# Autonomous Distribution of Programmable Multiqubit Entanglement in a Dual-
Rail Quantum Network

J. Agustí1,2,3, X. H. H. Zhang1,2,3, Y. Minoguchi,4, P. Rabl1,2,3,4 1Technical
University of Munich, TUM School of Natural Sciences, Physics Department,
85748 Garching, Germany  2Walther-Meißner-Institut, Bayerische Akademie der
Wissenschaften, 85748 Garching, Germany  3Munich Center for Quantum Science
and Technology (MCQST), 80799 Munich, Germany  4Vienna Center for Quantum
Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria

###### Abstract

We propose and analyze a scalable and fully autonomous scheme for preparing
spatially distributed multiqubit entangled states in a dual-rail waveguide QED
setup. In this approach, arrays of qubits located along two separated
waveguides are illuminated by correlated photons from the output of a
nondegenerate parametric amplifier. These photons drive the qubits into
different classes of pure entangled steady states, for which the degree of
multipartite entanglement can be conveniently adjusted by the chosen pattern
of local qubit-photon detunings. Numerical simulations for moderate-sized
networks show that the preparation time for these complex multiqubit states
increases at most linearly with the system size and that one may benefit from
an additional speedup in the limit of a large amplifier bandwidth. Therefore,
this scheme offers an intriguing new route for distributing ready-to-use
multipartite entangled states across large quantum networks, without requiring
any precise pulse control and relying on a single Gaussian entanglement source
only.

_Introduction._ —As quantum computing and quantum communication systems with
an increasing number of coherently integrated components become
technologically available, a growing demand for efficient schemes to transfer
quantum states or distribute entanglement across different parts of such
networks will arise [1, 2, 3, 4]. While basic protocols to do so are well
known and have already been successfully implemented in a variety of platforms
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14], it is envisioned that in future quantum
devices, entanglement must be generated and interchanged among many thousands
of qubits within a limited coherence time. In view of this challenge, there is
a strong motivation to go beyond a serial application of existing protocols
and search for more efficient quantum communication strategies that are fast,
parallelizable, and, ideally, require a minimal amount of classical control.

In this Letter, we describe a fully autonomous entanglement distribution
scheme, which exploits an intriguing physical effect, namely the formation of
multipartite entangled stationary states in a cascaded dual-rail quantum
network. Specifically, we consider a configuration as shown in Fig. 1, where
spatially separated qubits located along two photonic waveguides are
illuminated by the correlated output of a nondegenerate parametric amplifier
[15]. Previously, it was proposed to use broadband squeezed reservoirs for
generating bipartite entanglement between separated qubit pairs [16, 17, 18,
19, 20, 21, 22] or, for specific arrangements, between qubits along a 1D
channel [23, 24] or in coupled arrays [25, 26]. Here we show, first of all,
that this concept can be generalized to produce, under ideal conditions, an
arbitrary number of maximally entangled qubit pairs over large distances.
Moreover, we find that the entanglement shared between different sets of
qubits can be adjusted by simply changing the local qubit-photon detunings.
This provides a convenient way to “program” different classes of multipartite
entangled states without the need for any time-dependent control or additional
nonlocal operations.

Figure 1: Sketch of a dual-rail quantum network, where qubits along two
separated waveguides are driven by the correlated output of a nondegenerate
parametric amplifier and relax into a pure steady state
$|\psi_{0}(r,\vec{\delta}_{A},P)\rangle$. As shown in the inset, the qubits in
waveguide $A$ ($B$) are detuned from the central photon frequency $\omega_{A}$
($\omega_{B}$) by $\delta_{A,i}$ ($\delta_{B,i}$) and the qubit-waveguide
coupling is assumed to be fully directional. See text for more details.

To evaluate the scalability of this approach, we simulate the formation of
these multipartite entangled states under more realistic conditions, taking in
particular a finite bandwidth of the squeezing source into account. We find
that the maximal number of entangled qubit pairs, $N_{\rm ent}$, remains
rather robust under the influence of experimental imperfections and that the
total preparation time, $T_{\rm prep}\sim N_{\rm ent}$, scales at most
linearly with the system size, independently of the complexity of the prepared
state. In the limit of a large amplifier bandwidth, the intrinsic
parallelization of the preparation scheme can be exploited to further reduce
$T_{\rm prep}$, which shifts the technological requirements for scalability
from the control of many qubits to the optimization of a single Gaussian
squeezing source. This can be advantageous for many applications in optical,
microwave, or hybrid [27, 28, 29, 30] quantum networks, where such photonic
devices are currently developed [31, 32, 33, 34, 35, 36, 37, 14].

_Model._ —We consider a dual-rail quantum network as depicted in Fig. 1, where
two sets of qubits $\eta=A,B$ are coupled to two separate photonic channels.
The waveguides are connected to a common nondegenerate parametric amplifier,
which we model by a two-mode squeezing interaction $(\hbar=1)$
$H_{\chi}=ig(a_{A}^{\dagger}a_{B}^{\dagger}-a_{A}a_{B})$ for two local modes
with bosonic annihilation operators $a_{A}$ and $a_{B}$. These photons then
decay into the respective waveguides with rate $\kappa$ and drive the qubits
into a correlated state. For the following analysis, we assume that the qubit-
waveguide coupling is fully directional [38, 39, 40] and label the qubits by
the index $i=1,\dots,N$ along the direction of propagation. Such conditions
can be realized by using circulators [41, 42, 43, 44, 45, 46], chiral
waveguides [40], or other schemes for directional coupling [47, 48, 49, 50].

We first focus on the limit of a broadband amplifier,
$\kappa\rightarrow\infty$, in which case the dynamics of the photons can be
adiabatically eliminated to obtain an effective master equation (see [51] for
more details)

$\dot{\rho}_{\rm q}=-i[H_{\rm casc},\rho_{\rm
q}]+\sum_{\eta=A,B}\gamma\mathcal{D}[J_{\eta}]\rho_{\rm q}$ (1)

for the reduced qubit density operator $\rho_{\rm q}$. Here $\gamma$ denotes
the decay rate of each individual qubit and $\mathcal{D}[C]\rho=C\rho
C^{\dagger}-\\{C^{\dagger}C,\rho\\}_{+}/2$. In Eq. (1) we have already
rewritten the underlying directional qubit-qubit interactions in terms of a
coherent Hamiltonian evolution with

$H_{\rm
casc}=\sum_{\eta,i}\frac{\delta_{\eta,i}}{2}\sigma_{\eta,i}^{z}+i\frac{\gamma}{2}\sum_{\eta,j>i}\left(\sigma^{+}_{\eta,i}\sigma_{\eta,j}^{-}-{\rm
H.c.}\right),$ (2)

and purely dissipative processes with collective jump operators

$\displaystyle J_{A}=\cosh(r)L_{A}-\sinh(r)L_{B}^{\dagger},$ (3)
$\displaystyle J_{B}=\cosh(r)L_{B}-\sinh(r)L_{A}^{\dagger},$ (4)

where $L_{\eta}=\sum_{i=1}^{N}\sigma_{\eta,i}^{-}$. In this broadband limit,
the system is thus fully determined by the squeezing parameter
$r=2\tanh^{-1}(2g/\kappa)$, characterizing the degree of two-mode squeezing of
the photon source, and the two sets of qubit detunings,
$\vec{\delta}_{\eta=A,B}=(\delta_{\eta,1},\delta_{\eta,2},\dots,\delta_{\eta,N})$.

_Steady states._ —Equation (1) describes an open quantum many-body system with
competing coherent and dissipative processes, which in general drive the
qubits into a highly mixed and not very useful steady state. However, in the
following, we show that there exist specific conditions under which the steady
state of the network, $\rho_{\rm q}^{0}=|\psi_{0}\rangle\langle\psi_{0}|$, is
not only pure but also exhibits different degrees of multipartite entanglement
that can be controlled by the local detunings $\delta_{\eta,i}$.

We start our analysis by considering the simplest case of a single pair of
qubits ($N=1$) and $\delta_{A,1}=\delta_{B,1}=0$, as originally discussed in
Ref. [16]. In this case, one can explicitly show that the unique steady state
of Eq. (1) is $|\psi_{0}\rangle=|\Phi^{+}_{1,1}\rangle$, where

$|\Phi^{+}_{i,j}\rangle=\frac{\cosh(r)|0_{A,i}\rangle|0_{B,j}\rangle+\sinh(r)|1_{A,i}\rangle|1_{B,j}\rangle}{\sqrt{\cosh(2r)}}$
(5)

approaches a maximally entangled Bell state for $r\gg 1$. This state satisfies
the dark-state conditions $J_{\eta}|\psi_{0}\rangle=0$ and $H_{\rm
casc}|\psi_{0}\rangle=0$, which implies that once the qubits have reached the
steady state, they completely decouple from the squeezed photonic bath.
Consequently, they no longer affect successive qubits along the waveguide.

Importantly, this observation remains true even for finite detunings
satisfying $\delta_{A,1}+\delta_{B,1}=0$, which then allows us to
systematically identify also more complex multiqubit steady states by
proceeding in two steps. First, we set $\vec{\delta}_{B}=-\vec{\delta}_{A}$,
such that, according to the argument from above, qubits with the same index
decouple pairwise from the photonic reservoir. The network then relaxes into
the pure steady state $|\psi_{0}\rangle=|\Phi_{\parallel}\rangle$, where

$|\Phi_{\parallel}\rangle=\bigotimes_{i=1}^{N}|\Phi^{+}_{i,i}\rangle$ (6)

is the product of $N$ consecutive Bell pairs of the type given in Eq. (5).
Interestingly, this result is independent of the total number of qubit pairs.

In the second step, we make use of the form invariance of the cascaded master
equation in Eq. (1) under unitary transformations of the type [38]

$ U_{i,i+1}=e^{i\theta_{i,i+1}(\vec{s}_{B,i}+\vec{s}_{B,i+1})^{2}},$ (7)

where $\vec{s}_{\mu}=(\sigma_{\mu}^{x},\sigma_{\mu}^{y},\sigma_{\mu}^{z})/2$
and the mixing angle satisfies
$\tan(\theta_{i,i+1})=(\delta_{B,i}-\delta_{B,i+1})/\gamma$. Under these
transformations, one finds that
$U_{i,i+1}J_{\eta}U^{\dagger}_{i,i+1}=J_{\eta}$ and

$\displaystyle U_{i,i+1}H_{\rm
casc}(\vec{\delta}_{A},\vec{\delta}_{B})U^{\dagger}_{i,i+1}$ $\displaystyle=$
$\displaystyle H_{\rm casc}(\vec{\delta}_{A},P_{i,i+1}\vec{\delta}_{B}),$ (8)

where the permutation $P_{i,i+1}$ exchanges $\delta_{B,i}$ and
$\delta_{B,i+1}$. In other words, given a pure steady state $|\psi_{0}\rangle$
for a certain detuning pattern $\vec{\delta}_{B}$, the state
$|\psi_{0}^{\prime}\rangle=U_{i,i+1}|\psi_{0}\rangle$ is a pure steady state
of the same network with a permuted pattern of detunings,
$\vec{\delta}_{B}^{\prime}=P_{i,i+1}\vec{\delta}_{B}$.

This form invariance now allows us to construct a large family of multipartite
entangled steady states, which are parametrized by (i) the squeezing parameter
$r$, (ii) the set of detunings $\vec{\delta}_{A}$ for qubits in waveguide $A$
and (iii) a permutation $P$ that fixes the detunings in waveguide $B$ to be
$\vec{\delta}_{B}=-P\vec{\delta}_{A}$. By decomposing
$P=\prod_{\sigma}P_{i_{\sigma},i_{\sigma}+1}$ into a product of nearest-
neighbor transpositions, we can start with the state in Eq. (6) and then use
the relation below Eq. (8) to derive an explicit expression for the
corresponding steady state,

$|\psi_{0}(r,\vec{\delta}_{A},P)\rangle=\prod_{\sigma}U_{i_{\sigma},i_{\sigma}+1}|\Phi_{\parallel}\rangle.$
(9)

Importantly, this is also the unique steady state of the network, as discussed
in more detail in [51]. A graphical illustration of Eq. (9) is presented in
Fig. 2 (a).

Figure 2: (a) Graphical illustration of Eq. (9). Starting from
$\vec{\delta}_{B}=-\vec{\delta}_{A}$, the detunings in waveguide B are
reordered as $\vec{\delta}_{B}=-P\vec{\delta}_{A}$ through nearest-neighbor
transpositions, following the colored lines as a guide to the eye. Each
transposition maps into one of the unitary operations $U_{i,i+1}$ that
determine the final steady state. (b) Bipartite entanglement expressed in
terms of the concurrences $\mathcal{C}_{ij}$ for the four-qubit state in Eq.
(10) as a function of $r$, and in (c) as a function of $\Delta$ for $r=1$. (d)
Sketch of the detuning pattern for the family of multipartite states described
in the text and different partitions for evaluating the entanglement entropy.
(e) Entanglement entropy $\mathcal{S}_{n}$ as a function of $n$, for different
detunings $\Delta$ and $r=1$.

_Entanglement._ —To investigate the entanglement properties of the family of
states in Eq. (9), we start with the case $N=2$ and choose the only nontrivial
permutation $P=P_{1,2}$. We obtain

$\ket{\psi_{0}}=\frac{\gamma|\Phi^{+}_{1,1}\rangle|\Phi^{+}_{2,2}\rangle+i\Delta|\Phi^{+}_{1,2}\rangle|\Phi^{+}_{2,1}\rangle}{\sqrt{\gamma^{2}+\Delta^{2}}},$
(10)

where $\Delta=\delta_{A,1}-\delta_{A,2}$. In Figs. 2 (b) and (c) we visualize
the entanglement structure of this state in terms of the concurrences
$\mathcal{C}_{ij}\equiv\mathcal{C}(\rho_{A,i|B,j})$ [58, 59] of the reduced
bipartite qubit states, $\rho_{A,i|B,j}$. For $\Delta=0$, we find that for
parallel pairs $\mathcal{C}_{ii}\simeq 1$ already for moderate values of
$r\gtrsim 1$, consistent with the state $|\Phi_{\parallel}\rangle$. For
$|\Delta|\gg\gamma$ the same is true for diagonal pairs, i.e.,
$\mathcal{C}_{12}=\mathcal{C}_{21}\simeq 1$. For all intermediate parameters,
the state is a genuine four-partite entangled state [60], and belongs to the
set of locally maximally entanglable states [61] for $r\gg 1$.

For a larger number of qubits, we can use the entanglement entropy
$\mathcal{S}(\rho_{r})=-{\rm Tr}\\{\rho_{r}\ln\rho_{r}\\}$ for a reduced state
$\rho_{r}$ to study the entanglement between different bipartitions of the
network. First of all, this analysis shows that
$\mathcal{S}_{A}\equiv\mathcal{S}(\rho_{A})=-N\ln\left[x^{x}(1-x)^{(1-x)}\right]$,
where $x=\cosh^{2}(r)/\cosh(2r)$, only depends on the squeezing parameter $r$.
This can be understood from the fact that the unitaries $U_{i,i+1}$ only act
within subsystem $B$. Thus, with respect to this partition, the states in Eq.
(9) can be understood as generalized “rainbow states” [62, 63, 26] with a
volume-law entanglement $\mathcal{S}(\rho_{A})\simeq N\ln 2$ for $r\gtrsim 1$.
In contrast, for partitions along the chain, the entanglement entropy
$\mathcal{S}_{n}=\mathcal{S}(\rho_{[1,\dots,n]})$ depends not only on the
chosen permutation $P$, but also on the pattern of detunings
$\vec{\delta}_{A}$. This is illustrated in Figs. 2 (d) and (e), where we
consider as an example the detunings $\delta_{A,i}=(i-1)\Delta$ and the
reversed order, $\delta_{B,i}=-P_{\rm rev}\delta_{A,i}=-\delta_{A,N+1-i}$, in
waveguide $B$. For $\Delta\gg\gamma$ the unitaries in Eq. (9) correspond to
approximate SWAP operations and $\mathcal{S}_{n}\simeq 2n\ln 2$. Instead, for
$\Delta\lesssim\gamma$, the entangling unitaries $U_{i,i+1}\approx\sqrt{\rm
SWAP}$ generate more multipartite entanglement across the whole chain, which
reduces the block-entanglement $\mathcal{S}_{n}$ correspondingly. In general,
different choices for $\vec{\delta}_{A}$ and $P$ can be used to define certain
blocks of qubits that are entangled among each other, independently of their
physical location.

_Preparation time._ —So far we have shown that a single two-mode squeezing
source is in principle enough to entangle an arbitrary number of qubits.
However, for practical applications, we must still evaluate the time $T_{\rm
prep}$ that it takes to prepare this state. To do so we first continue with
the analysis of the ideal qubit master equation in Eq. (1) and study the
relaxation dynamics toward the steady state $|\psi_{0}\rangle$, assuming that
at $t=0$ all qubits are initialized in state $|0\rangle$. In Fig. 3 this
evolution is shown in (a) for the bi-partite entangled state
$|\Phi_{\parallel}\rangle$ with $\vec{\delta}_{A}=0$ and in (b) for the
multipartite entangled state considered in Fig. 2 (e).

Figure 3: (a) Relaxation into a bipartite entangled state for
$\vec{\delta}_{A}=0$ and (b) into a multipartite entangled state for
$\vec{\delta}_{B}=-P_{\rm rev}\vec{\delta}_{A}$ and $\Delta=\gamma/5$. In both
cases $N=5$. (c) Scaling of the preparation time $T_{\rm prep}$ for different
ratios $\Delta/\gamma$, where $\delta_{A,i}=\Delta(i-1)$ and
$\vec{\delta}_{B}=-\vec{\delta}_{A}$. We define $T_{\rm prep}$ via the
condition $[1-\mu(T_{\rm prep})]/N=0.001$, where $\mu={\rm Tr}[\rho_{\rm
q}^{2}]$ is the purity. For the examples in (a) and (b), $T_{\rm prep}$ is
indicated by the dashed vertical line. In all plots $r=1$.

In the bipartite case, we observe a successive, pairwise formation of Bell
states with a total time $T_{\rm prep}\sim N$. Interestingly, already for
$\delta_{A,i}=0$, this preparation time is faster than a sequential
preparation of $N$ independent Bell pairs, i.e., $T_{\rm prep}(N)<NT_{\rm
prep}(N=1)$. For detuned qubits the preparation time decreases further and
$T_{\rm prep}(N)\simeq T_{\rm prep}(N=1)$ for $\Delta\gtrsim\gamma$, i.e., all
pairs are prepared in parallel. For multipartite entangled states, where the
differences $|\delta_{A,i}-\delta_{A,j}|$ are necessarily small, a full
parallelization is not possible, but even in this case we obtain an intrinsic
advantage compare to a sequential distribution of entanglement, followed by
local gates. Note that for the same detunings $\vec{\delta}_{A}$, the
relaxation time $T_{\rm prep}$ is independent of the permutation $P$.

_Scalability._ —All the results so far have been derived within the infinite-
bandwidth approximation, which underlies Eq. (1) and assumes that correlated
photons are available at arbitrary detunings. Obviously, this assumption must
break down when $\delta_{\rm max}={\rm max}\\{|\delta_{A,i}|\\}\gtrsim\kappa$,
but even for $\delta_{A,i}=0$ it has been shown that any finite $\kappa$
limits the transferable entanglement [22]. Therefore, to provide physically
meaningful predictions about the scalability of the current scheme it is
necessary to go beyond the assumption of a Markovian squeezed reservoir [16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26] and take finite-bandwidth effects into
account. To do so we now simulate the dynamics of the state of the full
network, $\rho$, as described by the cascaded quantum master equation [51]

$\begin{split}\dot{\rho}=&-i[H_{\chi},\rho]+\sum_{\eta}\kappa\mathcal{D}[a_{\eta}]\rho\\\
&\sum_{\eta,i}\left(-i\frac{\delta_{\eta,i}}{2}[\sigma_{\eta,i}^{z},\rho]+\gamma\mathcal{D}[\sigma_{\eta,i}^{-}]\rho+\frac{\gamma_{\phi}}{2}\mathcal{D}[\sigma_{\eta,i}^{z}]\rho\right)\\\
&+\sum_{\eta,i}\sqrt{\kappa\gamma}\mathcal{T}[a_{\eta},\sigma_{\eta,i}^{-}]\rho+\sum_{\eta,j>i}\gamma\mathcal{T}[\sigma^{-}_{\eta,i},\sigma_{\eta,j}^{-}]\rho.\end{split}$
(11)

Here we have already included a finite dephasing rate $\gamma_{\phi}$ for each
qubit and introduced the superoperator
$\mathcal{T}[O_{1},O_{2}]\rho=[O_{1}\rho,O_{2}^{\dagger}]+[O_{2},\rho
O_{1}^{\dagger}]$ to model directional interactions between all nodes along
the same waveguide.

In Fig. 4 (a) we plot the steady state concurrences $\mathcal{C}_{ii}$ for the
case $\delta_{A,i}=0$ and different ratios $\beta=\kappa/\gamma$. We see that
a finite bandwidth $\kappa$ reduces the maximal amount of entanglement for the
first pair [22] and also results in a gradual decay of the entanglement along
the chain. By using a linear extrapolation, $N_{\rm
ent}=\mathcal{C}_{11}/(\mathcal{C}_{11}-\mathcal{C}_{22})$, we can use these
finite-size simulations to extract the maximal number of pairs that can be
entangled for a given $\beta$ and dephasing rate $\gamma_{\phi}$. These
results are summarized in Fig. 4 (b). We see that for otherwise ideal
conditions, rather large numbers of $N_{\rm ent}\sim 10-100$ can be entangled
for moderate $\beta$, while the presence of dephasing or other imperfections
sets additional limits on $N_{\rm ent}$. Note that these results are for
$\delta_{A,i}=0$, where the formation of the steady state is the slowest.
Thus, these results represent approximate upper bounds for $N_{\rm ent}$ also
for all other classes of multipartite entangled states. Additional plots for
number of entangled pairs for various experimental sources of imperfections
together with estimates for the achievable $N_{\rm ent}$ in state-of-the-art
microwave networks are presented in [51].

Figure 4: (a) Plot of the steady-state concurrences $\mathcal{C}_{ii}$ for
$\vec{\delta}_{A}=0$ and different amplifier bandwidths. (b) Maximal number of
entangled pairs, $N_{\rm ent}$ as a function of $\beta=\kappa/\gamma$ and
different dephasing rates $\gamma_{\phi}$. (c) Dependence of the concurrence
of a single qubit pair on the detuning $\Delta$, where
$\delta_{A,1}=-\delta_{B,1}=\Delta$ and different values of $\beta$ have been
assumed. (d) Plot of the concurrence $\mathcal{C}_{44}$ in a chain of $N=4$
qubit pairs with $\delta_{A,i}=(i-1)\Delta=-\delta_{B,i}$ and a finite
dephasing rate. This plot illustrates the initial gain from a parallel
preparation when $\Delta>0$, while the entanglement decreases again when
$\delta_{\rm max}=(N-1)\Delta\approx\kappa$, due to finite bandwidth effects.
In all plots $r=1$.

Finally, let us return to the observed speedup for far-detuned qubits, but
taking a finite amplifier bandwidth into account. In Fig. 4 (c) we
investigate, first of all, the dependence of $\mathcal{C}_{11}$ on the
detuning $\delta_{A,1}=\Delta$. As expected, this plot shows a significant
decay of the entanglement for $\Delta/\kappa>1$, from which we also deduce
that $\delta_{\rm max}<\kappa$ must be satisfied in the multiqubit case. Since
for a parallel preparation with $T_{\rm prep}(N)\sim\text{const}$ we require
$\delta_{\rm max}\approx\gamma N$, we conclude that the number of pairs that
can be entangled in parallel, $N_{\parallel}\approx N_{\rm ent}$, is actually
comparable to the total number of entangled pairs for $\vec{\delta}_{A}=0$. As
a minimal illustration of this behavior, we consider in Fig. 4 (d) the example
of $N=4$ pairs with $\delta_{A,i}=\Delta(i-1)$. We plot the concurrence of the
last pair, $\mathcal{C}_{44}$, for a fixed dephasing rate $\gamma_{\phi}$ and
increasing detuning $\Delta$. Up to $\Delta\sim\kappa$, entanglement increases
due to a reduced preparation time, while for larger detunings finite-bandwidth
effects set in and degrade the entanglement again. Note that for a parametric
amplifier with asymmetric decay rates, $\kappa_{A}\neq\kappa_{B}$, the
structure of the ideal qubit master equation in Eq. (1) remains the same [51],
but finite-bandwidth effects are determined by the minimal rate $\kappa_{\rm
min}={\rm min}\\{\kappa_{A},\kappa_{B}\\}$.

_Conclusions._ —In summary, we have presented a fully autonomous scheme for
distributing entanglement among two distant sets of qubits. Within the same
setup, states with varying degrees of bi- and multipartite entanglement can be
prepared by adjusting the squeezing strength and the local qubit detunings,
while retaining a preparation time that scales at most linearly with $N$.
Compared to related autonomous protocols discussed for single waveguides [38,
39, 23, 24], locally coupled chains [25, 26], or combinations thereof [64],
the use of a propagating two-mode entangled source offers the possibility to
entangle qubits that are arbitrarily far apart [51] and a systematic way to
parallelize the scheme by increasing the bandwidth of the amplifier. This
makes this approach very attractive for long-distance entanglement
distribution schemes with long-lived spins or narrow-bandwidth optical
emitters, but also for local area quantum networks [65, 66, 67, 68], where
multiple nodes can be simultaneously entangled with a limited amount of
control.

We thank Aashish Clerk, Matthias Englbrecht, Tristan Kraft, Barbara Kraus, and
Kirill Fedorov for many stimulating discussions. This work was supported by
the European Union’s Horizon 2020 research and innovation program under Grant
Agreement No. 899354 (SuperQuLAN) and the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) No. 522216022. Most of the computational
results presented were obtained using the CLIP cluster [69]. This research is
part of the Munich Quantum Valley, which is supported by the Bavarian state
government with funds from the Hightech Agenda Bayern Plus.

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Supplementary material for:
Autonomous Distribution of Programmable Multiqubit Entanglement in a Dual-Rail
Quantum Network

##  I Full model and derivation of the effective master equation

In this section, we present a more detailed discussion of the full model for
the cascaded quantum network shown in Fig. 1 of the main text and the
derivation of the effective master equation for the qubit state $\rho_{\rm q}$
in the broadband-amplifier limit.

###  I.1 Model

By assuming that the waveguides are sufficiently broadband and by moving into
a rotating frame with respect to the photon frequencies $\omega_{A}$ and
$\omega_{B}$, we can use the standard framework of cascaded quantum systems
[1, 2, 3] and model the dynamics of the full quantum network by a master
equation of the form ($\hbar$=1)

$\displaystyle\dot{\rho}=\left(\mathcal{L}_{\rm ph}+\mathcal{L}^{0}_{\rm
q}+\mathcal{L}_{\rm casc}\right)\rho.$ (S1)

Here, the first term describes the parametric amplifier with

$\displaystyle\mathcal{L}_{\rm
ph}\rho=-i[H_{\chi},\rho]+\sum_{\eta=A,B}\kappa_{\eta}\mathcal{D}[a_{\eta}]\rho,$
(S2)

where $H_{\chi}=ig(a_{A}^{\dagger}a_{B}^{\dagger}-a_{A}a_{B})$ and
$\kappa_{\eta}$ are the photon decay rates. The Liouville operator describing
the bare qubit dynamics reads

$\displaystyle\mathcal{L}^{0}_{\rm
q}\rho=\sum_{\eta,i}\left(-i\frac{\delta_{\eta,i}}{2}[\sigma_{\eta,i}^{z},\rho]+\gamma_{\eta,i}\mathcal{D}[\sigma_{\eta,i}^{-}]\rho+\frac{\gamma_{\phi}}{2}\mathcal{D}[\sigma_{\eta,i}^{z}]\rho\right),$
(S3)

where $\delta_{\eta,i}=\omega^{\rm q}_{\eta,i}-\omega_{\eta}$ is the detuning
of the qubit frequency $\omega^{\rm q}_{\eta,i}$ from the central photon
frequency $\omega_{\eta}$ and $\gamma_{\eta,i}$ is the qubit decay rate. We
have also included a dephasing of each qubit with rate $\gamma_{\phi}$.
Finally, the last term in Eq. (S1) accounts for the cascaded interaction
between all the nodes along each waveguide and is given by

$\mathcal{L}_{\rm
casc}\rho=\sum_{\eta,i}\sqrt{\kappa_{\eta}\gamma_{\eta,i}(1-\epsilon^{\eta}_{0,i})}\mathcal{T}[a_{\eta},\sigma_{\eta,i}^{-}]\rho+\sum_{\eta,j>i}\sqrt{\gamma_{\eta,i}\gamma_{\eta,j}(1-\epsilon^{\eta}_{i,j})}\mathcal{T}[\sigma^{-}_{\eta,i},\sigma_{\eta,j}^{-}]\rho.$
(S4)

Here, we have included the additional parameters
$\epsilon^{\eta}_{i,j}\in[0,1]$ to model losses in the system. In general,
$|\epsilon^{\eta}_{i,j}|$ is the loss probability for a photon propagating
between node $i$ and node $j$, where the index $i=0$ refers to the parametric
amplifier. In this way the $\epsilon^{\eta}_{i,j}$ can be adjusted to model
linear absorption losses, but also parasitic loss channels for the qubits.

Note that in Eq. (S1) we have absorbed all propagation phases into a
redefinition of the qubit operators. This means that up to local phase
rotations, all results presented in this work are independent of the precise
location of the qubits. This is in contrast to entanglement schemes in
bidirectional channels, which typically require specific arrangements [4, 5].
Note, however, that the validity of Eq. (S1) assumes that propagation times
between the nodes are negligible compared to the relevant timescales of the
system dynamics. We will relax this assumption later below.

###  I.2 Adiabatic elimination of the photonic modes

In the limit $\kappa_{\eta}\rightarrow\infty$ the dynamics of the parametric
amplifier modes can be adiabatically eliminated to derive an effective master
equation for the reduced density operator $\rho_{\rm q}={\rm Tr}_{\rm
ph}\\{\rho\\}$ of the qubits only. To do so, first, we rewrite the full master
equation as

$\displaystyle\dot{\rho}=\left(\mathcal{L}_{\rm ph}+\mathcal{L}_{\rm
q}+\mathcal{L}_{\rm ph-q}\right)\rho,$ (S5)

where

$\mathcal{L}_{\rm q}\rho=\mathcal{L}_{\rm
q}^{0}\rho+\sum_{\eta,j>i}\sqrt{\gamma_{\eta,i}\gamma_{\eta,j}(1-\epsilon^{\eta}_{i,j})}\mathcal{T}[\sigma^{-}_{\eta,i},\sigma_{\eta,j}^{-}]\rho$
(S6)

now includes all waveguide-mediated interactions among the qubits, while

$\mathcal{L}_{\rm
ph-q}\rho=\sum_{\eta=A,B}\sqrt{\kappa\gamma_{\eta}}\left([a_{\eta}\rho,\tilde{L}^{\dagger}_{\eta}]+[\tilde{L}_{\eta},\rho
a_{\eta}^{\dagger}]\right).$ (S7)

Here we have set $\gamma_{\eta}={\rm max}\\{\gamma_{\eta,i}\\}$ and introduced
the collective operators

$\tilde{L}_{\eta}=\sum_{i}\sqrt{\frac{\gamma_{\eta,i}}{\gamma_{\eta}}(1-\epsilon^{\eta}_{0,i})}\sigma_{\eta,i}^{-}.$
(S8)

In this form, the master equation in Eq. (S5) is identical to the master
equation of the 2-qubit setup considered in Ref. [6], but with the collective
operators $\tilde{L}_{\eta}$, instead of $\sigma_{\eta}^{-}$, appearing in the
photon qubit interaction in Eq. (S7). Therefore, for the adiabatic elimination
we can follow the same steps as in Appendix A of Ref. [6] and we obtain

$\begin{split}\dot{\rho}_{\rm q}(t)=\,&\mathcal{L}_{\rm q}\rho_{\rm
q}(t)-\sum_{\eta=A,B}\frac{\gamma_{\eta}}{2}N^{\eta}_{\rm
ph}\left([\tilde{L}_{\eta},[\tilde{L}_{\eta}^{\dagger},\rho_{\rm
q}]]+[\tilde{L}^{\dagger}_{\eta},[\tilde{L}_{\eta},\rho_{\rm q}]]\right)\\\
+&\sqrt{\gamma_{A}\gamma_{B}}\left(M_{\rm
ph}^{*}[\tilde{L}_{A},[\tilde{L}_{B},\rho_{\rm q}]]+M_{\rm
ph}[\tilde{L}_{A}^{\dagger},[\tilde{L}_{B}^{\dagger},\rho_{\rm
q}]]\right),\end{split}$ (S9)

where

$\displaystyle N^{\eta}_{\rm ph}$ $\displaystyle=$ $\displaystyle
2\kappa_{\eta}\mathrm{Re}\int_{0}^{\infty}\mathrm{d}t\,\langle
a_{\eta}^{\dagger}(t)a_{\eta}(0)\rangle,$ (S10) $\displaystyle M_{\rm ph}$
$\displaystyle=$
$\displaystyle\sqrt{\kappa_{A}\kappa_{B}}\int_{0}^{\infty}\mathrm{d}t\,\left(\langle
a_{A}(t)a_{B}(0)\rangle+\langle a_{B}(t)a_{A}(0)\rangle\right).$ (S11)

For our simple parametric amplifier, we obtain

$\displaystyle N_{\rm ph}=N^{\eta}_{\rm ph}$ $\displaystyle=$ $\displaystyle
g\frac{\sqrt{\kappa_{A}\kappa_{B}}}{2}\left[\frac{1}{\left(\frac{\sqrt{\kappa_{A}\kappa_{B}}}{2}-g\right)^{2}}-\frac{1}{\left(\frac{\sqrt{\kappa_{A}\kappa_{B}}}{2}+g\right)^{2}}\right],$
(S12) $\displaystyle M_{\rm ph}$ $\displaystyle=$ $\displaystyle
g\frac{\sqrt{\kappa_{A}\kappa_{B}}}{2}\left[\frac{1}{\left(\frac{\sqrt{\kappa_{A}\kappa_{B}}}{2}-g\right)^{2}}+\frac{1}{\left(\frac{\sqrt{\kappa_{A}\kappa_{B}}}{2}+g\right)^{2}}\right].$
(S13)

Since $M_{\rm ph}^{2}=N_{\rm ph}(N_{\rm ph}+1)$, we can further express these
quantities in terms of the squeezing parameter $r$, i.e.,

$N_{\rm ph}=\sinh^{2}(r),\qquad M_{\rm
ph}=\sinh(r)\cosh(r),\qquad\Rightarrow\qquad
r=2\tanh^{-1}(2g/\sqrt{\kappa_{A}\kappa_{B}}).$ (S14)

This relation allows us to write the effective master equation as

$\begin{split}\dot{\rho}_{\rm q}(t)=\,&\mathcal{L}_{\rm q}\rho_{\rm
q}(t)-\sum_{\eta=A,B}\gamma_{\eta}\mathcal{D}[\tilde{L}_{\eta}]\rho_{\rm
q}+\mathcal{D}[\tilde{J}_{A}]\rho_{\rm q}+\mathcal{D}[\tilde{J}_{B}]\rho_{\rm
q},\end{split}$ (S15)

where

$\displaystyle\tilde{J}_{A}=\sqrt{\gamma_{A}}\cosh(r)\tilde{L}_{A}-\sqrt{\gamma_{B}}\sinh(r)\tilde{L}_{B}^{\dagger},$
(S16)
$\displaystyle\tilde{J}_{B}=\sqrt{\gamma_{B}}\cosh(r)\tilde{L}_{B}-\sqrt{\gamma_{A}}\sinh(r)\tilde{L}_{A}^{\dagger}.$
(S17)

In the limit of negligible losses, $\epsilon^{\eta}_{i,j}\rightarrow 0$, we
can exploit the fact that for any set of operators $c_{i}$, the cascaded
waveguide interaction,

$\sum_{i}\mathcal{D}[c_{i}]\rho+\sum_{j>i}\mathcal{T}[c_{i},c_{j}]\rho=\mathcal{D}[C]\rho-i[H_{\rm
casc},\rho],$ (S18)

can be re-expressed in terms of a collective dissipation term with
$C=\sum_{i}c_{i}$ and a Hamiltonian evolution with

$H_{\rm
casc}=\frac{i}{2}\sum_{j>i}\left(c_{i}^{\dagger}c_{j}-c_{j}^{\dagger}c_{i}\right).$
(S19)

This then leads to

$\begin{split}\dot{\rho}_{\rm q}(t)=\,&-i[H_{\rm casc},\rho_{\rm
q}]+\sum_{\eta,i}\frac{\gamma_{\phi}}{2}\mathcal{D}[\sigma_{\eta,i}^{z}]\rho_{\rm
q}+\mathcal{D}[\tilde{J}_{A}]\rho_{\rm q}+\mathcal{D}[\tilde{J}_{B}]\rho_{\rm
q},\end{split}$ (S20)

with

$H_{\rm
casc}=\sum_{\eta,i}\frac{\delta_{\eta,i}}{2}\sigma_{\eta,i}^{z}+\frac{i}{2}\sum_{\eta,j>i}\sqrt{\gamma_{\eta,i}\gamma_{\eta,j}}\left(\sigma_{\eta,i}^{+}\sigma_{\eta,j}^{-}-\sigma_{\eta,j}^{+}\sigma_{\eta,i}^{-}\right).$
(S21)

For identical decay rates, this result reduces to the master equation given in
Eq. (1) in the main text. When general losses are included, we no longer
obtain such a simple form, but we can still use Eq. (S15) for numerical
simulations.

###  I.3 Propagation times

In the setup, we consider the photons with group velocity $c_{\eta}$ take the
time $\tau_{\eta,i}=d_{\eta,i}/c_{\eta}$ to propagate the distance
$d_{\eta,i}$ between the source and the $i$-th qubit in waveguide $\eta$. In a
strict sense, the validity of Eq. (S1) assumes that all these times are
negligible compared to the typical timescale of the system evolution. However,
due to the cascaded nature of the interaction, the cascaded master equation in
Eq. (S1) can still be used to evaluate expectation values for arbitrary large
networks. For example, to evaluate steady-state expectation values of the form
$\langle O_{\eta,i}O_{\eta^{\prime},j}\rangle_{0}$ in the case of non-
negligible $\tau_{\eta,i}$, we can use the prescription [6]

$\langle O_{\eta,i}O_{\eta^{\prime},j}\rangle_{0}=\langle
O_{\eta,i}(\tau_{\eta,i}-\tau_{\eta^{\prime},j})O_{\eta^{\prime},j}\rangle|_{\rm
Eq.\,(S1)},$ (S22)

where the two-time correlation function on the right-hand side is evaluated
with the help of Eq. (S1). The same applies to the effective qubit master
equation in Eq. (S20), which is only a reduced version of Eq. (S1). In
particular, this means that the qubits can be separated from the source by an
arbitrary distance, as long as the relative propagation times satisfy

$\gamma|\tau_{\eta,i}-\tau_{\eta^{\prime},j}|\ll 1.$ (S23)

We refer the reader to Ref. [6] for a more detailed discussion about the
influence of finite time delays on the achievable steady-state entanglement.

## Uniqueness of the steady state

In the main text we have studied the steady states of the effective master
equation for $N$ qubit pairs,

$\dot{\rho}_{\rm q}=\mathcal{L}_{N}\rho_{\rm q}=-i[H_{\rm casc},\rho_{\rm
q}]+\sum_{\eta=A,B}\gamma\mathcal{D}[J_{\eta}]\rho_{\rm q}.$ (S24)

It is clear that given a state $|\psi_{0}\rangle$ that satisfies the dark-
state conditions $J_{\eta}|\psi_{0}\rangle=0$ and $H_{\rm
casc}|\psi_{0}\rangle=0$, the density operator
$\rho_{0}=|\psi_{0}\rangle\langle\psi_{0}|$ is a pure steady state of this
master equation, i.e. $\mathcal{L}_{N}\rho_{0}=0$. However, this condition
does not guarantee that $\rho_{0}$ is the unique steady state and there could
be other mixed or pure states $\rho_{0}^{\prime}$ with
$\mathcal{L}_{N}\rho_{0}^{\prime}=0$. The actual steady state of an ideal
network would then depend on the precise initial condition, while in practice
residual imperfections would create an uncontrolled mixture of multiple
possible steady states. This would be detrimental for entanglement generation.

### N=1

To prove that the state $|\psi_{0}(r,\vec{\delta}_{A},P)\rangle$ defined in
Eq. (9) of the main text is indeed the unique steady state of the network for
a given detuning pattern $\vec{\delta}_{A}$ and permutation $P$, we start with
the case $N=1$ and $\delta_{A,1}=-\delta_{B,1}$. In this case, we can
calculate the eigenvalues of the Liouvillian $\mathcal{L}_{N=1}$ analytically
and verify that for any finite $r$ there is only a single eigenvalue
$\lambda_{0}=0$, which corresponds to the state
$\rho_{0}^{(N=1)}=|\Phi_{1,1}^{+}\rangle\langle\Phi_{1,1}^{+}|$. We also find
that the smallest non-zero eigenvalue is $\lambda_{1}=\gamma\cosh(2r)/2$ for
$r<r^{*}$ and $\lambda_{1}=\gamma(6\cosh(2r)-\sqrt{18\cosh(4r)-14})/4$ for
$r>r^{*}$, where $r^{*}\simeq 0.356$. This eigenvalue determines the gap in
the Liouvillian spectrum and, therefore, for any finite $r$ there is a finite
relaxation rate toward the steady state.

###  I.4 Induction step

We now assume that we already know that the product state
$\rho_{0}^{(N)}=|\Phi_{\parallel}(N)\rangle\langle\Phi_{\parallel}(N)|$
defined in Eq. (6) of the main text is the unique steady state of
$\mathcal{L}_{N}$ for $\vec{\delta}_{A,i}=-\vec{\delta}_{B,i}$ and that it
satisfies the dark-state conditions $J_{\eta}|\Phi_{\parallel}(N)\rangle=0$
and $H_{\rm casc}|\Phi_{\parallel}(N)\rangle=0$. We now show that under this
assumption it is also true that $\rho_{0}^{(N+1)}$ is the unique steady state
of the network with $N+1$ qubit pairs.

Let us first verify that $|\Phi_{\parallel}(N+1)\rangle$ is a dark state. The
conditions $J_{\eta}|\Phi_{\parallel}(N+1)\rangle=0$ are straightforward to
verify, since it holds for each qubit pair individually. For the second
condition, $H_{\rm casc}|\Phi_{\parallel}(N+1)\rangle=0$, we write the
cascaded Hamiltonian as

$H_{\rm casc}^{(N+1)}=H_{\rm
casc}^{(N)}-i\frac{\gamma}{2}\sum_{\eta}\left(L_{\eta}(N)\sigma_{\eta,N+1}^{+}-L^{\dagger}_{\eta}(N)\sigma_{\eta,N+1}\right),$
(S25)

and recall that
$|\Phi_{\parallel}(N+1)\rangle\sim|\Phi_{\parallel}(N)\rangle\otimes(\cosh(r)|0_{A,N+1}\rangle|0_{B,N+1}\rangle+\sinh(r)|1_{A,N+1}\rangle|1_{B,N+1}\rangle)$.
It follows that

$\begin{split}H_{\rm
casc}^{(N+1)}|\Phi_{\parallel}(N+1)\rangle\sim&\left[\cosh(r)L_{A}(N)-\sinh(r)L^{\dagger}_{B}(N)\right]|\Phi_{\parallel}(N)\rangle\otimes|1_{A,N+1}\rangle|0_{B,N+1}\rangle\\\
&+\left[\cosh(r)L_{B}(N)-\sinh(r)L^{\dagger}_{A}(N)\right]|\Phi_{\parallel}(N)\rangle\otimes|0_{A,N+1}\rangle|1_{B,N+1}\rangle=0.\end{split}$
(S26)

To prove that $\rho_{0}^{(N+1)}$ is also the unique steady state, we use the
fact that in a fully directional network the reduced steady state of the first
$N$ pairs of qubits, $\rho_{0}^{(N)}={\rm Tr}_{i=N+1}\\{\rho_{0}^{(N+1)}\\}$
is unaffected by adding an additional pair. Further, because $\rho_{0}^{(N)}$
is pure, there is no entanglement between the subsystems and we can write
$\rho_{0}^{(N+1)}=\rho_{0}^{(N)}\otimes\rho^{(x)}_{0}$, with a steady state
$\rho^{(x)}_{0}$ for the last pair, which still must be determined (to
simplify notation, we use the index $x$ to refer to the extra qubit pair with
index $i=N+1$). To do so we write

$\begin{split}\dot{\rho}_{0}^{(N+1)}=&\mathcal{L}_{N}\rho_{0}^{(N)}\otimes\rho^{(x)}_{0}+\rho_{0}^{(N)}\otimes\mathcal{L}_{x}\rho^{(x)}_{0}+\mathcal{L}_{N-x}\rho_{0}^{(N+1)}.\end{split}$
(S27)

Here $\mathcal{L}_{x}\hat{=}\mathcal{L}_{N=1}$ is the single-pair Liouville
operator acting on the state of the last qubit pair and

$\begin{split}\mathcal{L}_{N-x}\rho_{0}^{(N+1)}=&-\frac{\gamma}{2}\sum_{\eta=A,B}\left[
L_{\eta}(N)\sigma_{\eta,x}^{+}-
L^{\dagger}_{\eta}(N)\sigma_{\eta,x}^{-},\rho_{0}^{(N+1)}\right] \\\
&-\frac{\gamma}{2}\left\\{J^{\dagger}_{A}(N)\left(\cosh(r)\sigma_{A,x}^{-}-\sinh(r)\sigma_{B,x}^{+}\right)+J^{\dagger}_{B}(N)\left(\cosh(r)\sigma_{B,x}^{-}-\sinh(r)\sigma_{A,x}^{+}\right),\rho_{0}^{(N+1)}\right\\}_{+}\end{split}$
(S28)

accounts for the remaining cross terms. Note that here we have already used
that $J_{\eta}(N)\rho_{0}^{(N+1)}=\rho_{0}^{(N+1)}J_{\eta}^{\dagger}=0$ due to
the dark-state condition for $|\Phi_{\parallel}(N)\rangle$. By looking at all
the different contributions in Eq. (S28) we can collect terms such as

$\begin{split}\left[-L_{A}(N)+\sinh(r)J_{B}^{\dagger}(N)\right]
\sigma_{A,x}^{+}\rho_{0}^{(N+1)}=&\left[-L_{A}(N)+\sinh(r)\cosh(r)L_{B}^{\dagger}(N)-\sinh^{2}(r)L_{A}(N)\right]
\sigma_{A,x}^{+}\rho_{0}^{(N+1)}\\\
=&-\cosh(r)\underbrace{\left[\cosh(r)L_{A}(N)-\sinh(r)L_{B}^{\dagger}(N)\right]}_{=J_{A}(N)}
\sigma_{A,x}^{+}\rho_{0}^{(N+1)}=0,\end{split}$ (S29)

and find that they vanish independently of $\rho^{(x)}_{0}$. The same is true
for other combinations such that $\mathcal{L}_{N-x}\rho_{0}^{(N+1)}=0$.
Therefore, when tracing over the first $N$ qubit pairs, the steady state
$\rho_{0}^{(x)}$ satisfies

$\rho_{0}^{(x)}=\mathcal{L}_{x}\rho_{0}^{(x)}=0,$ (S30)

which has a unique solution given by
$\rho_{0}^{(x)}=|\Phi^{+}_{x}\rangle\langle\Phi^{+}_{x}|$.

### General detunings

Finally, we consider non-trivial detuning patterns
$\vec{\delta}_{B}=-P\vec{\delta}_{A}$ and show that also in this case the
steady state
$\rho^{(N)}_{0}=|\psi_{0}(r,\vec{\delta}_{A},P)\rangle\langle\psi_{0}(r,\vec{\delta}_{A},P)|$
is unique. This can be done by simply assuming that there is another steady
state $\rho_{0}^{\prime}\neq\rho^{(N)}_{0}$ of the Liouvillian
$\mathcal{L}_{N}$. Then we can simply invert the arguments about the form
invariance of the master equation presented in the main text below Eq. (8) and
obtain a steady state for the network with
$\vec{\delta}_{B}=-\vec{\delta}_{A}$,

$\rho_{0}^{\prime}(\vec{\delta}_{B}=-\vec{\delta}_{A})=\mathcal{U}^{\dagger}\rho_{0}^{\prime}\mathcal{U},\qquad\mathcal{U}=\prod_{\sigma}U_{i_{\sigma},i_{\sigma}+1}.$
(S31)

However, since we know that there is only one unique steady state for this
detuning pattern, it means that
$\rho_{0}^{\prime}(\vec{\delta}_{B}=-\vec{\delta}_{A})=|\Phi_{\parallel}(N)\rangle\langle\Phi_{\parallel}(N)|$
and
$\rho_{0}^{\prime}=|\psi_{0}(r,\vec{\delta}_{A},P)\rangle\langle\psi_{0}(r,\vec{\delta}_{A},P)|$.

## Scaling of entanglement in realistic networks

In the main text we have shown that in a realistic setup, the entanglement
along the chain is degraded by finite-bandwidth effects and qubit dephasing.
In this section, we provide some additional numerical results on the
dependence of $N_{\rm ent}$ on the squeezing strength and on the influence of
waveguide losses and imperfect chiral couplings.

### Squeezing strength

In Fig. S1 (a), we plot $N_{\rm ent}$ as a function of $\beta=\kappa/\gamma$
for a small squeezing parameter of $r=0.2$. Compared to Fig. 4(b) in the main
text, the decay of entanglement along the chain is much slower and also more
robust with respect to qubit dephasing. Note, however, that the maximal
concurrence of $\mathcal{C}_{11}\approx 0.38$ is also much lower than for
$r=1$. In Fig. S1 (b) we fix the value of $\beta=30$ and plot $N_{\rm ent}$ as
a function of $r$. This plot shows the expected trade-off between a high
degree of entanglement and the number of entangled pairs.

Figure S1: Maximal number of entangled pairs, $N_{\rm ent}$, as a function of
(a) $\beta=\kappa/\gamma$ for different dephasing rates $\gamma_{\phi}$ at
fixed squeezing strength $r=0.2$ and (b) as a function of the squeezing
strength $r$ for the same dephasing rates $\gamma_{\phi}$ at fixed $\beta=30$.
For both plots, we have assumed $\vec{\delta}_{A}$=0. Figure S2: (a) Effect of
finite propagation losses $\epsilon>0$ on the steady-state entanglement for
$N=4$ for (a) squeezing strength $r=1$ and on resonance $\Delta=0$, (b) for
$r=1$ and parallel configuration with detuning $\Delta=2\gamma$, and (c) for
smaller squeezing strength $r=0.5$ and on resonance $\Delta=0$. For a small
value of $\epsilon$, this plot predicts an approximately linear decay of the
entanglement along the waveguide. For all plots,
$\epsilon^{\eta}_{i,j}=\epsilon|i-j|$ and $\beta\rightarrow\infty$ have been
assumed.

### Waveguide losses

In Fig. S2, we consider a chain of $N=5$ pairs of qubits and simulate the
effect of losses along the waveguides. For concreteness, we assume
$\epsilon^{\eta}_{i,j}=\epsilon(i-j)$, which means that there is a fixed loss
probability $|\epsilon|$ between two successive nodes of the network.

### Nonideal chiral coupling

Throughout the main text, we have considered the waveguide to be completely
directional, that is, all the photons propagate along the same direction.
While this allows us to get analytical results, realizing such directional
interactions will only be possible with a certain fidelity. Here we present
additional numerical results for waveguides, the qubits along the waveguides
can decay into left-propagating modes with rate $\gamma_{L}$ and into right-
propagating modes with rate $\gamma_{R}$. The main results are then recovered
when $\gamma_{L}=0$ and $\gamma_{R}=\gamma$.

To extend our model to a bi-directional waveguide, we include the effect of an
additional left propagating channel into our effective qubit master equation
(see, e.g., Ref. [7]). We obtain

$\dot{\rho}_{\rm q}=-i[H_{\rm chiral},\rho_{\rm
q}]+\sum_{\eta=A,B}\gamma_{R}\mathcal{D}[J_{\eta}]\rho_{\rm
q}+\sum_{\eta=A,B}\gamma_{L}\mathcal{D}[L_{\eta}]\rho_{\rm q}.$ (S32)

The new modes contribute to both the coherent and incoherent interaction. The
coherent interaction now depends on the difference between left- and right-
modes and vanishes at a completely bi-directional waveguide,

$H_{\rm
chiral}=\frac{i(\gamma_{R}-\gamma_{L})}{2}\sum_{\eta,j>i}(\sigma^{+}_{\eta,i}\sigma^{-}_{\eta,j}-{\rm
H.c}).$ (S33)

For the incoherent term, we have assumed that the left-propagating modes decay
into vacuum modes with a collective jump operator
$L_{\eta}=\sum_{i=1}^{N}\sigma_{\eta,i}$. Therefore, only the right-
propagating modes are squeezed and correlated. Note that this form also
assumes that the qubits are spaced by multiples of the central wavelength,
such that all propagation phases cancel.

In Fig. S3, we numerically solve this master equation for different degrees of
chirality, $\gamma_{L}/\gamma_{R}$. We observe that the effect of a finite
$\gamma_{L}$ on the resulting steady state depends a lot on the type of
entanglement, which in turn is determined by the detunings. While multipartite
entangled states are strongly affected by a finite bi-directional coupling,
bipartite entangled states are more robust, and in the far-detuned regime, a
finite amount of entanglement survives up to $\gamma_{L}/\gamma_{R}=1$.

Figure S3: Plot of the bipartite concurrence of the steady state of the master
equation in Eq. (S32), which includes a decay into left-propagating waveguide
modes with rate $\gamma_{L}$. The plots in (a) and (d) assume parallel
detunings with $\vec{\delta_{B}}=-\vec{\delta_{A}}$ and
$\delta_{A,i}=\Delta(i-1)$, while the plots in (b), (c), (e) and (f) assume a
reversed detuning pattern $\vec{\delta_{B}}=-P_{\rm rev}\vec{\delta_{A}}$. In
all plots $r=1$.

### Example: Microwave quantum networks

To illustrate the performance of this protocol in a realistic setup, where all
types of imperfections are taken into account, we consider in this subsection
the example of superconducting qubits connected via microwave transmission
lines.

As a starting point, we use the parameters from a recent work by Joshi et al.
[8], which describes the realization of a chiral coupling of a superconducting
qubit to a microwave waveguide. From this reference, we deduce a qubit
dephasing rate of $\gamma_{\phi}/2\pi=50$ kHz (in accordance with other state
of the art experiments [9] ), a directional emission rate of
$\gamma_{R}/2\pi\simeq 1$ MHz and an unwanted decay into the opposite
direction with rate $\gamma_{L}/\gamma_{R}\simeq 0.01$. In addition, in this
experiment, there is a residual decay into non-guided modes with a rate
$\gamma^{\prime}/2\pi=364$ kHz. In our numerical simulations, we include this
process by adding a new term
$\gamma^{\prime}\sum_{\eta,i}\mathcal{D}[\sigma_{\eta,i}]\rho$ to our master
equation. For these parameters, $\Delta=0$ and assuming an ideal two-mode
squeezing source with $r=1$, we obtain $N_{\rm ent}\simeq 1$ and the
concurrence of the first pair is $\mathcal{C}_{11}\simeq 0.1$. Obviously, this
poor result is mainly related to the large residual decay rate
$\gamma^{\prime}$. By assuming that this decay channel can be eliminated in
future setups, $\gamma^{\prime}\rightarrow 0$ [10], the result improves to
$N_{\rm ent}\simeq 2$ and $\mathcal{C}_{11}\simeq 0.53$, now being primarily
limited by decoherence with rate $\gamma_{\phi}/\gamma\simeq 0.05$.

Let us now consider the same parameters, but assuming the finite detunings
$\delta_{A}=(i-1)\Delta$ with $\Delta=\gamma_{R}$. Consistent with Fig. 4 (d)
in the main text, we find that while keeping $\mathcal{C}_{11}\simeq 0.53$,
this modification would already boost the total number of entangled pairs to
about $N_{\rm ent}\simeq 20$ (assuming $\gamma^{\prime}=0$). Further, by
improving the ratio $\gamma_{\phi}/\gamma_{R}$ by a factor of ten (which is
well within the range of typical qubit coherence times) would boost this
number to about $N_{\rm ent}\simeq 120$ (with $\Delta=\gamma_{R}$) and $N_{\rm
ent}\simeq 6$ (with $\Delta=0$) and with $C_{11}\simeq 0.85$. At this stage
the effects of a finite rate $\gamma_{L}$ become relevant.

Let us now address the effect of a finite amplifier bandwidth. In the
microwave regime, two-mode squeezing sources are usually realized with
Josephson parametric amplifiers (JPAs) or travelling wave parametric
amplifiers (TWPAs). Typical bandwidths for these devices are $\kappa_{\rm
JPA}\simeq 2\pi\times 10$ MHz [11] and $\kappa_{\rm TWPA}\simeq 2\pi\times 1$
GHz [12, 13]. Combining the parameters from above with the JPA, the relevant
ratio between the amplifier bandwidth and the qubit decay rate is $\beta_{\rm
JPA}=\kappa_{\rm JPA}/\gamma_{R}\simeq 10$. In this case, the finite bandwidth
does not change the conclusion from above for $\Delta=0$ and we obtain $N_{\rm
ent}\simeq 2$ with $C_{11}=0.45$ for $\gamma^{\prime}=0$. For
$\Delta=\gamma_{R}$ our extrapolation predicts $N_{\rm ent}\approx 50$, but
since we must limit the maximum detuning to $\delta_{\rm max}<\kappa$, the
limit in this example is set by $N_{\rm ent}\approx\beta\simeq 10$.

To go beyond this limit, we can use a TWPA. In this case, the bandwidth ratio
can reach values up to $\beta_{\rm TWPA}=\kappa_{\rm TWPA}/\gamma_{R}\simeq
10^{3}$ and all the results for $N_{\rm ent}$ and $\mathcal{C}_{11}$ reduce to
the infinite-bandwidth results from above. Note, however, that this assumes an
ideal amplifier without any added noise.

In summary, these estimates show that while the preparation of highly
entangled multi-qubit states naturally requires sophisticated experimental
setups, existing experimental techniques in the field of superconducting
circuits are in principle already enough to demonstrate the simultaneous
entanglement of $N\approx 2-10$ qubit pairs or generate multipartite
entanglement among $\sim 4-8$ separated qubits.

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