final
interval lasting $t_{\mathrm{prep}}$ where it is switched off slowly. All
pulses used are symmetric, and $t_{\mathrm{prep}}$ is chosen so that
$t_{\mathrm{prep}}\gg w^{-1}$, in order for the process to be adiabatic with
respect to the energy gap between bulk and edge states. The same kind of
pulses, although in different models, were used in Zurita  _et al._ (2023).

If we set $t=0$ at the start of the pulse, the formula for the pulses we use
is:

$v(t)=\begin{cases}v_{0}\sin^{2}(\Omega t)&\textrm{if }0\leq t\leq
t_{\mathrm{prep}}\\\ v_{0}&\textrm{if }t_{\mathrm{prep}}<0\leq
t_{\mathrm{tr}}-t_{\mathrm{prep}}\\\
v_{0}\sin^{2}[\Omega(t-t_{\mathrm{tr}})]&\textrm{if
}t_{\mathrm{tr}}-t_{\mathrm{prep}}<t\leq t_{\mathrm{tr}},\end{cases}$ (14)

where the maximum $v_{0}$ can either be $v_{0}=v_{\mathrm{tr}}<w$, for the
topological domains that allow the transfer, or $v_{0}=v_{\mathrm{bar}}\gg w$,
for the trivial domains that stop the transfer (we will rename
$t_{\mathrm{prep}}$ to $t_{\mathrm{bar}}$ in that case). Both kinds of pulses
are nested, i.e. the barrier is lifted before ramping up $v_{\mathrm{tr}}$,
and it is lowered after switching $v_{\mathrm{tr}}$ off.

The links leading to the cavities with impurities, marked as $u_{k}$ in Fig. 2
(a), will either be switched off or tuned at the same time as a neighboring
domain, depending on whether that qudit is the start or end of the transfer.
This can be seen in detail for several transfer protocols in Fig. 9.

When the number of domains is $n_{d}>2$, the control parameters need to be
different in different domains for the transfer time to be minimal. The height
of the pulses are chosen symmetrically with respect to the center of the
transfer length, and they are found numerically once the $v_{\mathrm{tr}}$
(the height at the first and last domains) is fixed. This is explained in
depth in Zurita  _et al._ (2023).

For the qubit-cavity couplings $g_{k}(t)$ that create the $(n\pi/m)$-pulses,
we use the protocol above, but setting $t_{\mathrm{prep}}$ equal to half the
pulse duration, so there is no constant part. We fix $g_{0}=0.5$ in all cases,
and then find the pulse duration using the formula:

$t_{\mathrm{coup}}=\frac{n\pi}{mg_{0}}.$ (15)

We include below (Table 1) the parameters for all protocols used in this work,
when applicable: domain number $n_{d}$, domain length $\ell$, sites involved
in each transfer $L$, control parameter heights $v_{\mathrm{tr}}$ and
$v_{\mathrm{tr}}^{\prime}$ (for the central domains in the 4-domain case),
barrier height $v_{\mathrm{bar}}$, and the times for preparation
($t_{\mathrm{prep}}$), barrier preparation ($t_{\mathrm{bar}}$) and total
duration ($t_{\mathrm{tr}}$).

States | $n_{d}$ | $\ell$ | $L$ | $v_{\mathrm{tr}}$ | $v_{\mathrm{tr}}^{\prime}$ | $v_{\mathrm{bar}}$ | $t_{\mathrm{prep}}$ | $t_{\mathrm{bar}}$ | $t_{\mathrm{tr}}$
---|---|---|---|---|---|---|---|---|---
Bell 1 dom. | 1 | 12 | 16 | 0.5 | – | – | 25 | 15 | 304.0
Bell 4 doms. | 4 | 4 | 15 | 0.5 | 0.38 | – | 20 | 15 | 45.3
W, GHZ | 1 | 4 | 6 | 0.5 | – | 30 | 7 | 15 | 25.2
Table 1: Parameters for all protocols used, and the states prepared with it.

##  Appendix C Acquired phases and phase-shift protocols

The analytical form of the topological modes involved in a transfer is:

$\displaystyle|\mathcal{L}\rangle$
$\displaystyle=-\mathcal{N}_{\mathcal{L}}\sum_{j=1}^{\ell/2}\left(-\frac{w}{v}\right)^{-j}|j,A\rangle$
(16) $\displaystyle|\mathcal{P}_{k}\rangle$
$\displaystyle=\mathcal{N}_{\mathcal{P}}\left[\sum_{j=j_{0}-\ell/2}^{j_{0}-1}\left(-\frac{w}{v_{l}}\right)^{j-j_{0}+1}|j,\alpha(k)\rangle\right.$
$\displaystyle\left.-\sum_{j=j_{0}+1}^{j_{0}+\ell/2}\left(-\frac{w}{v_{r}}\right)^{j_{0}-j+1}|j,\alpha(k)\rangle\right]$
(17) $\displaystyle|\mathcal{R}\rangle$
$\displaystyle=-\mathcal{N}_{\mathcal{R}}\sum_{j=N_{c}-\ell/2}^{N_{c}}\left(-\frac{w}{v}\right)^{-j+N_{c}+1}|j,\alpha(N)\rangle$
(18)

where $j_{0}=\ell k/2+1$ is the unit cell where domain wall $k$ is located,
$\alpha(k)=A$ ($B$) for even (odd) $k$, $N$ is the number of domains and
$N_{c}$ is the total number of unit cells. In the $\mathcal{P}$ states, we
allow the control parameter on the left ($v_{l}$) and right ($v_{r}$) of the
wall to be different, as they will be in general.

If the initial or final states of the transfer are located in a domain wall
instead of the left or right ends (thus being $\mathcal{S}$ states instead of
$\mathcal{L}$ or $\mathcal{R}$ states), their form is identical except for a
translation to take them to said wall.

Using these and the Hamiltonian in Eq. (2), we obtain that, after a transfer
across $n_{d}$ domains with $\ell$ sites each, the final state will have a
phase factor of:

$\zeta(n_{d},\ell)=-[(-1)^{\ell/2+1}i]^{n_{d}},$ (19)

when compared to the original state.

As mentioned in the main text, shifts in the relative phases of the maximally
entangled states do not affect the amount of entanglement. However, they could
be changed natively if necessary by temporarily adding an on-site potential in
the relevant cavity after all transfers have been made. We use the same pulses
as for the control parameters $v$, fixing the height of the pulse
$\epsilon_{0}$ and the preparation time $t_{\mathrm{prep}}^{(\varphi)}$ and
then finding the total duration as:

$\displaystyle
t_{T}^{(\varphi)}=t_{\mathrm{prep}}^{(\varphi)}-\frac{|\varphi|}{\epsilon_{0}},$
(20)

where $\varphi$ is the desired phase shift. The sign of the pulse is
determined by the sign of $\varphi\in[-\pi,\pi)$. All this ensures the area
under the pulse provides the intended shift.

To show this, we distribute a photon equally between the four boundary
cavities of a 3-domain SSSH model, showing the acquired phases. Then, the
relative phases are dynamically shifted using on-site energies, until we
eliminate all relative phases (see Fig. 9). The final state is then
$(|1,A\rangle+|3,B\rangle+|5,A\rangle+|7,B\rangle)/2$. This phase-shift
operation can be used in any entanglement generation protocol to choose the
relative phases.

##  Appendix D Fidelity and bulk-edge decoupling

In this section, we study the average fidelity of the protocols. For a given
realization, it is obtained as:

$f=\langle\psi_{0}|\rho_{Q}|\psi_{0}\rangle,$ (21)

where $|\psi_{0}\rangle$ is the intended final state, and $\rho_{Q}$ is the
density matrix of the qudit system at the end of the realization, after
tracing out the photonic degrees of freedom. For every protocol considered in
this work, we obtain a fidelity higher than 0.99 for the pristine system,
which is the current error-correction threshold Fowler  _et al._ (2012). We
include the $\Phi$, W and GHZ states prepared as in the main text, and the
$\Psi$ state, prepared as detailed in Appendix E.

In order to get the full picture, we also study two more metrics: the
probability of the photonic lattice ending up empty,
$p(\varnothing_{\gamma})$, and the probability of finding the bulk subspace
completely empty, $p(\textrm{no bulk})$. Of course, $p(\textrm{no
bulk})>p(\varnothing_{\gamma})>f$, because the intended final states have no
photons left in the lattice, and if the lattice is empty, the bulk sites are
too. If we plot the average of these quantities over all 1000 realizations as
a function of disorder for each case, an interesting picture appears.

The average fidelity over 1000 realizations, shown in blue in Fig. 8, is
smaller in the general disorder case due to the absence of topological
protection, but also due to the fact that on-site fluctuations can affect Rabi
flopping protocols, like the ones we use, more than off-diagonal disorder can.
Additionally, when comparing Fig. 3(b) and Fig. 8, it can be observed that
when the fidelity drops down in the general disorder case for the single-
domain protocols, it retains a higher value than the concurrence does. This is
due to the fact that some failed protocols that do not create entanglement at
all can still return a nonzero fidelity. For example, a final qudit state of
$|00\rangle$ will still result in a fidelity of 0.5 when trying to implement
state $|\Psi\rangle$, even though no entanglement is created in the process.

We now consider the average probability of finding the photonic lattice empty
but not ending up at the intended state, which can be calculated as
$p(\varnothing_{\gamma})-f$, and is represented by the height of the purple
sector in Fig. 8. Given the protocols and the kind of disorder we consider
(i.e. only in the photonic lattice), this result can only be caused by errors
in the qudit-photon-coupling operations. It turns out these, which are a form
of Rabi flopping themselves, are much more affected by on-site fluctuations
than off-diagonal disorder, and so this sector increases greatly when
considering general disorder with respect to off-diagonal disorder. An
interesting research direction to follow would be to look for ways to protect
these operations against on-site disorder, using topology or other methods.

Finally, the orange sector of the figures represent the average probability of
finding a photon in the lattice at the end of the protocol, but only with
support on the topological state manifold. It has been obtained as:

$p(\textrm{edge})=(\sum_{k}\bra{k}\rho_{ph}\ket{k})^{2},$ (22)

where $\rho_{ph}$ is the density matrix of the photonic degrees of freedom.
The probability $p(\textrm{no bulk})$ is then found as $p(\textrm{no
bulk})=p(\varnothing_{\gamma})+p(\textrm{edge})$. We can see that it is
extremely close to 1, even in the presence of general disorder. This shows
that the bulk-edge decoupling caused by the topology of the system is
remarkably robust. This has been reported for other topological systems in the
literature Lang and Büchler (2017). The three-qudit cases show a slightly
smaller decoupling due to the use of trivial domains as barriers, which can
allow some photon amplitude to leak into the bulk. This problem could be
mitigated by decoupling the domains altogether by switching off some $w$
amplitudes, but we decided to keep them constant throughout in this work for
simplicity.

Finally, we want to acknowledge the existence of trivial boundary states in
the domain walls (in the fully dimerized case, for example, they are located
in the 3-site strong-strong defects, and spanned by the second site in them
and the 0-phase superposition of the first and third sites in them). These
states could, in principle, be a hindrance for the robustness of the
protocols, but their effect was shown to be negligible in all two-qubit
protocols, and extremely small in the three-qubit ones (with a probability of
less than 0.02 of finding a photon in those states with a general disorder of
0.1).

Figure 9: Partial transfer processes and phase shifts. (a) The photonic
lattice considered, a three-domain SSSH chain. (b) The legend used in (c),
with color labeling the magnitude and phase of the photon wavefunction at each
cavity. (c) Transfer processes needed to obtain
$|\psi\rangle_{\gamma}=(|1,A\rangle+|3,B\rangle+|5,A\rangle+|7,B\rangle)/2$,
chosen as an example. (d) The value of the control parameters during the
protocol, see (a). Each color refers to different parameters in each part,
separated with dotted lines. Orange lines ($v_{\mathrm{tr}}$) also include the
contiguous $u_{k}$’s, in order for the transfer to be induced properly; pink
lines ($v_{\mathrm{bar}}$) do not. The phase shifts are induced using on-site
energies $\epsilon_{k}$.

##  Appendix E GHZ-like protocol for Bell states

In this section, we include for completeness the concurrence results of the
preparation of the $|\Psi\rangle=|00\rangle+\zeta|11\rangle$ state using the
protocol described in the main text for the GHZ state, restricted to two
qutrits. We show the protocol and results in Fig. 10. Given that it only uses
a single transfer, just like the protocol used in the main text for
$|\Phi\rangle$, its results are remarkably similar.

Figure 10: (a) GHZ-like protocol, adapted to create state $|\Psi\rangle$.
States 0,1 and 2 are shown in blue, yellow and red, respectively. A photon in
a cavity is shown as purple shading. (b) Concurrence as a function of disorder
strength for off-diagonal disorder (left) and general disorder (right) for an
SSH chain with one domain and 14 sites (blue), and for a four-domain chain
with 15 sites (red).

##  Appendix F Generalization to any number of qubits

In this section, we generalize the W and GHZ protocols to prepare a $p$-qubit
generalized W or GHZ state. The pulses on the generalized W state protocol
depend on the number of qubits. The steps to prepare it are: (i) Start in
state $|1\cdots 0\rangle$, with an empty photonic lattice. (ii) Implement a
$(p-1)\pi/p$-pulse on the left qubit-cavity coupling parameter. (iii) Realize
a full transfer to the next qubit cavity. (v) Repeat steps (ii-iii) for each
qubit successively, now implementing a $(q-1)\pi/q$-pulse in step (ii), with
$q=p-1,\ldots,2$. (iv) After the final transfer, implement a $\pi$-pulse on
the last qubit, achieving a uniform probability distribution among all
intervening states. The final state is:

$\displaystyle|\textrm{W}_{p}\rangle=\frac{1}{\sqrt{p}}\left(|100\cdots
0\rangle-\zeta_{0,1}|010\cdots 0\rangle\right.$ (23)
$\displaystyle\left.-\zeta_{0,1}\zeta_{1,2}|001\cdots
0\rangle+\cdots-\zeta_{0,1}\cdots\zeta_{p-1,p}|000\cdots 1\rangle\right),$

where $\zeta_{k,k^{\prime}}$ is the phase acquired in the transfer from the
cavity housing qubit $k$ to the cavity housing qubit $k^{\prime}$, which can
be obtained using Eq. 19.

To prepare the generalized GHZ state we need qutrits instead of qubits, just
like in the three-qubit version, and we suppose that the gap between qutrit
states $|1\rangle$ and $|2\rangle$ can be tuned to the frequency of the
cavity. The steps required are: (i) Start in state $(|01\cdots
1\rangle+\eta|21\cdots 1\rangle)/\sqrt{2}$, with $|\eta|=1$, which has no
entanglement, and an empty photonic lattice. (ii) Implement a $\pi$-pulse on
the left qutrit. (iii) Realize a full transfer along the first domain. (iv)
Implement a $\pi$-pulse on the next qutrit. (v) Apply an $X$ gate to that
qutrit. (vi) Repeat operations (ii-v) for each additional qutrit $k$, that is,
$\Pi_{k-1}T_{k-1,k}\Pi_{k}X_{k}$. (vii) Finish the protocol by applying the
operation $\mathcal{A}=|1\rangle\langle 2|+h.c.$ to the last qutrit. The final
state is:

$\displaystyle|\textrm{GHZ}_{p}\rangle=\frac{1}{\sqrt{2}}\left(|00\cdots
0\rangle+(-i)^{2(p-1)}\eta|11\cdots 1\rangle\right).$ (24)

The value of $\eta$ in the initial state can be chosen to fix the relative
phase. Both protocols can be easily modified via the qudit-cavity coupling
operations to change the relative weights between their final components, and
the relative phases in the W state can be adjusted with phase gates or using
the phase-shift protocols in Appendix C.

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$\displaystyle N[5,60]=4-4=0.\hskip 101.17755pt$

###  5.3 Potential Over Multiple Wires

To demonstrate our last result, we consider a potential extending over both
wires, and suppose that the boundary conditions at $x_{1}=1$ and $x_{2}=1$ are
Dirichlet. Let

$V_{i}(x_{i})=\begin{cases}\nu,&x_{i}\in(0,s_{i})\\\
0,&x_{i}\not\in(0,s_{i})\end{cases}$

for both $i=1$ and $i=2$. To apply our theorem, we must construct the
corresponding Evans functions.

$E^{\Omega_{1}}_{\Gamma_{1}}(\lambda)=-\frac{\sin({\sqrt{\lambda}}(1-s_{1}))}{{\sqrt{\lambda}}},$
$E^{\tilde{\Omega}_{1}}_{\tilde{\Gamma}_{1}}(\lambda)=-\frac{\sin({\sqrt{\lambda}}(1-s_{2}))}{{\sqrt{\lambda}}},$

and

$E^{\tilde{\Omega}_{2}}_{\tilde{\Gamma}_{2}}(\lambda)=-\frac{\sin(\sqrt{\lambda-\nu}(s_{1}+s_{2}))}{\sqrt{\lambda-\nu}}.$

It can be shown that the determinant of $\mathcal{M}_{1}+\mathcal{M}_{2}$ is
given by

$\displaystyle|(\mathcal{M}_{1}+\mathcal{M}_{2})(\lambda)|=$
$\displaystyle\left(\sqrt{x+10}\cot\left(\sqrt{x+10}\right)+\sqrt{x}\cot\left(\frac{\sqrt{x}}{2}\right)+\frac{\sqrt{x+10}}{\sin\left(\sqrt{x+10}\right)}\right)$
$\displaystyle\times$
$\displaystyle\left(\sqrt{x+10}\cot\left(\sqrt{x+10}\right)+\sqrt{x}\cot\left(\frac{\sqrt{x}}{2}\right)-\frac{\sqrt{x+10}}{\sin\left(\sqrt{x+10}\right)}\right)$

It can also be shown that the function

$\displaystyle E(\lambda)=$
$\displaystyle\sqrt{x}\sqrt{10+x}\cos\left(\sqrt{10+x}\right)\sin\left(\sqrt{x}\right)$
$\displaystyle+$
$\displaystyle\left(-5+\left(5+x\right)\cos\left(\sqrt{x}\right)\right)\sin\left(\sqrt{10+x}\right)$

indicates the location of the positive eigenvalues of $H^{\Omega}_{\Gamma}$.
Once again, one can use a plot to confirm our formula accurately predicts the
location of eigenvalues, as demonstrated in Figure 7.

Figure 7: Two splits on two distinct wires with $s_{1}=s_{2}=\frac{1}{2}$ and
$\nu=-10$. The blue curve is $100E^{\Omega_{1}}_{\Gamma_{1}}$ as well as
$100E^{\tilde{\Omega}_{1}}_{\tilde{\Gamma}_{1}}$ (they are identical in this
case), the red is $100E^{\tilde{\Omega}_{2}}_{\tilde{\Gamma}_{2}}$, the purple
curve is $10|\mathcal{M}_{1}+\mathcal{M}_{2}|$, and the black curve is $E$.
These objects are vertically rescaled to accentuate the zeros and poles.

Again, this figure demonstrates proper cancellation between the subproblem
eigenvalues and the poles of the map determinant, and the proper
correspondence between eigenvalues of $H^{\Omega}_{\Gamma}$ and the zeros of
the map determinant. Examining the counting functions, we find that

$\displaystyle\mathcal{N}_{H^{\Omega}_{\Gamma}}[3,60]=4,\hskip
14.45377pt\mathcal{N}_{H^{\Omega_{1}}_{\Gamma_{1}}}[3,60]=1,\hskip
14.45377pt\mathcal{N}_{H^{\tilde{\Omega}_{1}}_{\tilde{\Gamma}_{1}}}[3,60]=1,\hskip
14.45377pt\mathcal{N}_{H^{\tilde{\Omega}_{2}}_{\tilde{\Gamma}_{2}}}[3,60]=1,$
$\displaystyle N[5,60]=4-3=1.\hskip 101.17755pt$

##  Appendix A Note on Linear Algebraic Concepts

Despite introducing separate spatial variables on each wire, we still wish to
often refer to notions of linear dependence and independence of solution sets.

######  Theorem A.1.

Let $\\{{\vec{u}}_{1},...,{\vec{u}}_{2n}\\}$ be a set of solutions to the
equation $(H^{\Omega}-\lambda){\vec{u}}=\vec{0}$, where $H^{\Omega}$ is the
differential expression from (7). Let $U$ be a $2n\times 2n$ matrix whose
$i$th column is ${\vec{u}}_{i}$ followed by ${\vec{u}}_{i}^{\prime}$. Then
$|U|$ is a function of $\lambda$ only.

###### Proof.

We can calculate $|U|$ using Laplace expansion by complementary minors. In
particular, we can choose to expand along the pair of rows $1$ and $n+1$,
which contain information about $({\vec{u}}_{i})_{1}$ and
$({\vec{u}}_{i})_{1}^{\prime}$, respectively. Let $H$ be the set of two-
element combinations $\\{s,t\\}$ from the set $\\{1,...,2n\\}$. Then the
determinant of $U$ is

$|U(\lambda)|=(-1)^{n-1}\sum_{\\{s,t\\}\in
H}\pi(s,t)\begin{vmatrix}({\vec{u}}_{s})_{1}(x_{1},\lambda)&({\vec{u}}_{t})_{1}(x_{1},\lambda)\\\
({\vec{u}}_{s})_{1}^{\prime}(x_{1},\lambda)&({\vec{u}}_{t})_{1}^{\prime}(x_{1},\lambda)\end{vmatrix}c_{s,t}\,,$
(139)

where $\pi(s,t)$ is the sign of a permutation sending $1$ to $s$, $2$ to $t$,
and all other numbers in $\\{1,...,2n\\}\setminus\\{1,2\\}$ are sent to the
elements of $\\{1,...,2n\\}\setminus\\{s,t\\}$ in increasing order, and
$c_{s,t}$ is the $(2n-2)\times(2n-2)$ complementary minor obtained from $U$ by
deleting the $1$st and $(n+1)$th rows, and the $s$th and $t$th columns. The
$(-1)^{n-1}$ term represents the row permutation sending $1$ to $1$, $n+1$ to
$2$, and all else paired in increasing order. Since the $2\times 2$
determinants involved in this expansion are solutions to a second order
ordinary differential equation with no first order term, by Abel’s Identity
they depend only on $\lambda$, and not on the choice of $x_{1}$. We can
clearly use a similar method to expand each $|c_{s,t}|$ by $2\times 2$ blocks
of functions sharing one spatial variable, so none of these terms depend on
${\vec{x}}$ either. Thus $|U|$ is ${\vec{x}}$ independent, just as in the
single variable case. ∎

As with a set of single variable vector functions, we say that
$\\{{\vec{u}}_{1},...,{\vec{u}}_{2n}\\}$ forms a linearly independent set if
the only solution to

$\sum_{i=1}^{2n}c_{i}{\vec{u}}_{i}({\vec{x}},\lambda)=\vec{0}\,,$ (140)

is the one with $c_{i}=0$ for all $i$. This definition needs no adjustment
from the single-variable case since vector addition acts componentwise, and
thus the separate spatial variables do not interact at all in this sum.
Indeed, we are free to vary each $x_{i}$ individually without altering the
result of the sum. Since the typical definition of linear dependence applies
for functions on a quantum graph, and since Abel’s Identity still holds, one
can use the standard proof to show that $|U|=0$ if and only if there is a
linearly dependent set of columns in $U$.

### Acknowledgements

A.S. acknowledges the support of an AIM workshop on _Computer assisted proofs
for stability analysis of nonlinear waves_ , where some of the key ideas were
discussed. A.S. acknowledges support from the National Science Foundation
under grant DMS-1910820.

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