light of recent success of applying network theory to correlation networks of
spin chains to predict their phase transitions, as explained in more detail in
the next Section. A related research avenue could be to consider probing the
partial, mesoscopic structure of a network in the ETH phase along the lines
of, e.g., [216, 217].

##  V Network representation of quantum systems

###  V.1 Networks for taming quantum complexity

Recently it has emerged that networks are a very powerful mathematical and
computational tool to tame quantum complexity. Therefore in this Section we
shift perspective with respect to the previous Section. Indeed, while in the
previous Section networks have been used to encode for the physical
interactions of quantum systems, here networks are adopted as their
mathematical and abstract representations. This research line corresponds to
the quantum-applied block of Fig. 1. The works summarized in the Section
generally assume that a quantum system can be represented as a network
according to a suitable rule of the form

$G=G(\mathbf{M},\mathbf{H},\mathbf{t},\boldsymbol{\rho}_{0}),$ (10)

where $\mathbf{M}$ can indicate a set of measurements or instruments,
$\mathbf{H}$ a set of relevant Hamiltonians, $\mathbf{t}$ a set of relevant
interaction times and $\boldsymbol{\rho}_{0}$ a set of initial states. It
should be stressed that Eq. (10) is intended to be illustrative in nature,
rather than a rigorous definition. Whereas complexity of the physical networks
in the previous Section is encoded by the network $G$ characterizing the
interactions captured by the Hamiltonian, here the graph $G$ is a
representation of the quantum system itself whose complexity might arise in a
nontrivial and sometimes even surprising manner. Of particular interest are
cases where the topology of the network reflects some properties of interest
of the underlying physical system. In this case network theory can help tame
the complexity of the considered system.

Due to the general nature of Eq. (10) the networks do not need to indicate
physical interactions and might represent more abstract relations. For
example, a linear optical setup can be described as a network by taking the
nodes to be states and the links to be optical paths weighted by state
transition amplitudes. Hence this approach can naturally lead to a directed
network with complex link weights [218]. Moreover, one could also adopt a
colored network approach where networks describing different experiments might
be colored according to the used optical setup [219], phase shift [220] or
mode number [221]. Forming instead a network out of pairwise correlation
measures to represent some quantum state leads to real but typically
nonuniform weights which might reflect both the structure of the Hamiltonian
and the phase of the system [24, 14, 222], or give rise to multiplex networks
where each layer is associated with a particular measure [223]. It should be
noted that in such cases the physical network determined by the Hamiltonian
need not to be complex. In fact it will be seen that even chains and square
lattices of quantum systems can give rise to complexity in the ground state
that can be tackled with the help of a suitably defined network. The networks
might also be constructed in such a way that certain transformations of the
underlying state translate to simple transformation rules of the network, thus
demonstrating the utility of networks to represent quantum dynamics.

The motivations to introduce such networks virtually always revolve around
using the networks as a convenient mathematical tool to characterize, explain
or simplify the quantum system under consideration. In the following we
present examples of such research line to further illustrate the concept.

###  V.2 Applications and examples

####  V.2.1 Network description of states

A complete description of a generic quantum state $\rho$ depends on the
Hilbert space dimension which grows rapidly with the number of systems
involved. In recent years networks have been proposed as an alternative ways
to describe a quantum state. Most of the approaches use networks that encode
pairwise correlations in the weights of the links. Such a possibility is
attractive not only because the networks scale only quadratically with the
number of their nodes, typically taken to be the systems, but also because in
the case of pairwise measures tomography of the correlation network is
drastically cheaper than full state tomography [223]. Typically a transition
in the network structure, captured by some appropriate network measure, can be
linked to a physical transition. Such a link may then be used to both detect
known transitions or discover novel phenomena and characterize them through
the network picture. Although information will be lost as the network is not
formed using the full state, a plethora of cases have been identified where
the network approach is accurate.

Such use of weighted networks based on (von Neumann) mutual information was
introduced in [24], which considered detecting quantum critical points of the
transverse field Ising chain from the properties of the correlation network of
its ground state. Specifically, the link weight $\mathcal{I}_{ij}$ between
some qubits $i$ and $j$ reads

$\mathcal{I}_{ij}=\frac{1}{2}(S_{i}+S_{j}-S_{ij})=\frac{1}{2}(-\mathrm{Tr}(\rho_{i}\log_{2}\rho_{i})-\mathrm{Tr}(\rho_{j}\log_{2}\rho_{j})+\mathrm{Tr}(\rho_{ij}\log_{2}\rho_{ij}))$
(11)

where $S_{i}$, $S_{j}$ and $S_{ij}$ are marginal von Neumann entropies and
$\rho_{i}$, $\rho_{j}$, $\rho_{ij}$ reduced density matrices of the qubits
individually and together, respectively. This work considered as network
measures disparity, (global) clustering coefficient, density and similarity
between nodes quantified by Pearson correlation coefficient and found that
their first and second derivatives or local minima revealed the points with
state-of-the-art performance when compared to standard measures for both
transverse field Ising and Bose-Hubbard models and three classes of quantum
phase transitions. The behavior of the measures against a model parameter is
shown in Fig. 9. For example, in the ferromagnetic phase of the Ising model a
spin in the chain is correlated with many close spins with similar strength,
whereas in the paramagnetic phase a spin is correlated mostly with its nearest
neighbors. This transition to short range correlations is captured by the
disparity which was shown to exhibit an abrupt change in behavior near the
critical point, remaining very small in the ferromagnetic phase but starting
to grow with field strength in the paramagnetic phase.

\begin{overpic}[width=433.62pt,unit=1mm]{figures/Figure9A.pdf}
\put(2.0,2.0){(a)} \end{overpic}

\begin{overpic}[width=433.62pt,unit=1mm]{figures/Figure9B.pdf}
\put(2.0,2.0){(b)} \end{overpic}

Figure 9: Normalized network measures calculated from the mutual information
network of the ground state as a function of a parameter in the Hamiltonian
for several system sizes. Shown are link density (dashed black line),
clustering coefficient (solid red line), Pearson correlation coefficient
(dashed green line) and disparity (dot-dashed blue line); arrows indicate the
direction in which system size is increased. $(a)$ The transverse-field Ising
chain passes from para- to ferro-magnetic phase as field strength increases.
All measures display an abrupt change in behavior at the transition which here
takes place near $g=1$. $(b)$ Bose-Hubbard chain passes from Mott insulator to
superconducting phase as the ratio between particle tunneling $J$ and on-site
particle interaction $U$ increases, however the transition is not sharp at the
considered system sizes. Nevertheless the measures are clearly useful for
identifying the boundary, which should be near $J/U=0.3$. Reprinted figure
with permission from [24]. Copyright 2017 by the American Physical Society.

Mutual information networks have been applied also in the case of thermal
states. In [222] mutual information networks were applied for a Fermi-Hubbard
model on a square lattice considering the same measures as in [24]. The
behavior of the measures were suggested to be connected to the appearance of
the pseudogap phase. In [14] a transverse field Ising chain was considered
using also other correlation measures such as Rényi mutual information,
concurrence and negativity and as network measures the ones of [24] except
node similarity and additionally betweenness centrality, average geodesic
distance and diameter. The gradients of these measures were found to exhibit
extrema at the transition when temperature and field strength were varied.

Concurrence in particular was applied to study entanglement networks of the
ground state of an XX spin chain in [25]. Concurrence $C(\rho)$ is a measure
of entanglement between two two-level systems with the density matrix $\rho$.
It reads

$C(\rho)=\max(0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4})$ (12)

where $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$ and $\lambda_{4}$ are the
eigenvalues, in decreasing order, of the matrix

$\mathbf{R}=\sqrt{\sqrt{\rho}(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})\sqrt{\rho}}$
(13)

where $\sigma_{y}$ is one of the Pauli spin matrices and $\rho^{*}$ is the
complex conjugate of $\rho$. This work considered both weighted and unweighted
variants of degree and local clustering coefficient as well as disparity, and
going beyond microscopic structure also the communities as shown in Fig. 10.
The network approach was found to reveal new phenomena such as instability of
pairwise entanglement with respect to perturbations in the magnetic field
strength or community structure in the entanglement network reflecting a
global symmetry in the system. Results in a similar vein have been reported
also for the quantum critical points of the Kitaev chain in [224]—whereas the
clustering of the mutual information network witnessed the previously known
transition from topological to trivial phase, the clustering in the
concurrence network revealed previously unknown critical points where
entanglement no longer decays with distance.

Figure 10: Concurrence networks calculated from the ground state of the XX
spin chain for different values of the magnetic field strength $B_{k}$. Each
node is a spin and links indicate concurrence; node size is proportional to
the total weighted degree and link width and color to magnitude of
concurrence. Nodes are colored according to detected community. The
relationship between the community structure and $B_{k}$ is evident. Used with
permission of The Royal Society (U.K.), from [25]; permission conveyed through
Copyright Clearance Center, Inc.

The different networks arising from multi-qubit states were unified into a
single mathematical object in [223], which introduced the concepts of pairwise
tomography networks and quantum tomography multiplexes as well as an efficient
scheme to construct them with measurements. In a pairwise tomography network
nodes are qubits and the links are the associated two qubit reduced density
matrices. The presented scheme allows the construction of such a network using
a number of measurement settings that scales only logarithmically with the
number of qubits. The pairwise tomography network determines then the quantum
tomography multiplex where in each layer the links are weighted by some
pairwise quantifier computed from the corresponding two qubit density matrix.

Although the continuous-variables case has received less attention, mutual
information networks were considered as early as 2013 for quantum harmonic
oscillators in [194] which considered the interplay between the correlation
and interaction networks and established how the latter leaves its fingerprint
on the former. Although the study focused on ground states the relation was
found to be robust to small temperatures. More recently, the impact of local
operations on the structure of a correlation network has been considered
[225]. The starting point was a Gaussian cluster state with an embedded
network structure. If $\mathbf{A}$ is the weighted adjacency matrix of the
network, then the corresponding ideal continuous variable cluster state can be
made from the product state of $N$ momentum eigenstates with eigenvalue $0$ by
acting on some modes $i$ and $j$ with the $C_{Z}$ gate
$\exp{(\mathrm{i}gq_{j}q_{k})}$ if they are connected and where $g$ is the
link weight in $\mathbf{A}$, i.e.

$\left|\psi_{\mathbf{A}}\right>=C_{Z}[\mathbf{A}]\left|0\right>_{p}^{\otimes
N}=\prod_{j,k}^{N}\exp{\left(\frac{\mathrm{i}}{2}\mathbf{A}_{jk}q_{j}q_{k}\right)}\left|0\right>_{p}^{\otimes
N}.$ (14)

While networks of arbitrary topology can be created deterministically, the
states $\left|0\right>_{p}$ must in practice be approximated. For instance, a
way to achieve this is by approximate them by squeezed vacuum states with
small but finite variance for $p$. Experimental demonstration of a
computational advantage with such states as a resource has been achieved with
non-Gaussian measurements [226], however with only Gaussian operations
efficient classical simulation is always possible [227]. Here multi-photon
subtractions were considered, de-Gaussifying the state. The state both before
and after the operation can also be presented in terms of photon number
correlations between the modes; each gate creates such correlations between
also the modes adjacent to the target modes—next nearest neighbors—and unlike
the cluster state this network has continuous weights between $0$ and $1$.
Firstly, the effect of moving from cluster state to correlation network was
analyzed. Increase of local clustering coefficient was observed for the
Barabási-Albert network, whereas for the Watts–Strogatz network increasing the
rewiring probability was found to decrease both degrees and clustering. In
general, local multi-photon subtraction was found to increase both the mean
degree and the variance and have limited range as biggest effect was on nodes
up to two hops away and beyond four hops there was no effect. The impact on
higher moments of the degree distribution was sensitive both to the network
class, parameter values and the choice of the node however, highlighting also
the importance of the topology of the local neighborhood. For example,
choosing a low degree node causes a large increase in the correlations in a
relatively small subgraph whereas choosing a high degree node modifies a large
subgraph but the effect on each individual link is smaller.

We also mention recent results linking the squeezing cost of setting up a
Gaussian cluster state to the spectrum of the matrix $\mathbf{A}$ [228], which
are an exact generalization of earlier results of [229] from the large
squeezing limit to any squeezing. Importantly, the results imply that co-
spectral networks have the same cost and consequently form an equivalence
class of cluster states that can be changed into each other applying only
passive linear optics. The relationship between cost and topology was also
studied, revealing how the scaling with size is strictly topology dependent.

Going beyond correlations, networks have been also constructed based on
pairing amplitude in topological superconductors [230] and electron wave
function overlap in quantum dot systems [167]. To the best of our knowledge
the former work, Ref. [230], was in fact among the earliest to investigate
network measures on induced weighted networks as a tool to facilitate
understanding, namely to detect topological phase transitions. Instead, the
latter work, Ref. [167], proposes to model quantum dots randomly distributed
on a plane as random geometric graphs and considers network measures such as
degree distribution, clustering and average geodesic distance to explain
phenomena such as the emergence of transport.

Before concluding we mention the tensor networks formalism, a wholly different
approach to network characterization of states. The main idea is to use
networks of tensors connected by contractions to efficiently represent
physically relevant states [231, 232, 233, 234], such as low-energy
eigenstates of Hamiltonians with local interactions and a finite gap between
the ground state energy and first excited state energy. The formalism takes
advantage of the limited amount of entanglement in these states, and it may be
argued that the overwhelming majority of the inapplicable states are in fact
of little practical interest. Such tensor networks lend themselves to
diagrammatic manipulation which can be used to reason about the state and find
ground states of suitable Hamiltonians when combined with suitable numerical
techniques. Although immensely useful, widely applicable and enjoying a strong
interest tensor networks are typically lattices and the complex network aspect
at least in the research carried out so far is overall weak, making them a
borderline case with respect to the classification proposed in Fig. 1 which is
why they are not considered further here.

####  V.2.2 Network description of experimental data sets

To apply the results of the previously introduced networks on an unknown
state, one needs to perform tomography to estimate with sufficient accurary a
suitable bipartite correlation measure that then constitutes the links. In
practice one accumulates experimental data by carrying out measurements from a
tomographically complete set on a large ensemble of identically prepared
systems such that one approaches asymptotically the actual values as the
ensemble size increases, and therefore the actual network.

Very recently a more direct approach has been introduced in Ref. [15] where
the network is constructed directly out of the experimental data set based on
just a single measurement setting, i.e. projecting a pure many-body state into
a fixed basis. The outcomes are called wave function snapshots and they can be
probed experimentally as well as numerically; for qubits each is a binary
string. The wave function network is constructed out of the snapshots by
treating them as nodes, defining a suitable metric (such as the Hamming or the
Euclidean distance) and an upper limit $R$ for distance and then creating a
metric network where the nodes are connected if their distance is less than
$R$ and disconnected otherwise.

Importantly, these choices were shown to generate nontrivial and informative
networks. This was exemplified with an Ising model undergoing a quantum phase
transition from disordered to ferromagnetic phase where consequently the
network degree distribution experiences a transition from a Poisson to scale-
free distribution. When applied to data obtained from a Rydberg quantum
simulator the network description facilitated the estimation of the Kolmogorov
complexity of the simulator output: the degree distribution remained scale-
free in which case efficient algorithms can be used on the network, showing a
non-monotonic evolution of the complexity. This is remarkable because for
generic strings, finding the Kolmogorov complexity—essentially quantifying how
difficult it would be to generate them with a classical simulator—is a NP-hard
problem. Finally, a cross-certification method based on network similarity was
proposed, allowing one to determine whether two devices sample from the same
probability distribution by comparing the network degree distributions. The
method was demonstrated by comparing the outcomes of two experiments as well
as an experiment and a simulation. An interesting research question arises
from the comparisons of these networks with the recently introduced IsingNets
networks [235] constructed from configuration snapshots of classical Ising
models. In particular this comparison will be key to determine the exclusive
signature of quantumness in the quantum wave function networks.

####  V.2.3 Network description of experimental setups

Previously little attention was given to how the state was or could be
created. Shifting focus to this quite naturally leads to the concept of
interpreting experimental setups as networks associated with graphs. This can
facilitate intuitive and convenient ways to determine how the state transforms
from the initial to the final form, somewhat akin to Feynman diagrams.
Moreover, this approach can link experiments to graph theory. In particular,
graph theoretical methods may be used to answer experimental questions and
experimental methods used to answer graph theoretical questions. Both avenues
have been followed particularly in the case of quantum optics.

For an example of the former, Ref. [236] has introduced a method to transform
a linear lossless device consisting of beam splitters and interferometers into
a directed tree connecting input field operators (the roots) to output field
operators (the leaves) by optical paths weighted by complex probability
amplitudes. Specifically, assuming two input ports and two output ports and
monochromatic light in a pure state for simplicity, the input state can be
expressed as

$\left|\psi_{in}\right>=f(a^{\dagger}_{0},a^{\dagger}_{1})\left|0\right>$ (15)

where the subscripts stand for input port $0$ and input port $1$. Suppose the
output ports are labeled $N$ and $N+1$. If one can find functions such that
$a^{\dagger}_{0}=g_{0}(a^{\dagger}_{N},a^{\dagger}_{N+1})$ and
$a^{\dagger}_{1}=g_{1}(a^{\dagger}_{N},a^{\dagger}_{N+1})$, then one can write
the output state as

$\left|\psi_{out}\right>=f(g_{0}(a^{\dagger}_{N},a^{\dagger}_{N+1}),g_{1}(a^{\dagger}_{N},a^{\dagger}_{N+1}))\left|0\right>.$
(16)

After replacing the optical elements with their corresponding graph elements,
an output operator can be computed by simply following every directed path
from the roots to it multiplying the amplitudes on a path and summing the
products of amplitudes of different paths. Hence, this procedure facilitates
the extraction of the sought functions $g_{0}$ and $g_{1}$. Inverting the
orientation of the graph allows the extraction of their inverses, as shown in
Fig. 11. Generalizations to non-monochromatic light and mixed states are
discussed in the reference. This method was supplemented with graph elements
for nonlinear optical devices performing spontaneous parametric down-
conversion in [237] and demonstrated by explaining three previous experimental
results on such setups. It is worth noting that the nonlinear optical elements
were presented by directed hyperedges, pointing from two nodes to a third one
in the graph. Finally in [238] optical resonators were treated as directed
cycles. Although such a cycle creates infinitely many directed paths to the
output the amplitudes are such that the resulting infinite sum converges.

Figure 11: A simple example of the application of the graphical method
developed in Refs. [236, 237, 238]. The symmetrical beam splitter on the left
has input ports $0$, $1$ and output ports $2$, $3$. The graph depicts the
action at the level of creation operators $a^{\dagger}_{i}$ indexed by port.
The input operators can be either transmitted with weight $T$ or reflected
with weight $R$. Here output operators point to input operators, facilitating
the computation of the former as functions of the latter:
$a_{2}^{\dagger}=Ta_{0}^{\dagger}+Ra_{1}^{\dagger}$,
$a_{3}^{\dagger}=Ra_{0}^{\dagger}+Ta_{1}^{\dagger}$. Reversing the orientation
inverts the functional relationship. Figure reproduced from Ref. [238], doi:
https://doi.org/10.1088/2399-6528/aab50f, license:
https://creativecommons.org/licenses/by/4.0/.

Another closely related approach to interpreting experiments as networks for
computational convenience was introduced in [218], with applications to
homodyne linear optical setups. Here the nodes are states linked by optical
paths, and the links are weighted by state transition amplitudes. Much like
previously, the input and output states have a special role as nodes with zero
in-degree and out-degree, respectively. Moreover, the overall transition
amplitudes from the input to the output are computed by tallying the directed
paths. Loops may be present, leading to infinitely many allowed paths which
however keeps the computed amplitude finite. Importantly, this work introduces
graph simplification rules which can be applied to eliminate intermediate
states, parallel paths, loops and the like. In this way the goal is to finally
arrive at a graph featuring only the input and output states connected by a
single link weighted by the overall amplitude.

The other approach was adopted in [219] which concerned post-selected states
prepared by experiments involving probabilistic photon pair sources and
nonlinear down-conversion crystals. Such setups were associated with a graph
where the nodes are optical paths and links are the crystals, colored by
layer. Notably, the terms in a superposition state created by the setup
correspond to perfect matchings of the graph. This means limitations on what
kind of terms are possible in a given setup translate then to what kind of
matchings are possible in the corresponding graph, facilitating the
application of graph theoretic methods to answer such questions. The problem
was explored further in [221] where the existence and construction of
experimental setups for generating different entangled states was solved by
finding the graph with the required properties.

Weighted networks were used in [220], concerning post-selected states prepared
by experiments involving linear optics elements, nonlinear crystals and
probabilistic 2-photon sources, and generalized to $n$-photon sources in
[239]. In the former the graph consists of photonic modes linked by photon
pair correlations weighted by probability amplitudes for photon pair creation.
The weights can be used to account for interference effects. In the latter the
graph consists of optical output paths playing the role of nodes, grouped by
$n$-photon sources playing the role of hyperedges and weighted by probability
amplitudes. In this work in particular the use of experiments to solve graph
theoretic problems was proposed, as detection of an $n$-fold coincidence event
reveals the existence of a perfect matching in the corresponding hypergraph.
Whether a hypergraph admits perfect matchings is in general a difficult
problem [240].

This line of research was recently generalized beyond the post-selected case
in [241] and proposed to be used for the design of new quantum optics
experiments. The nodes are photonic paths and links correlated photon pairs
colored by mode numbers and weighted by complex coefficients. The key novelty
over previous works is defining the weights in such a way that they contain
the full information of the associated state and the introduction of a
function that maps the weights of the network to the corresponding state
preparation operator. The presentation was further applied in an algorithm
based on optimizing an objective or a loss function depending on the weights,
which was found to have superior performance over alternatives in benchmark
tasks involving both entangled state enumeration and identifying high-
dimensional C-NOT gates. Finally, it was proposed that thanks to the network
presentation the algorithm produces human understandable solutions,
potentially allowing the user to understand and generalize the concept beyond
particular cases.

####  V.2.4 Network description of dynamics

Characterizing the properties of a quantum system with a network emerging from
its state is appealing in particular when the network is much simpler than the
state. But not every state of interest is amenable to a description by a
number of parameters quadratic in system size, and therefore not by a network
unless the transformation to a network is many-to-one. When it is, all the
information is not in the network which makes it unsuited for evolving states
since one would have to determine the evolution in the Hilbert space. Indeed,
the previously presented Refs. [24, 14, 222, 25] consider only stationary
states. Alternatively a network interpretation can be assigned to an
experimental setup for convenience or to benefit from the toolbox of graph
theory but the state itself usually remains in conventional form. Sometimes
such networks might include special nodes for sources of allowed initial
states, as in for example Ref. [218].

The network description of the dynamics becomes possible when both the state
and its transformations can be represented by the network. That is to say
given the state and the operations on it one can express the resulting
dynamics as a time ordered sequence of networks, or a temporal network. The
final state can then be transformed back to conventional formalism if
necessary. This can be achieved by defining a network presentation for the
state and a rule to express the operations as network rewrite rules from the
set of admissible networks to itself, or creating a graphical calculus.
Although both the set of states and the set of operations are typically
restricted such approaches have been used both to facilitate classical
simulation of the dynamics. For example, this approach can be used for
translating tasks involving entanglement distribution to maximally entangle
given systems into network rewiring problems. As a rule of thumb, the more
operations one includes the less elegant the rewrite rules become; obviously
any transformation not taking us outside the set of admissible states is
possible but the way the network transforms might then elude an intuitive
interpretation.

A graphical calculus might be created simply for convenience. A prime example
of this is the one introduced in [242] which accounts for all pure Gaussian
states and all unitaries that preserve the Gaussianity as well as quadrature
measurements. Whereas most are given as transformations of the adjacency
matrix, some admit simple graph rewrite rules. The graphs are undirected and
complex weighted, featuring also loops. At the unphysical limit of infinite
squeezing the weights become real however, corresponding to ideal Gaussian
cluster states of Eq. (14); this is tied to the chief motivation of
quantifying, informally speaking, the distance of any physical and therefore
approximate cluster state to its ideal limit. Any state that can actually be
prepared has finite squeezing and is therefore only an approximation, but such
approximations were lacking a network description up until this work was
published. Here it was applied to finding suitable Hamiltonians for their
adiabatic preparation, supplementing the previously mentioned similar result
derived by other means. Its power was also illustrated by finding graphical
rules to compute bipartite entanglement for certain states.

In a somewhat similar vein, Ref. [243] considered the behavior of qubit
cluster states under local complementation. A qubit cluster state,
corresponding to unweighted adjacency matrix $\mathbf{A}$, is

$\left|G\right>=CZ[\mathbf{A}]\left|+\right>^{\otimes
N}=\prod_{j,k}^{N}\mathbf{A}_{jk}CZ_{jk}\left|+\right>^{\otimes N}$ (17)

where $CZ$ is the controlled $Z$ gate and
$\left|+\right>=\frac{1}{\sqrt{2}}(\left|0\right>+\left|1\right>)$. Local
complementation with respect to some node $\alpha$ toggles every link in the
subgraph induced by its neighborhood $n(\alpha)$; if the link was present it
is deleted and otherwise it will be inserted. The adjacency matrix
$\mathbf{A}$ changes according to

$\mathbf{A}\mapsto\mathbf{A}\oplus\mathbf{K}_{n(\alpha)}$ (18)

where $\oplus$ is addition modulo two and $\mathbf{K}_{n(\alpha)}$ is the
adjacency matrix of the complete graph of the nodes adjacent to $\alpha$.
Importantly, local complementation of the graph can be achieved by applying
local gates on the qubits. Note that, since entanglement cannot increase under
local operations, local complementation cannot increase it either. Here it was
shown how repeated applications of local complementation creates orbits in the
set of qubit cluster states, implying that in fact the entanglement in every
state of the orbit must be the same, constituting a graph entanglement class.
Several examples are shown in Fig. 12. Other connections between the
properties of the orbits and entanglement and also preparation complexity were
identified as well, paving the way for follow-up studies where graph theory
could perhaps be applied to understand entanglement.

Figure 12: Orbits of qubit cluster states under local complementation. Orbit
edges indicate which qubit is chosen. Directed edges indicate that isomorphic
graphs are treated as equivalent. $(a)$ Local complementation with respect to
some node $\alpha$ can be achieved with local operations on $\alpha$ and its
neighborhood, which has implications on its effect on the entanglement in the
state as explained in main text. Orbits starting from the completely connected
graph of $4$ qubits are shown in $(b)$ and $(c)$. Orbits of the cycle graph of
$4$ qubits are shown in $(d)$ and $(e)$. An orbit featuring $6$ qubits is
shown in $(f)$. Figure reproduced from Ref. [243], doi:
https://doi.org/10.22331/q-2020-08-07-305, license:
https://creativecommons.org/licenses/by/4.0/.

Cluster states in general, both in the continuous and discrete variable case
given respectively by Eqs. (14) and (17), can be viewed as a network
presentation of dynamics when used for computation. Specifically, a
computation achieved by a quantum circuit featuring also measurements can be
emulated by suitable pattern of just local measurements and operations on such
a state [244]; whether the overall evolution is deterministic for a given
cluster state and measurement pattern depends on a graph property called
g-flow [245] or CV-flow [246] for discrete and continuous variable cases,
respectively. Cluster states themselves are a special case of more general
graph states where the requirements for the gates playing the role of links
are relaxed somewhat, however in such a way that the state can still be
desribed by a simple graph; the discrete variable case is covered in [247].
Conventions vary, however, and sometimes the terms are used interchangeably.

Using a graphical calculus for simulation purposes has seen use especially for
so called (qubit) stabilizer states. These states arise from qubit cluster
states of Eq. (17) via local Clifford operations and have applications, for
example, in quantum error correction and fault tolerant quantum computing. Up
to a global phase of the from $e^{\mathrm{i}\phi}$, a local Clifford operator
can be generated by the Hadamard gate $\mathbf{H}$ and phase gate
$\mathbf{P}$, given in Eq. (1). The first such software is called GraphSim,
introduced in 2006 [248]. Despite its respectable age for a piece of software,
it still boasts the fastest speed in specific tasks—albeit not in general—when
compared to some contemporary alternatives [249], namely IBM’s Qiskit [250],
Google’s Cirq [251], and a very recent powerful simulator called Stim [249].
As explained earlier, cluster states are in one-to-one correspondence with
graphs. GraphSim uses the corresponding adjacency list and the list of applied
local Clifford operators, which may be thought of as node weights or weighted
loops although this point of view was not adopted. Any circuit consisting of
gates from the Clifford group of operators and local measurements in the
computational basis can be simulated. Applying a local gate amounts to just
updating the node weight, whereas more complicated operations include also a
combination of local complementations and toggling links on or off. The
approach has recently been improved by introducing a novel canonical form for
stabilizer states as graphs as well as more advanced graph rewrite rules in
Ref. [252], which also features an excellent compact introduction to the
stabilizer formalism. Importantly, by introducing both a canonical form and a
canonicalization algorithm this work avoids completely the need to test
whether two graphs represent the same stabilizer state or not. As a side note,
while Stim is both very powerful and not based on graph presentation, it
greatly benefits from tallying the action of the circuit essentially in
reverse. It might be wondered if the same approach could further benefit the
approaches that do use graphs.

Besides simulation, graph theoretic methods can also be used in conjunction
with diagrammatic languages to simplify quantum circuits. Here we mention some
relevant examples based on ZX-calculus [253, 254]. It was applied in Ref.
[255] to circuits acting on a registry of qubits by expressing them as
measurements and local operations on a graph state, optimizing, for example
the time taken or measurements made, and then returning to circuit formalism.
In particular, the simplification took advantage of graph theoretic notions
such as local complementation. When applied to Clifford circuits, this
approach produced the graph of the cluster state augmented with the local
Clifford operators. The difficulty of extracting the circuit from the ZX-
diagram limited the approach to measurements in a specific plane of the Bloch
sphere only, however this limitation has since been lifted in a follow-up work
[256]. Local complementation was also one of the workhorses of an approach to
minimize the number of gates not in the Clifford group of operators in a
circuit [257]. Very recently a circuit extraction method for the continuous
variable case has been introduced, perhaps paving also the way for similar
applications [246].

As will be seen in Sec. VIII, network description of dynamics has applications
in communications. Entanglement percolation and the entanglement distribution
primitives of Fig. 24 can be thought of as relatively simple examples of this.
Assuming that the communication network can be initially prepared in a cluster
state leads to new opportunities. Starting from an arbitary and possibly
complex $G$, Ref. [258] studied how to manipulate the network state to
distribute a Bell pair between two given nodes using linear optics operations.
On the contrary, in [259] this was achieved via local complementation achieved
by applying local Clifford operations on the nodes, and was observed to lead
to fewer measurements than a conventional entanglement distribution protocol.
Establishing the initial large-scale cluster state was discussed in [260] and
stabilizer formalism was used to describe both entanglement distribution and
error correction. Importantly, the power of the approach was demonstrated by
connecting several performance metrics to the topology of the underlying graph
and then optimizing them. Such large-scale applications aside, judicious
manipulation of suitable cluster states can be used to realize all-optical
entanglement distribution schemes which trade the challenge of requiring
powerful quantum memories to the challenge of efficiently preparing and
measuring the states [261, 262].

###  V.3 Avenues for further research

Several authors have suggested applying a network presentation beyond
stationary states [14, 223] to study temporal correlations in the evolving
network. Such research would be separate from work focusing on network
description of dynamics, as the network at some point of time would not in
general determine uniquely the network at later times. It could have
applications for example in efficient extraction of nontrivial information of
the evolution, providing an alternative to process tomography. Ref. [223] in
particular suggested studying the topological correlations in multiplex
networks formed by associating each layer with a different correlation
measure. Aside from generalizing the approach, one can also simply apply it to
novel models or classes of states. As for network description of data sets
such as wavefunction snapshots, there should be much room for further work as
the concept itself is still very new.

Correlation networks in particular can be connected to interaction networks.
When such a relation exists it can be applied for example in the preparation
of special resource states where both the state and the Hamiltonian that
prepares it have a network structure [263], or in their adiabatic preparation
using Hamiltonians which have them as ground states [264]. Connections like
these might warrant further investigation in the general complex quantum
networks context.

Network description of experimental setups such as the one exemplified in Fig.
11 have already been further developed by expanding the set of elements
covered by it. Similar expansion of such methods in general can be expected to
continue. Conversely, one can ask what are the limitations of graphical
approaches, especially in terms of convenience. Systematic exploration of such
limitations might help characterize the best use cases of these methods,
guiding future work and applications.

While a network description of dynamics can be applied for simulation
purposes, the alternatives that do not use it featured for example in Ref.
[249] are in general more powerful. Since several theoretical advancements
have been made in graphical methods there might very well be room to also
introduce simulation software exploiting them. Moreover, the research aiming
to shed light on non-classical properties of quantum systems via a network
description of their dynamics as in the exploration of cluster state orbits
shown in Fig. 12 still seems to be quite sparse, perhaps warranting more
attention.

##  VI Emergence in network models

###  VI.1 Introduction to emergent quantum network models

In the previous Section we have seen that networks can be used to encode the
information of a quantum system. Here we discuss how classical network models
including the Bianconi-Barabasi model [13, 265], the growing Cayley tree
network [266] and the Network Geometry with Flavor (NGF) model [267, 268, 40]
can represent quantum statistics and how the Bose-Einstein condensation of a
Bose gas can predict their topological phase transitions.

In network models quantum statistics can either emerge from a non-equilibrium
network dynamics [13, 266, 267, 268, 40], or it can be a characteristic
property of network ensembles [269, 270, 271] defined following a parallel
construction to the statistical mechanics ensembles of quantum particles. All
these models can be classified as quantum-generalized according to the
classification we have proposed in Fig. 1.

Of special interest are the models in which quantum statistics emerges
spontaneously from a non-equilibrium network evolution. These models can be
considered as network representations of quantum statistics. In particular a
network can be mapped to a Bose gas or to a Fermi gas. Interestingly the
network mapped to the Bose gas can undergo a topological phase transitions,
called the Bose-Einstein condensation in complex networks, [13] in
correspondence to the Bose-Einstein condensation of the Bose gas.

Emergence is a key property of complex systems and refers to the manifestation
of properties that cannot be explained by considering the elements of the
complex systems in isolation. Examples of key emergent properties are
cognition which cannot be explained by neurons taken in isolation or life
itself that cannot be explained by considering separately the constituents of
a cell. In physics, and in particular in quantum gravity it is widely believed
that space-time itself should be emergent, and this line of though is nicely
summarized by the Roger Penrose quote [272]: My own view is that ultimately
physical laws should find their most natural expression in terms of
essentially combinatorial principles, $[\ldots]$ . Thus, in accordance with
such a view, should emerge some form of discrete or combinatorial spacetime.
In this Section it is not our goal to cover the intense activity on quantum
gravity approaches at the interface with network science; rather here we would
like to discuss the relevance of network science for models in which quantum
statistics emerges spontaneously from the network dynamical rules and briefly
cover their relation to questions arising in quantum gravity.

Interestingly since the beginning of network science, with the formulation of
the Bianconi-Barabasi model [13, 265] and the growing Cayley tree model [266,
273] it was realized that non-equilibrium models of networks can display the
emergence of quantum statistics. Indeed quantum statistics characterize the
statistical properties of the structures of these networks and can determine a
topological transition called Bose-Einstein condensation in complex networks.
More recently is has been found that models in which quantum statistics are
emergent include not only (pairwise) network models but also higher-order
network models (evolving simplicial complexes). In the simplicial complex
models also called NGFs [267, 268, 40] we observe the remarkable phenomena
that different quantum statistics can describe the statistical property of the
same higher-order network structure. In particular the degrees of the nodes
and the generalized degrees of the links and triangles have statistical
properties that are captured by different quantum statistics. This implies
that a single higher-order network can represent different quantum statistics
at the same time, encoding them in the statistical properties of simplices of
different dimension. Interestingly these higher-order networks display also
the emergence of hyperbolic geometry [267] and the emergence of a non-
universal spectral dimension [274] characterizing the diffusion properties of
classical and quantum walkers on these structures and their spectral dimension
can be inferred using quantum probes [196].

This section will provide a guide to all the models representing quantum
statistics, emphasizing the research questions that arise in network science
as well the relation with fundamental questions in emergent spacetime. We note
that however, due to space limitations, we cannot cover all the works at the
interface between network science and quantum gravity, a field in which
research interest is recently growing (see for instance Refs. [275, 276, 277,
278, 279, 280, 281, 282] ).

Figure 13: Graphical visualization of two evolving networks representing
respectively the Bose (panels (a),(c)) and the Fermi statistics (panels
(b),(d)). The network representing the Bose statistics evolves through the
Bianconi-Barabasi model [13] and grows through the consecutive addition of one
node connected to the exiting network by $m^{\prime}=2$ links, following a
generalized preferential rule. The other model representing the Fermi
statistics is the Cayley tree defined in Ref. [266] that grows by the
subsequent selection of nodes giving rise to $m^{\prime}=2$ offspring nodes.
In both models nodes are characterized by intrinsic scalar properties
indicating their energies $\epsilon$ (here indicates with colours of the
nodes). The mapping to the quantum statistics is performed by placing a
particle to an energy level $\epsilon$ if in the first model the new node
attaches to a node of energy $\epsilon$, and in the second model if a new node
of energy $\epsilon$ is selected to give rise to new offsprings. Reprinted
figure with permission from [266] ©Copyright (2002) by the American Physical
Society.  Figure 14: Visualization of the Bose-Einstein condensation in
complex network. The Bianconi-Barabasi model [13] representing the Bose-
Einstein statistics is displayed above (panel (a), $\beta<\beta_{c}$) and
below (panel (b), $\beta>\beta_{c}$) the critical temperature for the Bose-
Einstein condensation occurring at $\beta=\beta_{c}$. Reprinted figure with
permission from [283] ©Copyright (2003) by the American Physical Society.

###  VI.2 Quantum statistics and Bose-Einstein condensation in complex
networks

Network Science is a field that has benefited greatly from statistical
mechanics approaches that have been key to characterize both equilibrium
(maximum entropy) and non-equilibrium (growing) network models. The maximum-
entropy models [269, 270] define ensembles of networks which are maximally
random while preserving some network properties. Non-equilibrium network
models are instead models of growing networks dictated by simple rules that
are able to generate non-trivial complex network topologies including for
instance the BA model. Non-equilibrium models are actually the most promising
models for studying emergent properties. Indeed in non-equilibrium models the
network evolution, implementing simple combinatorial rules can self-organize
leading to macroscopic network structures with emergent macroscopic
properties. In this context it has been shown that quantum statistics can
represent and encode the statistical properties of non-equilibrium growing
networks. In particular the Bianconi-Barabasi [13, 265] model of complex
networks describes the emergence of network topologies that represent quantum
Bose-Einstein statistics while the growing Cayley tree with energy (and
fitness) of the nodes [266] represents the Fermi-Dirac statistics. Both models
[273] describe the growth of a network by the addition of new nodes and links.
Moreover both models are characterized by a dynamical rule that is not only
determined by the topological characteristics of the existing network, but is
also dependent on an additional feature associated to the nodes called energy
characterizing the quality of the nodes. Each node $i$ has an energy
$\epsilon_{i}\geq 0$ drawn from a distribution $g(\epsilon)$ which determine
the so called node fitness given by

$\displaystyle\eta_{i}=e^{-\beta\epsilon_{i}},$ (19)

where $\beta>0$ is a model parameter called inverse temperature. The
definition of the node’s fitness implies that nodes with low energy have high
fitness. In the considered models nodes with high fitness have either a higher
ability to attract new links (in the Bianconi-Barabasi model) or a higher
ability to give rise to off-springs (in the growing Cayley tree model). The
dynamics of the Bianconi-Barabasi model includes a preferential attachment of
new nodes to nodes with both high-degree and high-fitness. The growing Cayley
tree with fitness of the nodes define the growth of a Cayley tree by the
subsequent branching of nodes into a constant number of new nodes. In this
case the branching nodes are chosen among the nodes that have not yet
branched, with a probability proportional on their fitness. Interestingly both
models [273] can be shown mathematically to represent quantum statistics when
nodes are mapped to energy levels and links pointing to old (for the Bianconi-
Barabasi model) or to new (for the growing Cayley tree model) nodes are mapped
to occupation number of energy levels (see Figure 13).

This mapping is not only an interesting mathematical result of these models
but is actually a powerful tool to discover an important topological phase
transition. Specifically the Bianconi-Barabasi model displays an important
topological phase transition in correspondence to the Bose-Einstein
condensation, which is called Bose-Einstein condensation in complex networks.
Indeed when the inverse temperature of the model $\beta$ exceed the critical
temperature $\beta_{c}$ the network structure is dominated by succession of
super-hub nodes that significantly change the topology of the network (see
Figure 14). These super-hub nodes are clear leaders of the network acquiring
new links linearly in time (albeit eventually with logarithmic corrections)
until the emergence of the next leader. Since the Bianconi-Barabasi model is
considered a stylized model which capture salient feature of the evolution of
the World-Wide-Web these super-hubs have been usually identified as major
players such as Google, Facebook etc. These results have been confirmed by
numerous studies and mathematical rigorous results [284, 285, 286, 287, 288,
289, 290]

At the network level, quantum statistics have been also shown to describe
equilibrium network ensembles, such as Exponential Random Graphs [269, 270,
271]. In particular the marginal probability of a link takes the form of a
Fermi-Dirac occupation number for unweighted networks while takes the form of
the Bose-Einstein occupation number for weighted networks. As opposed to the
emergence of quantum statistics in the non-equilibrium growing network models
discussed before, in equilibrium network ensembles [269, 270, 271] the
fundamental reason for the emergence of the quantum statistic is not very
surprising. Indeed the network ensembles define statistical mechanic models
determining the unweighted and the weighted adjacency matrix entries which can
take in one case only values zero or one, and in the other case can take any
non-negative integer values and are effectively treated as occupation numbers
of energy states.

Figure 15: Schematic representation of the evolution of the “Network Geometry
with Flavor” (NGF) [267, 268] with flavor $s=-1$ and inverse temperature
$\beta=0$ for dimension $d=2$ (panel (a)) and dimension $d=3$ (panel (b).
Reprinted figure from Ref. [267]. Figure 16: Visualization “Network Geometry
with Flavor” (NGF) with flavor $s=-1$ model in dimension $d=2$ with intrinsic
energies of nodes, links and triangles (visualized with colours) above
($\beta<\beta_{c}$) and below ($\beta>\beta_{c})$ their topological phase
transition. The network displays emergent hyperbolic geometry and for $s=-1$
generates a random Farey graph. For $\beta>\beta_{c}$ the Farey graph develops
in a fairly distributed way in every direction of the hyperbolic plane,
whereas for $\beta>\beta_{c}$ the evolution proceed along a spine changing the
scaling of the network diameter.

###  VI.3 Emergent quantum statistics in higher-order networks

Quantum statistics emerge as well in higher-order networks revealing new
unexpected interplay with the higher-order network structure. The higher-order
(simplicial complex) model where quantum statistics emerge are the NGFs[267,
268, 40]. The NGFs generalize the network models covered in the previous
paragraph and reduce to the Bianconi-Barabasi model for dimension $d=1$.
However they greatly extend this model as they can be formed by gluing
triangles $d=2$, tetrahedra $d=3$, etc as well as regular polytopes such as
squares, cubes, icosahedra, orthoplexes, etc. [274]

NGF are simplicial or cell complexes that grow in time by the subsequent
addition of their building blocks (triangles, tetrahedra, etc.) which are
attached to the existing simplicial complex by a combinatorial rule that
depends on a parameter $s$ called flavor in addition to the the inverse
temperature $\beta$ defining the fitness defined in the previous paragraph. An
interesting property of NGFs is that although their growth obeys a purely
combinatorial rule the NGFs display an emergent hyperbolic geometry (obeying
Gromov $\delta$-hyperbolicity condition for any value of the flavor $s$
[291]).

For observing the emergence of the quantum statistics, we need to assign the
energies not only to each node of the simplicial complex but also to each
link, triangle, tetrahedra and so on. This is done by assigning random
energies to the nodes and attributing to the link the sum of the energy of its
two end nodes, to the triangle the sum of the energies of its three nodes and
so on. In this way, using the definition given by Eq.(19) a fitness value is
assigned to each node, link, triangle etc. Interestingly in this set-up the
same higher-order network can represent several statistics at the same time,
each of them characterizing the statistical properties of the $\delta$
dimensional faces of the NGF (see Table 1). For instance in dimension $d=3$
and for flavor $s=-1$ the statistical properties of triangles, links, and
nodes are representing respectively the Fermi-Dirac, the Boltzmann and the
Bose-Einstein statistics, [268] whereas in $d=2$ and flavor $s=-1/m$ with
$m\in\mathbb{N}$, and $m>1$ we have that the statistical properties of links
and nodes represent respectively the Fermi-Dirac and the Bose-Einstein
statistics [292].

Interestingly NGF can also undergo a topological phase transition if the
temperature is lowered below a threshold $T_{c}=1/\beta_{c}$. In this phase
transition the diameter of the network changes scaling [293] and while for
$s=1$ the diameter grows slower than logarithmically with the network size
(the network is highly compact) for $s=1,0$ the diameter grows polynomially
with the network size for $s=-1$ developing a so called spine (see Figure 16).

While mathematical rigorous results have confirmed the emergence of quantum
statistics in the high-temperature regime [294], the rigorous mathematical
characterization of the topological phase transitions of NGF is quite
challenging and many mathematical research questions remain still open
requiring further in depth investigation.

Table 1:  In a $d$-dimensional NGF of flavor $s$, [268, 267, 292] the
statistical properties of the generalized degrees of $\delta$-dimensional
simplices with energy $\epsilon$ follows either the Fermi-Dirac, the Boltzmann
or the Bose-Einstein statistics depending on the values of the dimensions $d$,
$\delta$ and $s$.

flavor | $s=-1$ | $s=0$ | $s=1$
---|---|---|---
$\delta=d-1$ | Fermi-Dirac | Boltzmann | Bose-Einstein
$\delta=d-2$ | Boltzmann | Bose-Einstein | Bose-Einstein
$\delta\leq d-3$ | Bose-Einstein | Bose-Einstein | Bose-Einstein

flavor | $s=-1/m$
---|---
$\delta=d-1$ | Fermi-Dirac
$\delta\leq d-2$ | Bose-Einstein

###  VI.4 Relation to quantum gravity research questions

An interesting question is to what extent network models representing quantum
statistics relate to quantum gravity approaches. As we discussed in the
beginning of this Section, Roger Penrose was the first to postulate a discrete
and combinatorial spacetime. Currently a large variety of quantum gravity
approaches describe a discrete spacetime [295, 296, 297]. This reflects in
some cases a fundamental belief that spacetime should be discrete at the
Planck scale. Alternatively, a discrete spacetime may be chosen even if the
spacetime is assumed to be inherently continuous. Indeed a discrete spacetime
is mathematically convenient as dealing with discrete structures enforces a
cutoff that allows to regularize the theory escaping the dangers of non-
renormalizability. An important scientific problem that arise in this context
is the identification of the characteristics of discrete spacetimes that
correspond to the different quantum gravity approaches. In particular great
attention has been addressed in characterizing the spectral properties of
these emergent discrete geometries defining the effective spectral dimension
—typically considered to be the measure of dimension— of the networks. This
research line has lead to the development of new concepts and ideas such as
the fractal dimensionality of spacetime and the scale-dependent spectral
dimension [298, 299]. Interestingly the characterization of the spectral
dimension of discrete spacetimes emerging from different quantum gravity
models has been recently considered important to classify quantum gravity
approaches and to determine whether these different theories define universal
predictions valid across different approaches [300]. Importantly the NGFs do
not only show the emergence of the quantum statistics and hyperbolic geometry
but they also display the emergence of a (non-universal) spectral dimension
[274] that can be inferred by quantum probes [196]. The emergence of a finite
spectral dimension is observed not only in simplicial but also in cell
complexes, i.e. higher-order networks not only formed by nodes and links
triangles but also formed by squares, tetrahedra and so on. Moreover in the
framework of the NGFs it has also been shown that the notion of the spectral
dimension extends also to the higher-order network level and can be used to
characterize the spectrum of higher-order Laplacians [301]. Interestingly in
the NGFs the spectral dimensions depend on the order of the Laplacian [301],
the dimension of the simplicial complex [274] and the nature of building
blocks of the cell-complexes [291]. Therefore while the presence of a finite
spectral dimension seems to be a universal property of all the different
variants of NGFs, the value of their spectral dimension is highly non-
universal [40]. It is still an open question whether these results are due to
the highly heterogeneous structure of the NGFs.

###  VI.5 Discussion and future directions

The Barabasi-Bianconi model has attracted large interest in the network
science and in the mathematical community. In these communities most of the
attention has been addressed to the characterization of the networks in the
Bose-Einstein condensation phase. Indeed while above the critical temperature
the mapping to the Bose gas completely captures the statistical properties of
the network, below the critical temperature there are some important
differences. In fact the networks in the Bose-Einstein condensed phase are
dominated by a succession of super-hub nodes that dominate the network
structure, while in the Bose gas it is a single energy level, the fundamental
state that acquires a finite occupation number. In terms of the application of
this model to describe the competition for links in a network what happens in
this condensed phase is of major importance. Major questions that arise in
this context are: is it always the best (lower energy) node to win? is today’s
winner destined to be overcome by an even better node arising in the future?
Is the fraction of links connected to the winner node really extensive or
there are logarithmic corrections to the linear scaling due to this temporal
effects?

Although much progress has been made on these very challenging questions many
questions remain open. In particular, if the transitions observed on networks
displaying emergent quantum statistics is still posing interesting
mathematical questions, the transition observed on the NGFs are mostly
unexplored so far.

While all these questions are related to the classical consequences of Bose-
Einstein condensation for network structures, an important open question is
whether the representation of quantum statistics embodied by these networks
can be harnessed by quantum gravity approaches or even quantum technologies or
applications.

##  VII Quantum algorithms for network inference

###  VII.1 Quantum concepts useful for complex networks

In the last decade there has been increasing attention devoted to the
formulation of quantum algorithms and observables that can reveal important
properties of classical networks [302]. In this Section we will cover this
very innovative research direction which could flourish in the next years with
the new generation of quantum computers. This research direction corresponds
in our classification to the quantum-enhanced research line (see Fig. 1).

Recently, quantum algorithms have been proposed to capture a wide range of
structural properties of classical networks starting from the quantum degree
distribution all the way up to quantum community detection and quantum link
prediction. In this class of works the focus is not always to formulate
algorithms that are faster than their classical counterpart, rather more often
the goal is to capture structural properties that can be neglected or not
sufficiently highlighted by classical network measures.

Even more recently with the increasing attention addressed to higher-order
networks, it has become clear that quantum concepts are also key to treat
higher-order dynamics. In particular the discrete topological Dirac operator
[27] can be used to formulate a gauge theory [303] for topological spinors,
which have an explicit geometrical interpretation and capture the dynamics of
simple and higher-order networks defined on nodes, links, and even triangles
and higher-dimensional simplices of simplicial complexes. This gauge theory
can be used to define an emergent mass of simple and higher-order networks
which depends on their geometry and topology [304].

The topological Dirac operator can also be used to characterize the dynamics
of coupled (classical) topological signals [59]. In particular the Dirac
operator allows to define a new class of dynamical processes on network and
simplicial complexes, revealing new physical phenomena as demonstrated by its
application to Dirac synchronization, Dirac Turing patterns and Dirac signal
processing [305, 306, 307, 308].

The topological Dirac operator hence can be quite transformative in our way to
treat dynamics on networks, usually only associated to the nodes of the
network.

###  VII.2 Quantum inference of networks

####  VII.2.1 Quantum inference of network structure

Large scientific attention has been addressed to define algorithms that probe
the network topology using quantum random walks. For instance in Ref. [140] a
quantum definition of the degree of a node of the network is proposed starting
from the long-time probability distribution for the location of a quantum
walker on the complex network. In this work it is shown that for low-energy
quantum walkers this probability distribution coincides with the one of the
classical random walk, and is hence closely related to the degree distribution
of the network. However for higher-energy states the classical and the quantum
distributions differ, providing a quantum generalization of the concept of
degree distribution. Also based on quantum walks in Ref. [153] a quantum
algorithm for community detection is proposed. This algorithm is a
hierarchical clustering algorithm where the similarity between the nodes are
based on quantum transport probability and state fidelity. The proposed
algorithm is tested on light-harvesting complex finding good agreement with
the partitioning of nodes used in quantum chemistry.

Figure 17: Example of network embedding generated extracting information from
the eigenvector of the magnetic Laplacian in which directed edges are
associated to a complex angle $2\pi/3$. Reprinted figure from Ref. [309].

An alternative approach for detecting community structure in networks that is
becoming increasingly popular uses instead embeddings generated from the
eigenvectors of the magnetic Laplacian. The magnetic Laplacian [310, 311, 312,
313, 309] is a complex valued Hermitian variant of the Laplacian that captures
non trivial mesoscale network structures. In particular the magnetic Laplacian
[310, 311, 312, 313, 309] can capture community structure in directed networks
and can detect cyclic patterns of connections such as the ones that are formed
by three communities of nodes where the nodes of each community link
preferentially with the nodes of other communities forming effectively a
mesoscale “triangle” formed by the three communities (see Figure 17). Having
complex matrix elements the eigenvectors of the magnetic Laplacian display a
complex phase. For instance in the case of the cyclic community structure
described before the phases on the different communities will differ by an
angle $2\pi/3$ reflecting the three communities that are present in the
network. The interest on the magnetic Laplacian has recently further fuelled
an interesting line of research aimed at exploiting complex weights in network
science. In this framework new network dynamical processes have been proposed
including consensus models [314] that generalize the Schrödinger-Lohe model
[122, 121], complex weights random walks [315] and quantum Hopfield model
[316].

Link prediction is one of the most challenging and used classical inference
algorithm for reconstructing missing or hidden interactions. In Ref. [162] a
quantum algorithm based on a quantum walk is used to infer missing links
extracting information from paths of even and odd path length. Here the
emphasis is on the efficiency and speedup of the quantum algorithm with
respect to the classical counterparts while showing that the quantum algorithm
retains a performance comparable with classical algorithms.

Network symmetries are fundamental properties of lattices and tree structures
traditionally studied in condensed matter and in quantum information. Indeed
lattices are characterized by well known symmetry groups. Moreover, symmetric
structures formed by two binary trees connected at their leaves have been
shown by Farhi and Gutmann [147] to allow an exponential speed up of the
quantum random walk with respect to the classical random walk. Interestingly
also complex networks have non trivial symmetries [317, 318] and detecting
isometries is an important NP hard problem of computer science. Recently
continuous time random walks have been used to detect symmetries or quasi
symmetries in network structures and isometries of quasi-isometries among
different networks. In Ref. [319] the symmetries of a given network are
studied. To this end, for every pair of nodes two states localized on them are
prepared corresponding to amplitudes either in phase or antiphase. The Jensen-
Shannon divergence between the density of states corresponding to the two
random walks is proved to achieve its maximum value if the two selected nodes
are symmetrically placed (note however that is a necessary by not a sufficient
condition of ensuring symmetry). Interestingly the Jensen-Shannon divergence
can be used also to detect or infer quasi-symmetries because it is a measure
that is robust to the introduction of random perturbations of the symmetries.
In Ref. [320] this research line is extended to compare and reveal symmetries
between two different networks. In particular given two unconnected networks,
the Authors suggest to construct a joined network where links between the
nodes of the two networks are inserted so that each node originally in network
1 is connected to all the nodes originally in network 2 and vice versa. On
this joined networks two continuous random walk walks are studied, which
evolve from initial states constructed in such a way that the amplitude
corresponding to the states associated to the nodes of the two networks are
either in phase or antiphase. The Jensen-Shannon divergence between the two
different random walks is then used to detect isometries and more in general
to construct kernels between the two networks that can then be used by machine
learning approaches to classify networks.

####  VII.2.2 Quantum centrality measures (Quantum PageRank)

PageRank [321] is undoubtedly the most successful Network Science algorithm.
It is the original algorithm used by the Google search engines and since then
its use has been extended to rank in order of decreasing importance nodes in a
variety of networks, including social, technological and biological networks.

The classical algorithm runs polynomially and scales well with the network
size, however the PageRank algorithm in practice is run on a continuous basis
to rank all the pages on WWW and to address time-dependent changes of the
webpage content and the network topology.

From the quantum perspective two major questions arise. The first research
question regards the possible speedup that a quantum algorithm to calculate
the classical PageRank can achieve [322]. The second research question regards
the formulation of the Quantum PageRank that can extend the classical
definition and retain its good performance while being of potential use to
rank quantum webpages. Quantum webpages indicate the nodes of a quantum
network with quantum capabilities such as reading in/out quantum states. In
other words quantum webpages do not require a fully fledged quantum computer
and can be realized by quantum storage devices and quantum memories.

Interestingly the answer to these questions is based on the definition of the
same matrix, the Google matrix $\mathbb{G}$ defined as

$\displaystyle\mathbb{G}=\alpha E+(1-\alpha)\frac{{\bf 1}}{N}$ (20)

where $\alpha\in(0,1)$ is a parameter of the model analogous of the
“teleportation” parameter of the classical PageRank, ${\bf 1}$ is the matrix
of all elements equal to one, $N$ is the network size and $E$ is the
transition matrix of a classical random walk where every zero row
corresponding to a node with zero out-degree is substituted with a row in
which all elements have values $1/N$. The Google matrix can be shown to be
both irreducible and primitive.

The classical PageRank can be obtained by applying $k$ times the Google matrix
to an initial guess for the ranking of the nodes and going in the limit
$k\to\infty$. In Ref. [322] the Authors have proposed a quantum annealed
algorithm to speed up the calculation of the classical PageRank showing that
the improvement on the performance of the classical algorithm is more
significant if the out-degree distribution of the network is broad (as it is
the case for our current —classical— WWW).

The quantum PageRank [323] provides instead an alternative ranking of the
nodes providing a Quantum PageRank class on which the classical PageRank can
be embedded and allowing for a classical computation belonging to the
complexity class P. The Quantum PageRank is based on the Google matrix and
uses Szegedy procedure to quantize the Markov chain algorithm that provides
the classical PageRank. The resulting quantum Pagerank of the node of the
network is a ranking that fluctuates as it depends on the time duration of the
quantum evolution (see Figure 18). Therefore it is possible to characterize
the fluctuating instantaneous ranking given by the quantum PageRank and the
ranking provided by the average of the instantaneous quantum PageRank over a
suitably large time window. Several works [323, 324, 325] have investigated
the performance of the quantum PageRank algorithm on real WWW network data and
on network models including the ER model, the Barabási-Albert model, and
random scale-free networks. The rankings obtained with the quantum PageRank
are compared with the ranking of the classical PageRank showing that the
classical ranking of the top ranked nodes is always within the range of
fluctuation of their corresponding quantum PageRank. However the quantum
PageRank in particular for networks with broad degree distribution has the
advantage that it can distinguish better secondary hubs and removes the
degeneracy in the nodes with low ranks. Moreover, comparing the results
obtained on different network topologies it has been shown that the quantum
PageRank can really capture their differences. In particular scale free
networks are characterized by corresponding random walks that are localized
[324, 325, 326] while random ER networks are characterized by random walks
that are not localized.

An alternative approach [163] uses the theory of quantum open systems to
define a Quantum Page Rank algorithm with a single stationary state. The
approach allows to interpolate between purely classical and purely quantum
Page Ranks. It is shown that a certain level of quantumness is beneficial to
speed up the algorithm.

Pioneering works are using CTQWs with a non-Hermitian Hamiltonian given by the
directed (asymmetric) adjacency matrix of the network to define quantum
centralities of the nodes of the network that generalize well the classical
eigenvector centrality. The benefit of these simplified definitions of node
centralities is that they can be experimentally implemented via linear optics
circuits and single photons [154, 327].

Figure 18: Quantum Page Rank of nodes in the root, in the first level and in
the second level of a binary tree plotted as a function of time. Reproduced
figure from Ref.[323].Creative Commons Attribution-NonCommercial-ShareALike
3.0 Unported License http://creativecommons.org/licenses/by-nc-sa/3.0/

####  VII.2.3 Von Neumann entropy of networks

The von Neumann entropy allows the investigation of the structure of complex
network using tools of quantum mechanics based on the spectral properties of
the network. Given a definition for the density matrix of a network which is
positive semi-definite and normalized to one, the von Neumann entropy of a
network has the usual definition and the quantum Jensen–Shannon divergence to
measure the dissimilarity between two networks can be defined as well.
Therefore the crucial point for defining the von Neuman entropy is to make a
suitable choice for the density matrix. Originally it was proposed [26] to
consider a density matrix given by the Laplacian, normalized with the sum of
degrees i.e.

$\displaystyle\rho=\frac{L}{\langle{k}\rangle N}.$ (21)

The resulting von Neumann entropy can be highly affected by the degree
distribution if the network is scale-free and it correlates well with other
classical entropy measures of the network [270, 328]. This definition of the
von Neumann entropy has been adapted to multiplex networks in [329] and the
corresponding Jensen-Shannon divergence has been used to compare and
compress/clusterize different layers of real multiplex networks [330]. Other
works have also considered the use of the normalized Laplacian ${\bf\hat{L}}$
instead of the unormalized graph Laplacian L [331].

Later it was proposed to use the alternative expression for the density matrix
of a network [332]

$\displaystyle\rho=\frac{e^{-\beta L}}{\mbox{Tr}e^{-\beta L}}.$ (22)

In Figure 19 we show the von Neumann entropy of key network models as a
function of the inverse temperature $\beta$ as reported in Ref. [332]. This
Gibbs like definition of the density matrix is significantly affected by the
low eigenvalues of the Laplacian relating to the long time diffusion dynamics
on the network. In particular with this latter definition of the density of
states, the von Neumann entropy has a classical interpretation as the number
of eigenmodes that are important for paths that evolve up to time $\tau=\beta$
[333]. The derivative of the von Neumann entropy with respect to $\ln\beta$
has been recently proposed as a measure characterizing the temporal scale at
which diffusion processes display significant dynamical transitions or cross-
overs [333, 334].

Figure 19: The von Neumann entropy of key network models (from left to right:
ER networks, SW networks and $K$-regular networks) is shown as a function of
the temperature. Figure reproduced from Ref. [332], doi:
https://doi.org/10.1103/PhysRevX.6.041062.

Although the von Neumann entropy and its variation is the most popular
definition of quantum entropy of a network, alternative definition exist
including the one formulated in Ref. [335] where the Authors first introduce a
mapping between the adjacency matrix of a network and pure quantum bipartite
states and subsequently show that the associated entanglement entropy captures
important structural properties of the graph.

###  VII.3 Quantum higher-order networks and the topological Dirac operator

The research on quantum higher-order networks [40] constitute a new promising
field of research given the increasing interest of the network science
community on higher-order interaction networks. Early works include the
extension of graph quantum states [71, 247] to hypergraph quantum states [336,
337] prepared from single qubits by performing operations between $k$
connected qubits, with $k\geq 2$. In Ref. [337] it is shown that hypergraph
states are in one-to-one correspondence with real-equally-weighted (REW)
states that are essential for quantum algorithms while the graph states in
which $k$ is fixed to be equal to two,(i.e. $k=2$) only constitute a subsets
of REW states.

Higher-order networks and in particular simplicial complexes are also
interesting because they can shed light in the interplay between network
topology on dynamics [40, 59] revealing new physics and phase transitions.
Topology is currently gaining significant attention and offers new paradigms
for describing classical dynamics inspired by quantum mechanics, which
includes among the other applications the characterization of edge currents in
biological synthetic biology [338] and game theory [339].

Higher-order networks provide a mathematical framework in which the interplay
between topology and dynamics is transformative. In particular higher-order
networks can sustain topological signals, i.e. dynamical variables not only
associated to the nodes of the networks, but also to their links, and in a
simplicial complex even triangles, tetrahedra and higher-order simplices.
These topological signals can undergo collective phenomena [63, 340] which
display very new physics with respect to the associated dynamics defined
exclusively on nodes. These topological signals can be studied using Hodge-
Laplacian operators [40] that describe diffusion from $n$-dimensional
simplices to $n$-dimensional simplices through either $n-1$ or $n+1$
dimensional simplices. Indeed while the graph Laplacian describes diffusion
from nodes to nodes occurring through links, the $1$-Hodge Laplacian can
describe diffusion from link to link occurring through nodes or through
triangles. The spectral properties of the Hodge Laplacian encode for important
topological features such as the Betti numbers and allow generalized higher-
order diffusion [341]. The research in the field is rapidly growing and
interestingly the Hodge-Laplacians have also been used to define a higher-
order von Neumann entropy that encodes relevant higher-order network
properties [334].

A very active and very promising research direction for treating the dynamics
of coupled topological signals involves the discrete topological Dirac
operator. The discrete topological Dirac operator [27] of networks and
simplicial complexes is a topological operator, rooted in quantum physics,
that has the ability to couple topological signals of different dimension. The
discrete topological Dirac operator has first been proposed in non-commutative
geometry [342] and then its spectral properties have been further investigated
in the framework of quantum graphs [343, 344].

Figure 20:  Visualization of two eigenstates of the topological Dirac equation
defined on the nodes and links of a real network. Figure reproduced from Ref.
[27]. The real network used for this figure is published in Ref. [345].

The topological Dirac operator can be used to formulate a topological discrete
Dirac equation [27] in which the spinor acquires a geometrical interpretation
and is defined on each node and link of the network. Moreover the topological
Dirac equation can be also generalized to simplicial complexes by considering
topological spinors defined on nodes, links, triangles, tetrahedra and so on.
The topological spinors are determined by dynamical variables (cochains)
defined on each simplex of the simplicial complex. In particular on a network
the topological Dirac operator acts on the topological spinor by coupling the
dynamics on the links to the dynamics of the nodes. As the continuous Dirac
operator, the discrete topological Dirac operator can be considered as a
square-root of the Laplacian operator and admits both positive and negative
eigenvalues. Interestingly for each positive eigenvalue there is a
corresponding negative eigenvalue corresponding to the matter-antimatter
symmetry. The corresponding eigenvectors are related by chirality. However an
interesting aspect of the discrete topological Dirac equation is that in
general the harmonic eigenvectors of the topological Dirac operator do not
display the matter-antimatter symmetry [27].

The topological Dirac equation can be adapted to capture different directions
of regular lattices, and can be used to study the interplay between topology
and quantum dynamics on multiplex networks and simplicial complexes.
Interestingly the topological Dirac operator allows to define a lattice gauge
theory in which the fermion fields are taking values on both nodes and
directed links which play the role of the fiber bundle of the network [303].

The discrete Dirac operator is currently gaining increasing attention in non-
commutative geometry [346] and in quantum graph literature [347, 348, 349,
350, 351].

Recently the Dirac operator has been proposed to define classical higher-order
dynamical models displaying relevant new physics. In particular in Refs. [305,
306] Dirac synchronization is proposed. Dirac synchronization is a higher-
order Kuramoto model with phases associated to nodes and links of a network,
coupled to each other by the Dirac operator which display a discontinuous
synchronization and an emergent rhythmic phase. Moreover in Ref. [307] the
Dirac operator is used to reveal novel mechanisms for the emergence of Turing
patterns on both nodes and links.

The topological Dirac operator is also useful to define inference algorithms.
In [352] the topological Dirac operator is used to formulate a quantum
algorithm that calculates the homology of simplicial complexes. This algorithm
is based on a representation of the simplicial complex as a quantum state
having as basis the set of the simplices of the simplicial complex. The
proposed quantum algorithm is shown to display an exponential speed-up over
the best known classical algorithms for calculating homology. Recently in
Refs. [353, 354] this approach has been extended also to propose a quantum
algorithm for the calculation of persistent homology of simplicial complexes.

Another inference algorithm using the Dirac operator is Dirac signal
processing [308] that allows to jointly process signals defined on simplices
of different dimensions.

###  VII.4 Discussion and future directions

Quantum algorithms to infer relevant information from networks are only in
their infancy however the field has already obtained important results that
provide good foundations for further development.

One of the important aspects of quantum algorithms for complex networks is
their strong connection to the network spectral properties. Indeed the most
important operators and observables that have been defined, from the magnetic
Laplacian, and the quantum Google matrix, to the Dirac operator and the von
Neumann entropy, are based on the spectral properties of the discrete network
structures under investigation.

However the algorithms differ significantly from their classical counterparts.
The magnetic Laplacian introduces complex valued weights of the links
capturing for their direction. The quantum Google matrix obeys dynamics that
is not dissipative like the classical PageRank algorithm. Finally the Dirac
operator provides a fundamental change of the understanding of dynamical
processed on networks because it describes a topological coupling between
dynamical variables associated to nodes, links, and higher-dimensional
simplices of higher-order network structures.

These different ways to use quantum concepts to model, understand and extract
information from networks have already shown their clear advantage as
demonstrated in particular by the large success of von Neumann entropy
measures of networks and the important role of the Dirac operator to describe
new physics in networks and simplicial complexes.

One open problem that emerges in this context are whether quantum algorithms
can further transform the landscape of combinatorial algorithms on networks,
providing progress for instance in the graph isomorphism problem.

Additionally one important question is whether new mathematics is needed to
treat networks and simplicial complexes. In particular in the continuous Dirac
equation, the Dirac operator is coupled to the algebra of gamma matrices. An
interesting question is whether also to analyse networks coupling the Dirac
operator to a group could be key to capture the full geometry and topology of
the data and to model coupled topological signals.

##  VIII Quantum communication networks

###  VIII.1 Fundamentally different communication

In quantum communication networks, photons are exchanged between distant nodes
to facilitate distribution of cryptographic keys and entanglement as well as
transmission of quantum information [355]. Such networks have applications
beyond those of their classical counterparts [3], making this very vibrant
research area fall into the quantum enhanced class in Fig. 1, although certain
aspects can also be considered to be network-generalized. They may be roughly
divided into within reach and theoretical networks, which now in the early
2020s still correspond to quantum key distribution (QKD) and quantum
information (QI) networks, although not as strongly as for example just in mid
2010s. Both are subject to the same fundamental limitations arising from the
properties of quantum information, particularly to communication rate limits
that decrease rapidly with distance travelled in optical fiber, making them
metric networks. Indeed, much of the research has focused on designing
architectures that tolerate the limitations [356, 357] or on improving the
basic building blocks [358, 359, 360]. There are many excellent contemporary
treatises on the topic [3, 361, 362, 363], however here a more concise account
of the field is provided with a unique point of view emphasizing the network
aspect. In particular, we focus on research where the architecture is fixed
and it is asked what kind of topology it tends to lead to [364, 365] or how a
given topology controls its performance [366, 367, 368, 369]. To this end we
will also briefly introduce the relevant architectures.

By QKD networks we mean specifically the case where the transmission of
photonic quantum states is limited by the total distance between participants
and inbound photons can be only measured or forwarded. Then the actual
messages will be classical and transmission of the states merely facilitates
its encryption with quantum secure keys; consequently such networks have also
been called semi-classical [370] or partially quantum [260]. Examples of past
and present QKD networks include the DARPA network in Boston [371], SECOQC
network in Vienna [372], the Tokyo quantum network [373], the Hefei quantum
network [374, 375], the London Quantum-Secured Metro Network [376] and the
Beijing-Shanghai backbone quantum network [377]. This latter network, shown in
Fig. 21, has since been used to connect four metropolitan area networks in
Shanghai, Hefei, Jinan and Beijing and has been supplemented by a satellite
link connecting two ground stations around 2600 km apart [378]. In contrast,
direct fiber links achieving reasonable rates cover distances of around 100 km
[379, 380]. There is an interest in the integration with existing optical
fibres both to save costs and because the communication protocols virtually
always require classical communication as well [381, 382]; the feasibility of
such co-existence of both quantum and classical layers in the same fiber
network has been demonstrated in particular in the Madrid Quantum
Communication Infrastructure [383]. Although significant as testbeds for
research and development which might later benefit QI networks as well, it is
the case that fiber based QKD networks are either limited to a small service
area or lack end-to-end security, inhibiting their growth. Recent theoretical
results [365, 384, 385] for satellite based networks are quite encouraging,
however, as are novel fiber schemes connecting next nearest neighbors [386,
387] which have been reported to achieve reasonable experimental rates beyond
400 km [388]. The boundaries are moving.

Figure 21: The Beijing-Shanghai backbone network connects four metropolitan
area QKD networks and also features a long distance quantum satellite link
covering 2600 km between Xinlong and Nanshan (not shown). Reprinted figure
with permission from [380]. Copyright 2020 by the American Physical Society.

In QI networks also quantum information can be transmitted over long
distances. While they form a broad class with many subcategories, the holy
grail is the quantum Internet [29]: a global public commercial network capable
of storing, processing and transmitting quantum information and entanglement.
Photonic qubits can be transformed to stationary qubits stored in quantum
memories and back as necessary—a process called quantum transduction [389]
achieved by quantum interconnects—and the entire network can be prepared into
a nonclassical state. In conventional proposals the quantum layer is divided
into physical and virtual layers, where the former is used only to distribute
entanglement. All quantum information is transmitted over the virtual layer
consisting of teleportation channels which consume the distributed
entanglement as fuel. The network must then be able to at least generate and
distribute, but ideally also store and accumulate entanglement. As networks,
they would be not only metric but also multiplex and the virtual layer would
be temporal. Capable of much more than just QKD, such networks could
revolutionize the world much like classical Internet did. For now, they face
daunting technological challenges related in particular to sufficiently
powerful quantum memories.

Proposals to achieve QI networks in the near-term future feature prominently
all-optical schemes [261, 262, 390, 391, 385] as this could eliminate the need
for such memories, albeit at the cost of some applications. At the other end
of the spectrum are all solid state schemes, which however are envisioned to
for example facilitate state transfer on a chip with minimal control and less
sources of errors. This is achieved by the coherent dynamics of interacting
but stationary carriers of information, much like in CTQW described in Sec.
IV.2.2, as opposed to physically moving, i.e. shuttling, the systems.
Originally proposed in the early 2000s to facilitate communication between
nearby quantum processors [135] it has since been studied in both spin [392,
393, 394, 395, 396] and oscillator systems [397, 398, 399, 400, 401] with or
without some limited control or additional operations. More recently a scheme
for transferring logical qubits in quantum dot arrays has been proposed and
found to be favorable over shuttling in terms of energy cost [402]. Results
concerning complex networks are scarce, however, and consequently this
otherwise very important and highly active area of research will not be
discussed further here; we recommend instead Refs. [159, 160, 161].

In the following we first focus on networks within reach of current
technology. Although quite different in terms of applications, as networks
they share for example the natural weights with theoretical networks and
provide an opportunity to introduce them in a simpler setting. We then present
two common approaches for achieving entanglement distribution, considering
mostly ideal conditions for the sake of simplicity, and briefly review
network-generalized nonlocality. Finally, we consider the quantum Internet and
its applications at various stages of development and along the way present
results concerning what could be called noisy intermediate scale quantum
(NISQ) networks, covering some of the vast and still somewhat unexplored
landscape between within reach and ideal networks. As we focus on the network
aspect we refer the reader to: [362] for motivation and applications and to
[403, 404, 28, 405] for QKD in particular, [361, 363, 406] for quantum
repeaters, quantum memories and their candidate platforms, [360] for quantum
interconnects, [407] for quantum error correction, [408] for quantum Internet
protocol stack, [355] for classical simulation of the networks and [379, 384,
409, 410, 411] for the use of quantum satellites with their simulation
discussed in particular in [411].

###  VIII.2 Metric quantum networks

####  VIII.2.1 Within reach: quantum secure networks

Considering only technology mature enough to be deployed in the field now, the
basic building blocks are nodes capable of generating, detecting or possibly
forwarding quantum states, connected by quantum channels constituted by
optical fiber or free space. The ideal carriers for the transmitted quantum
states are photons. Besides moving at the speed of light and being highly
resilient to decoherence, non-classical photonic states can nowadays be
routinely generated, transmitted and measured. As anticipated, such networks
already suffice for QKD. The working principle is to use the states to share a
random string of classical data, upper bound the amount of leaked information
by taking advantage of certain fundamental properties of quantum information,
and finally to distill a secret key from the data that can then be used to
encrypt the actual messages. QKD is attractive because its security is based
on the laws of Nature, implying for example that it is future technology proof
as long as for example experimental imperfections can be accounted for. In
contrast, conventional cryptography protocols are based on plausible
assumptions about the difficulty of inverting certain functions and are secure
only if the computational power of the adversary is limited and the
assumptions indeed hold.

Unlike conventional protocols, QKD is greatly limited by distance however.
Specifically, the secret key bit rate achieved via point-to-point transmission
depicted in Fig. 22 of quantum systems over lossy bosonic channels (see Sec.
II.4) is limited by the channel capacity, or in network terms the maximum
flow. It is in units of the average number of (already distilled) secret key
bits transmitted per channel use, which in turn can in principle reach but
never exceed a fundamental, protocol-independent limit known as the
Pirandola–Laurenza–Ottaviani–Banchi (PLOB) bound [412]. For such channels this
ultimate capacity is

$\mathcal{C}(\eta)=-\log_{2}(1-\eta)$ (23)

but approximately $1.44\eta$ for $\eta\ll 1$, where transmissivity $\eta$—the
fraction of photons that survive the transmission—drops exponentially with
distance. Assuming state-of-the-art optical fiber of length $d$, one would
typically use $\eta=10^{-\gamma d/10}$ where $\gamma=0.2$ dB/km quantifies
losses per distance. In free space one would include for example a factor
accounting for the geometric position of the source with respect to the
receiver [384]. In principle, the rate merely decreases rapidly with distance
but in practice collapses abruptly to zero due to detector noise washing out
the quantum signal [413], making distance a hard limit. This is further
exacerbated by high consumption rates: it is natural to pair QKD with
encryption providing the highest security, where each key must be as long as
the message and used only once. From now on this is assumed unless stated
otherwise.

Figure 22: End-to-end transmission concerns two nodes communicating over a
network using, for example, single or multiple paths. Point-to-point
transmission takes place when two adjacent nodes communicate directly and the
rest of the network plays no role. Upper bounds for end-to-end capacities may
be found by considering the point-to-point capacities where the the PLOB bound
of Eq. (23) applies directly.

This on the one hand provides the links with their natural weights, i.e. the
physical distance and the resulting values of $\eta$ and $\mathcal{C}(\eta)$,
and on the other hand strongly limits the networks. In the following we
present generalizations of the ultimate capacity or maximum flow to both
chains and arbitrary networks [412, 414]. Conveniently, they are also
applicable to QI networks as seen later.

Under the considered limitations Eq. (23) can be applied to a transmission
over some path $P$ by simply considering the total transmissivity

$\eta_{P}=\prod_{e\in P}\eta_{e},$ (24)

where $\eta_{e}$ is the transmissivity of link $e$. For constant $\gamma$, it
coincides with that of a direct optical link. Therefore only nodes connected
by a short enough path can directly share a secret key.

Covering larger areas can be achieved with trusted nodes or relays. Such a
trusted chain $P_{t}$ could operate for example as follows. First every link,
i.e. adjacent pair of nodes, generates locally stored secret key bits for $n$
rounds. In the large $n$ limit the number of bits in some link $e$ tends to
$n\mathcal{C}(\eta_{e})$. Next, secret bits are transmitted end-to-end (see
Fig. 22) by encrypting and decrypting them using locally stored bits. Since
strongest encryption is assumed each transmitted bit consumes a local bit from
a link, and once the link with the smallest $\mathcal{C}(\eta_{e})$ runs out
we are done. Therefore the rate per use of the chain tends to
$\mathcal{C}(\eta_{P_{t}})$ where

$\eta_{P_{t}}=\min_{e\in P_{t}}\eta_{e},$ (25)

making the rate limited by the bottleneck rather than the total distance. It
turns out that $\mathcal{C}(\eta_{P_{t}})$ is the ultimate upper limit for the
capacity. $P_{t}$ can be thought of as a chain of classical
repeaters—conversely, Eq. (24) is also known as the repeaterless PLOB bound. A
conventional repeater receives, amplifies and repeats a signal to extend its
range; here each intermediate node increases the bit rate between end nodes in
a scalable manner but deals with classical information. In the example $P_{t}$
follows a continuous generation protocol where links constantly accumulate
resources, allowing operation near the theoretical maximum rate as any
fluctuations caused by the probabilistic loss of some of the photons average
out. Indeed, this is the standard operation mode of QKD links in experimental
networks since DARPA and SECOQC networks [28].

Generalization to QKD networks of arbitrary topology is straightforward,
although their ultimate capacity is of limited practical interest as discussed
in Ref. [28]. Like before, each link accumulates secret bits which are then
consumed by the transmission. When a link is out it may be discarded; when the
considered nodes are disconnected we are done. Any set of links $C$ that
disconnects the network is called a cut(-set), which here is understood to
specifically disconnect the sender from the receiver. The minimum cut

$C_{\mathrm{min}}=\operatorname*{arg\,min}_{C}\sum_{e\in
C}\mathcal{C}(\eta_{e})$ (26)

formalizes the bottleneck, and the ultimate capacity is given by the
corresponding minimum sum. This is of course just a network flow problem, with
maximum flow of secret bits achieved by flooding the network such that every
unique path from the source to the sink is utilized at the highest possible
capacity.

The enormous drawback of trusted networks is that every link can know the
secret bits it transmitted. As any of the nodes involved could in principle
leak them they must be assumed to be isolated from any unauthorized parties,
which is the trusted node hypothesis. Although there are ways to mitigate this
somewhat [28], the lack of end-to-end security makes real-world QKD networks
strongly gravitate towards private and non-commercial, inhibiting their
growth—conversely, this is why having few or no trusted nodes is significant.
The properties of complex QKD networks are best analyzed using suitable random
network models. In particular, it turns out that under reasonable additional
assumptions and a fixed protocol a fiber network leads to a Poisson degree
distribution but a satellite network to a log-normal distribution. We also
present results showing how satellites are a game changer in long haul
communications, making them a strong candidate for a backbone that connects
smaller networks together [415, 369].

The case of a fiber network was recently considered in [364]. The nodes are
embedded in a disk and the fibers are distributed according to the Waxman
model [416]. Then the probability of a link decreases exponentially with
distance but is adjusted by parameters controlling the maximum distance,
typical distance and average degree, chosen here to mimic the U.S. fiber-
optics network. Next each fiber connecting some nodes $i,j$ is assigned a
probability

$p_{i,j}=1-(1-q_{i,j}(d_{i,j}))^{n_{p}}$ (27)

of a successful photonic link where its length $d_{i,j}$ controls the
transmissivity $q_{i,j}(d_{i,j})=10^{-\gamma d_{i,j}/10}$ with $\gamma=0.2$
dB/km and where $n_{p}$ is a free parameter controlling the number of attempts
made. Although a fixed value $n_{p}=1000$ was used for main results it was
reported that the properties were not sensitive to the value of $n_{p}$. The
degrees were found to follow a Poisson distribution controlled by the density
of nodes with a giant component appearing at relatively low densities. The
model was found to exhibit large clustering, but perhaps unsurprisingly not
the small-world property as far away nodes required many intermediate nodes to
reach one another.

This model was compared in [365] to a network where a satellite shares Bell
pairs to ground stations uniformly distributed in a disk, playing the role of
the nodes. This kind of architecture where a central node merely generates and
distributes Bell pairs is known as entanglement access network. Remarkably,
the central node can be untrusted as the secret bit is created when the halves
are measured, not when the state is prepared. Here the cost is requiring a
simultaneous line of sight which provides a hard limit to the size of the
disk; in experiments, such satellite links have achieved 1200 km [417]. The
satellite is assumed to be stationary at $h_{\mathrm{sat}}=500$ km above the
disk’s center, which could correspond to a sun-synchronous orbit—daily
transmission bursts can be imagined. The probability that an entangled photon
is received by some ground station $i$ is $p_{i}(d_{i})\in(0,\eta_{0}]$ which
decreases exponentially with distance $d_{i}$ to the satellite and where
$\eta_{0}\approx 0.1$ is an empirical value accounting for various
imperfections. Two nodes $i$, $j$ are taken to be connected if after $n_{p}$
trials, at least once both nodes receive their half. The probability is

$\Pi_{i,j}=1-(1-p_{i}(d_{i})p_{j}(d_{j}))^{n_{p}}.$ (28)

Crucially, here each node has its own distance. The smaller the distance
$d_{i}$ is for some node $i$ the higher the probability $\Pi_{i,j}$ for any
$j$, making nodes near the center more attractive than nodes in the periphery.
This bias was found to lead to the appearance of hubs as well as the the
small-world property, and the degree distribution was found to be closely
approximated by a log-normal distribution. The satellite network was also
found to cover large areas with less nodes for a fixed number of trials
$n_{p}$ whereas the hubs increased robustness to random failures but decreased
it against targeted attacks.

Fiber based local or metropolitan area entanglement access networks have been
envisioned. Many have also been built [418, 419, 420], however scaling such
networks to a large number of users is challenging as discussed at length in
the cited works. This and the limited reach have been proposed to be
alleviated by a hybrid architecture where many such networks are connected by
a single shared trusted user [421]. Importantly, this would still leave all
the other nodes untrusted.

Figure 23: Comparison of daily secret key rate achievable with a repeater
chain (black solid lines) and a satellite (red lines). The capacity in the
former case is essentially based on Eq. (25) and increases with separation
between repeaters, whereas the satellite operates as a classical trusted
repeater. As it passes over the stations once a day the daily rate is distance
independent. A clock speed of $10$ MHz is assumed. Figure reproduced from Ref.
[384], doi: https://doi.org/10.1103/PhysRevX.6.041062, license:
https://creativecommons.org/licenses/by/4.0/.

A thorough approach to satellites was taken in [384] which derived practically
achievable daily secret key rates between two distant ground stations
connected by a single sun-synchronous satellite. Importantly, although the
rate is still limited by the PLOB bound modified by the effect of the
geometric position, the rate-distance scaling is more favorable [422]. As one
round always takes a day the rate is distance independent, however as there is
no simultaneous line of sight the satellite must be trusted. It should be
stressed however that it is remarkable how a global distance can be covered
just by a single untrusted node which is hard for unauthorized parties to
directly access as it is in orbit. These rates have been benchmarked against
two ideal fiber based alternatives: a chain [384, 368] and a lattice like
network [368] utilizing ideal quantum repeaters. As anticipated, the chain
achieves $\min_{e\in P}\mathcal{C}(\eta_{e})$ and the network $\sum_{e\in
C_{\mathrm{min}}}\mathcal{C}(\eta_{e})$ with end-to-end security. For any
fixed number of links $L$ in the chain there is a total distance beyond which
the satellite is superior [384] as seen in Fig. 23, whereas to reach a
superior distance independent rate the maximum link length should be around
$200$ km or less [368]. The network was taken to be degree regular with
restrictions on neighbor-sharing properties of adjacent nodes to facilitate
analytical treatment. Distance independent rate requires that the minimum cut
$C_{\mathrm{min}}$ is distance independent, which in this case can be
connected to both maximum link length and nodal density, and critical values
to beat the satellite may be derived for different unit cells. All in all it
was found that for long distances, a single trusted satellite can already
achieve rates that would be very costly to beat even with highly idealized
fiber networks.

Before concluding we highlight two exciting and potentially disruptive avenues
to push networks within reach further: trusted node free QKD between next
nearest neighbors [386] and long distance transmission of quantum states with
a chain of co-moving untrusted satellites equipped with reflecting telescopes
[385]. Remarkably, the former scheme can already break the PLOB bound of Eq.
(23), achieving a secret bit key rate that scales with $\sqrt{\eta}$. Although
the node is not a repeater, meaning that the rate cannot be boosted further by
introducing more such nodes, the scheme can be realized with existing
technology and recent experimental results are very promising [388], achieving
a record distance of 830 km in fiber. The satellite train on the other hand
could receive a photonic state from a ground station and reflect it from
satellite to satellite, bending with the surface of the Earth, finally
reflecting it to the receiving ground station. Simulations are encouraging,
predicting acceptable losses over global distances. Together with other
presented results this underscores the indispensability of satellites for
achieving such coverage in the near-term future.

####  VIII.2.2 Entanglement distribution: prerequisite for quantum information
networks

Moving from secret bits to qubits prevents the use of classical trusted
repeaters. A QI network utilizing quantum repeaters can be imagined, but such
a network is then subject to the no-cloning of quantum information which rules
out signal amplification and also prevents making back-up copies: the
transmission of a single unknown qubit can only ever be attempted once. Under
these circumstances the network would need a perfect quantum channel which is
noiseless, always succeeds and can cover as much distance as classical
channels. Teleportation can achieve this; given pre-shared entanglement it can
be consumed to swap the qubit to the receiver via local operations and
classical communication (LOCC). This requires entanglement distribution,
namely preparing entanglement between two marked nodes in a network. Due to
non-increase of entanglement under LOCC [423], this unavoidably involves
transmitting entanglement bits, or halves of a Bell state. Importantly, there
is a crucial difference between unknown qubits and entanglement: we are free
to prepare as many Bell states as we like and use them only as fuel for the
virtual teleportation channels that will handle the actual communication of
quantum information.

While fundamental rate limits of quantum channels have been quantified in many
ways [424, 425, 426, 427], here we still focus on the PLOB bound. It turns out
that for lossy bosonic channels considered here the ultimate capacities for
secret bit, qubit and entanglement bits all coincide. Indeed, a shared Bell
state can either be converted into a secret bit or a qubit. Importantly, these
rates correspond to exact Bell states which can be expected to require
entanglement distillation where many sufficiently entangled noisy states can
be probabilistically converted into less states with stronger entanglement and
higher purity via LOCC, not increasing it on average. The PLOB bound is
closely related to ultimate entanglement distillation rates, which in
particular require an unlimited mean photon number to be achieved; this is why
$C(\eta)\xrightarrow[\eta\to 1]{}\infty$. Remarkably, an explicit distillation
protocol achieving these limits has very recently been introduced [428].
Initial links are created by transmitting entangled photons and is known as
remote entanglement generation. Once distilled, short entanglement links can
be converted into longer ones with entanglement swapping, which replaces two
incident links by a longer link, effectively ”detaching” from the shared node.
At this point nodes adjacent in the entanglement layer no longer need to be
adjacent in the channel layer. These common entanglement distribution
primitives are depicted in Fig. 24. Some more recent proposals consider
quantum error correction which might reduce the classical communication
overhead [358], but the same ultimate capacities still hold.

Figure 24: Common entanglement distribution primitives. Dashed lines depict
bipartite entangled states shared by the nodes. $(a)$ Entanglement
distillation (or purification) turns several states with relatively weak
entanglement and low purity into a state with stronger entanglement and higher
purity. $(b)$ Entanglement swapping turns two incident links into a single,
longer link connecting the end points, in general at the cost of purity. $(c)$
Elementary link creation, also called remote entanglement generation, creates
the initial short distance links with the help of, e.g, optical fiber and
distillation. The rate cannot exceed the repeaterless PLOB bound.

Some form of quantum memory is typically assumed to facilitate the repeated
use of the primitives. For simplicity we assume that the memories can store an
arbitrary number of qubits and have infinite coherence time, any local
operations can be carried out, and there can be unlimited classical
communication. Now we are in a position to relatively easily introduce quantum
repeaters. In fact under such strongly ideal conditions they can operate
analogously to the classical trusted repeaters with shared random string links
replaced by entanglement links, secret key distillation replaced by
entanglement distillation and secret bit swapping—transmission of secret bits
by consuming local secret bits—by entanglement swapping. Such networks can
achieve the capacities of Eqs. (25) and (26) for example by operating in
continuous generation mode with the crucial difference that the capacities now
concern also entanglement bits and qubits and the repeaters can remain
untrusted—consequently the networks could be public and commercial, fostering
growth. It is highly nontrivial that the capacities cannot be exceeded; this
was proven in full generality in Ref. [414] which considered also other types
of channels.

One may ask what kind of capacity distributions can be expected for the links
and the nodes; the latter is just the total capacity of incident links, or the
weighted degree. Considering expected end-to-end capacity, it can be argued
why both unusually high and unusually low capacity links might be absent. For
former any capacity in excess of the bottleneck will be wasted, whereas for
latter the link is a bottleneck at worst and not particularly useful at best.
The capacity distribution can then be expected to be relatively narrow around
the mean value, as in for example a Poisson distribution. The expected node
capacity is arguably the simplest upper bound for the expected end-to-end
capacity since the bottleneck cannot be larger. Similar arguments apply also
here. If link capacities are indeed all rather similar then it follows that
not only node capacities but also (unweighted) degrees will be distributed
close to the mean value. These speculations are in line with recent results
comparing Waxman and scale-free networks [370], where for the latter the
probability of a new link was $p_{i,j}\propto k_{i}/d_{i,j}$ where $j$ is a
node to be added and $k_{i}$ the current degree of an old node $i$. Each new
node is connected to $m$ old ones. This results in hubs, which however were
found to inhibit the expected end-to-end capacity since they attract links
from great distances which leads to an abundance of low capacity links and
nodes. This is exacerbated by limiting the number of links to $m$ per node
which means that every low capacity link is one less decent to high capacity
link. Indeed, for scale-free networks the capacity was found to saturate as
node density was increased, but for the Waxman networks it increased linearly.
For both the expected capacity was found to abruptly start increasing after a
critical node density which importantly was higher than the density required
for the giant component. One may also consider the robustness of such networks
to different imperfections such as loss of nodes or links. This was done in
[429] where it was found that while the capacity decreased linearly under
random breakdowns for both, the scale-free network was very vulnerable to
targeted attacks as the loss of only a relatively few hubs in terms of either
capacity or degree significantly decreased the average end-to-end capacity.
The results hold as is also for the ultimate secret bit capacities of fiber
based trusted repeater networks.

First proposed in 1998 [356], the original and later repeater protocols made
various assumptions about imperfections but not about memory until recently.
Unfortunately, an imperfect memory is both unavoidable in realistic models and
arguably the Achilles’ heel of repeater networks as they have been designed
assuming scalable accumulation of resources to facilitate a repeat-until-
success approach for every subtask, as will be elaborated on in the next
Section. For now, we introduce entanglement percolation, proposed in 2007
[357] as an alternative for repeater networks designed specifically to operate
entirely on-demand to ease the memory requirements. In short, starting from a
given initial state it makes a single attempt at distributing the entanglement
such that there is a phase transition in the success probability where it
abruptly becomes distance independent at a critical value of initial
entanglement. If it fails the protocol must start from scratch.

Assuming an initial state for the network where each link shares an identical
pure but non-maximally entangled state, entanglement percolation focuses on
singlet conversion probability (SCP), or the probability to reach from a given
initial state a Bell state shared by given nodes—including adjacent nodes as a
special case—using distillation and swapping. The links have some
$\mathrm{SCP}=p<1$; if conversion fails the link is lost. Swapping preserves
SCP but not purity [357, 430, 431], and swapping the resulting mixed state
again is not done as this would decrease SCP [430]. The goal is then to use
probabilistic conversion permitted for any link and deterministic swapping
permitted for pure state links to form at least one path of maximally
entangled states between the given nodes. The nodes may then be directly
connected using swappings. The central question concerns the sufficient amount
of preshared short range entanglement, as quantified by $p$, for entanglement
distribution to beat the exponential scaling of direct transmission.

Figure 25: An example of quantum entanglement percolation. The lattice in
panel $(a)$ is transformed into the triangular lattice of panel $(b)$ by
swapping the entanglement at the circled nodes, lowering the percolation
threshold. Reprinted figure with permission from [366]. Copyright 2008 by the
American Physical Society.

In a strategy called classical entanglement percolation (CEP), first
simultaneous conversion of all links is attempted, which divides the network
into connected components where links are now maximally entangled. CEP
succeeds if the target nodes are in the same component and otherwise fails.
The anticipated phase transition occurs at $p\geq p_{\mathrm{th}}$ where
$p_{\mathrm{th}}$ is the network percolation threshold, since for $p\geq
p_{\mathrm{th}}$ a giant component appears and $\mathrm{SCP}=\theta(p)^{2}$
where $\theta(p)$ is the probability that a node is in the giant component.
For $p<p_{\mathrm{th}}$ SCP decreases exponentially with distance, making CEP
useless. Remarkably, often the percolation threshold $p_{\mathrm{th}}$ may be
lowered by first reshaping the network with swapping, facilitating
entanglement distribution even when $p$ is not enough for CEP; this strategy
is called quantum entanglement percolation (QEP). Typically when QEP is used
each link is assumed to be a product state of two identical states to
facilitate reshaping the network, whereas in the reshaped network links have
only one state.

It was shown in the seminal work [357] that in open chains CEP is not optimal
but gives the correct asymptotic scaling which is exponential for all $p<1$; a
single failed conversion is fatal. In 2D lattices the possibility of QEP was
demonstrated with a honeycomb lattice which was reshaped into a triangular
lattice as shown in Fig. 25. Optimizing SCP in lattices was considered in
[366]. QEP was successfully generalized beyond lattices in [367] where it was
shown that reshaping could be done based on local information only and
moreover the advantage of QEP over CEP can be significantly larger in random
networks. Vulnerability of such networks to attacks was considered in [432].
The framework has also been generalized to $n$-partite maximally entangled
states and generalized swapping called $n$-fusion. The case $n=3$ was shown to
lead to advantages over $n=2$ (Bell states) in lattices in [433], and recently
it has been shown that for $n\geq 3$, distance independence of the success
probability remains possible even if the $n$-fusion is probabilistic and
sometimes fails [434]. Going beyond two states per link, a strong advantage of
QEP may be achieved even in chains but at the cost of more LOCC operations per
node [435]. Finally, the related problem of when various subgraphs can appear
as $p$ changes was considered in [436], where a quantum strategy was
introduced such that all possible subgraphs appear at the same threshold
value.

Entanglement percolation has recently been reviewed in great detail in Ref.
[437] which also compares it to a novel approach called concurrence
percolation [438, 439, 440]. Switching from SCP to concurrence, a measure of
bipartite entanglement, serves as a basis for a new type of percolation that
still uses essentially swapping and distillation—conversion of series and
parallel links to single links—but in general no longer attempts to convert
any of the states to Bell states. Informally speaking, this leads to a more
economical use of the available resources. Indeed, concurrence percolation has
been found to achieve a lower critical threshold for success than other
approaches in many lattices [438], whereas in random networks the advantage is
supported by numerical evidence [439]. Furthermore, a non-trivial saturation
point can appear where a non-maximal amount of initial entanglement can
suffice for entanglement distribution to both succeed with certainty and lead
to a Bell state between distant nodes. In contrast, CEP/QEP have only the
trivial saturation point at $p=1$. As noted in the review, there are still
open questions and work continues.

Like repeater networks, percolation networks are not ready for deployment.
Whereas early repeater protocols took perfect memories for granted, early work
on CEP/QEP took a pure initial resource state for granted. A more realistic
initial state would be mixed but as will be seen in the next Section this
leads to problems. Furthermore, conventional proposals require a high
percolation threshold whereas compensation with more states per link might
require some accumulation as the creation of each initial entanglement link
must still respect the PLOB bound of Eq. (23). For the same reason the
physical link length is still limited for CEP/QEP to work. Concurrence
percolation is promising and its generalization to mixed states is an
important open research direction.

We conclude by pivoting from entanglement distribution to network-generalized
nonlocality. Consider two nodes receiving a Bell state from an untrusted
source. They can measure it in either, say, basis
$\\{\left|0\right>,\left|1\right>\\}$ or
$\\{\left|+\right>,\left|-\right>\\}$. If the two nodes happened by chance to
choose the same basis they have shared a secret bit because there can be no
local hidden variable involved in the preparation of the state that, if known,
would allow the prediction of the measurement outcomes before the measurements
have been carried out. This is in fact a consequence of the Bell state
violating the celebrated Clauser–Horne–Shimony–Holt Bell inequality [441]
which must be obeyed by all bipartite correlations with binary outcomes and
two possible measurement settings amenable to a local hidden variable model.
In general, Bell inequalities separate local and nonlocal correlations and
under certain mild assumptions are amenable to a geometric interpretation as
hyperplanes that define all local correlations as their convex hull [442], the
so called local polytope. The set of quantum correlations contains the local
polytope as a proper subset, meaning that some of them violate a general Bell
inequality.

In the past decade it has been recognized that in a more general network where
links are Bell pairs generated by independent sources between each adjacent
pair of nodes, qualitatively new inequalities arise that separate local and
nonlocal correlations at the network level, as recently reviewed in great
detail in [22]. Importantly, the set of network local correlations is
contained inside the local polytope as a proper subset, meaning that there are
correlations which violate a network Bell inequality without violating any of
the ordinary Bell inequalities—if correlations are all assumed to be local
then this means that the assumed network structure must be false. Furthermore,
the set of network local correlations is not convex, complicating its
characterization. One may consider even more general scenarios if multipartite
entanglement is introduced. As pointed out in [22], the field is still facing
many open questions.

Very recently also quantum steering has been generalized to networks [23].
Unlike previously, here some of the nodes are trusted and one considers the
conditional states that the untrusted nodes can prepare for the trusted nodes
by performing local measurements. In the absence of any correlations the
states of the trusted nodes are of course independent of any local operations
the untrusted nodes perform, whereas in the case of a shared Bell state the
untrusted node can choose to project it to an arbitary basis simply by
measuring in that basis. Between these two extremes one may consider whether
the effect can be explained with local hidden variables, and for quantum
correlations in particular this is not always the case. When it is not, it is
said that the shared state is steerable. In networks and under certain
conditions, it was found that the set of steerable and network steerable