states are not necessarily the same, however several no-go results where also
introduced forbidding network steering in many scenarios.

These research avenues are closely related to the study of the relationship
between the structures of the fiber and entanglement layers. This in turn
naturally depends very strongly on the assumptions one makes about the initial
short range correlations and the allowed operations. On the one hand they may
lead to forbidden correlation structures that the underlying fiber network
simply cannot create as in, e.g., [443] and as mentioned above. On the other
hand one may conclude that in fact even a physical chain suffices for the
creation of a variety of entanglement networks such as lattices, random
networks and small-world networks [444].

####  VIII.2.3 Road to quantum Internet: public commercial quantum information
network

The most important takeaway of a recently proposed roadmap [3] towards a full
blown quantum Internet is that we may reap benefits not only at the end but
also continuously along the way, with each new stage unlocking previously
unavailable applications. This is great news because the road is long and
rocky and our maps unreliable. Some of the different activity sectors
benefiting from the developing quantum Internet were identified in [411] as
industry, critical infrastructures, finance, administrations and operational
as well as fundamental science. Importantly, different sectors were proposed
to have different requirements; for example, whereas industry and science
might tolerate relatively high latencies and low entanglement distribution
rates, administrations would not. Returning to the roadmap, it envisions three
stages of networks capable of QKD and some related protocols not discussed
here, followed by another three for QI networks.

The former achieve QKD with trusted nodes, without trusted nodes and with
device independence. Although not exactly corresponding to the proposed
architecture the first two have already been reached by fiber based trusted
repeater and entanglement access networks, respectively. If the end nodes of
the latter could also carry out deterministically any single qubit
measurements they could switch to device independent protocols which both
relax certain conventional assumptions and close some loopholes related to
experimental imperfections or vulnerabilities as presented, e.g., in [380]. We
are certainly at least at stage one but trusted node QKD cannot be expected to
be valuable enough for the networks to grow to their theoretical continental
[364] or even global service area [384]. It is arguable whether we are past it
already for example because the diameter of an entanglement access network is
limited by the repeaterless PLOB bound, which in fiber translates to roughly
100 km, and the number of users by some technical difficulties. Some recent
developments and proposals discussed in the end of Sec. VIII.2.1 might push
the limit of trusted node free networks much farther than this in the near-
term future however, which might be interpreted as reaching stage two or
three.

The QI networks might be described as teleportation, distributed quantum
computing and quantum computing networks. Reliable teleportation of unknown
qubits can be achieved if the network is equipped with quantum memories and is
capable of arbitrary local unitary operations. While the network diameter and
service area could be limited by imperfect memories and lossy operations it
could provide for example secure cloud quantum computing where clients with
limited capabilities outsource demanding computations to an untrusted service
provider. Using so called homomorphic encryption [445] the data provided by
the client remains private; using blind quantum computing [446] even the
algorithm remains private, a feat which cannot be achieved for all algorithms
in classical computing. Other proposed applications include improved clock
synchronization [447] and extending the baseline of telescopes [448]. The
final stages introduce fault tolerant quantum computing to all end nodes,
performing classically intractable computations either at the network level
with distributed quantum computing or also at the single end node level. In
the latter case the network could perform tasks related to facilitating
efficient co-operation of local computers with an advantage over their
classical counterparts [449, 450].

Aside from the development of its abilities, one may also consider how the
quantum Internet could develop as a network. Distance is crucial as even in
ideal conditions it controls the overhead. Tentatively three different regimes
may be identified: short, intermediate and long. Point-to-point optical links
are feasible only for the first, whereas for intermediate distances the
overhead and cost of using entanglement distribution in fiber would still be
acceptable. Long distances require solutions where even the overhead is
(almost) distance independent. This could be achieved by powerful quantum
memories moving on board a satellite or as freight [451, 362]; the latter case
is known as the quantum sneakernet. Crucially, provided that local
entanglement stores are maintained the bottleneck would be the time it takes
to carry out a teleportation protocol, i.e. it would be limited mainly by the
classical communication rate. As a side note, the classical capacities are
still very rarely taken into account although they affect both use cases [411]
as well as efficiency of QKD [452] and presumably other applications. Both
sneakernet [451, 362] and satellite links [415, 369] have been proposed as the
backbone for the quantum Internet. The backbone would stitch together networks
where distances are in the short or intermediate regime.

Figure 26: Examination of how trusted node free QI networks could evolve for
certain generic architectures as elementary link creation probability, memory
coherence times and fidelities increase. See text for details.

For the remainder of this Section we very briefly examine possible evolution
of fiber networks based on entanglement access, end-to-end transmission and
entanglement percolation as well as satellite based entanglement access
networks, shown in Fig. 26. Specifically, an entanglement percolation network
first attempts to create a large entanglement cluster which is then
manipulated to distribute entanglement where possible and needed, whereas in
end-to-end transmission the nodes that would like to communicate are
predetermined and the network might opt for the single highest probability
path connecting them. For simplicity, only a single satellite is considered;
using many offers new possibilities as shown in, e.g., [453, 454, 455, 385].

In the near term rates are still restricted by the repeaterless bound which
then strongly restricts the diameter. The appeal of entanglement access
networks is that the end nodes do not need to be able to generate the initial
short distance entanglement whereas in end-to-end case they do. Using
entanglement percolation might increase the range especially if the physical
links are short enough to keep the states almost pure, at the cost of
requirements for the percolation threshold which should translate to
requirements for the link and node density. Satellite links have more lenient
rate-distance scaling but without memory an untrusted satellite requires a
simultaneous line of sight. This leads to perhaps the largest but still
limited area of service. The case of networks approaching the ultimate limits
is not too difficult to envision as we may now assume strongly ideal
conditions similar to those in the previous Section. For entanglement access
networks a modular structure where several such networks are connected
together by a shared user may be envisioned, as already proposed for networks
running QKD and related protocols [421]. The line between end-to-end
transmission and percolation networks blurs as both could switch to flooding;
fiber backbones could appear, limited mostly by cost-effectiveness since now a
satellite could connect any two nodes near its ground track.

The NISQ case introduces both soft and hard diameter restrictions arising from
lossy memories, noisy probabilistic operations and mixed states of limited
fidelity and energy. It must be stressed that it both covers an enormous leap
in technology—even assuming that one day we will have a fully developed
quantum Internet, it might be expected that QI networks will stay in the NISQ
regime much, much longer than in the near term regime—and has features which
are missing from both near term and nearly ideal networks. At best, they limit
how much the service area can grow from near term regime; for both fiber and
satellite based entanglement access networks at least a second end node
cluster should become possible, in the latter without a simultaneous line of
sight.

Considering percolation, for simplicity we may imagine a kind of 0th order
approximation for imperfect memories: the entangled resource states are
maintained perfectly up to some maximum time and then vanish, forcing us to
minimize temporal overhead by relying on parallelism. But considering mixed
states in general leads to a $\mathrm{SCP}=0$—although in special cases it may
survive if the number of resource states is increased and adapted strategies
are used [456, 457, 458]—makes QEP impossible and the final fidelity path
length dependent [430, 459]. We will not consider imperfect repeaters with
perfect memory, which are covered quite well in Ref. [358]. When states in
memory decohere, both maximum [460] and minimum [262] distances where the
repeaters can beat the repeaterless PLOB bound may appear. On-demand
generation is commonly assumed, leading to complicated waiting time
distributions [427] as then stochasticity dominates, which in particular
complicates the prospects of a neat network description. Decoherence
introduces a maximum waiting time [461] after which resources can no longer be
distilled and the entire protocol might have to be restarted. When also the
operations such as swapping are probabilistic the physical overhead quantified
for example in terms of memory qubits grows rapidly [462, 262].

These challenges have in part prompted the introduction of hybrid strategies.
One approach is to replace solid state memories used in conventional repeaters
with highly entangled resource states called repeater graph states [261];
proof-of-principle experiments have already been carried out [390, 391]. Like
in percolation, temporal overhead is reduced by massive parallelism and losses
are managed by introducing many alternative paths. Also like in percolation
the challenge is shifted to the generation and manipulation of the resource
states, although there has been some recent progress regarding the former
[463]. As noted in Ref. [262], this approach in particular suffers also from a
poorly scaling physical overhead and a minimum distance to beat the
repeaterless bound. Combining entanglement percolation with lossy repeaters
was considered in [464] where it was found that allowing for some repeated
attempts at the subtasks lowered the critical probability of initial link
creation for the emergence of a giant entanglement component.

At this stage, the growing networks create a need for concrete and practical
protocols for distributing the entanglement. Focusing on end-to-end case,
conventional path finding and routing algorithms cannot be used directly but
with suitable modifications may achieve good [465] or even optimal performance
[466], however in general optimality may require specific properties from the
network [467]. Efficient algorithms for finding the shortest path are possible
even under quite general conditions [468]. If there are many overlapping
requests the network might need to operate on-demand to decrease the average
latency [469]. If demand is low or higher latency is acceptable the requests
may be handled one at a time, in which case multipath routing may more easily
allow to beat the repeaterless rate [470], or in batches, in which case
computationally efficient and optimal routing algorithms have been proposed
for specific architectures [466]. Routing multipartite entanglement may also
be considered. Ref. [471] introduced multiple such routing algorithms and in
particular one which simultaneously optimizes both the rate and final fidelity
for GHZ states; here it was also demonstrated how focusing only on the
bottleneck in a non-ideal network can lead to drastically worse results,
underscoring the importance of taking into account the imperfections in any
practical routing algorithm. Optimizing the repeater scheme itself has also
been considered [472] and practical figures of merit proposed, such as average
connection time and largest entanglement cluster size [464].

Robustness of routing with mixed states was considered in [473] where a finite
amount of mixed initial resource states was considered. Since the cost in
resource states to satisfy a given target fidelity is path length dependent,
transitivity in who can reach who may then be broken: if Alice can achieve a
non-vanishing rate at target fidelity with Bob and Bob with Charlie, it does
not ensure that also Alice can achieve one with Charlie. Indeed it was found
that under these circumstances networks can experience an abrupt transition to
overlapping connected components in terms of such a rate as a function of both
the amount of initial resources and the probability of random link failures.
Assuming identical initial resources in all links, critical router
efficiencies were derived to suppress such transitions for various topologies.
Although scale free networks were found to be the most promising, the ones
considered here would in practice require links with both long distance and
relatively high capacity; adding satellite links to a fiber network was
tentatively proposed.

To conclude, it can tentatively be said that with improving efficiency the
possible variation in local node density and complexity could increase
rapidly. What kind of complexity should be expected would depend on the growth
principles of these networks, which in turn should depend on the one hand on
the fundamental limitations and on the other hand on the incentives such as
service requirements and interest.

###  VIII.3 Avenues for further research

Scalability requires managing the impact of losses and operation errors and in
particular either the development of sufficiently powerful quantum memories or
significant improvements in memoryless alternatives. Proof-of-principle
experiments in the near term should demonstrate beating the repeaterless PLOB
bound up to a few lossy repeaters, trusted node free QKD networks with a
diameter in this regime and the use of a quantum satellite equipped with
memory. Percolation and related approaches have experienced somewhat of a
renaissance in the form of hybrid approaches [261, 464] and concurrence
percolation [438]. This very promising theoretical framework opens new
important research questions for the years to come. For instance it would be
crucial to incorporate in this framework the preparation of the initial
entanglement links, which quite naturally makes the network metric, and to
investigate how well the main results tolerate mixedness. Meanwhile,
standardisation of mainly trusted node QKD networks but also others is pursued
by several organizations (see Ref. [411] for a recent summary) and in
particular the interplay between the network structure, key management policy
and degree of practical security might benefit from further applications of
network theory. One can also never quite rule out disruptive novel ideas such
as the recently introduced QKD protocol able to break the repeaterless PLOB
bound with a single memoryless intermediate node [386, 474, 475, 387];
although the advantage is limited to that of a single repeater, this is
already substantial and ranges of 830 km have been reported [388]. We also
mention in passing emerging new directions such as going beyond definite
causal order [476, 477, 478], generalizing to entanglement-assisted LOCC by
introducing short distance entangled catalyst states to the network [479],
quantum network coding [480] and pursuing hybrid technology applying, e.g,
both quantum theory describing qubits and continuous variable states of light
[481, 482].

So far a great deal of attention has been given to restrictions concerning
both near term and ideal networks, however there is still much room for
further work. For example, whereas entanglement distribution and simulation of
intermediate stage networks has been considered, random network models
incorporating their limitations and objectives such that they could be used
without deep understanding of the microscopic theory are still missing. A good
example of what would be needed to build such models are ideal capacity
weighted networks: they can be readily applied with just a superficial
understanding of the physics and the engineering, have a clear interpretation
as networks of ideal quantum repeaters connected by pure loss channels and
provide meaningful benchmarks. Introducing something similar for intermediate
stage networks is of course a great challenge for the quantum networks
community as, e.g, deriving the waiting time distributions or the final
fidelity can become involved already in chains [483]. Furthermore, more
research on growth principles for networks at all stages is needed. Such
principles can be expected to consist of both limitations and incentives that
together govern the evolution of future quantum networks, and work on
especially the incentives is scarce, with some notable exceptions such as Part
VIII of Ref. [362].

Interesting connections between these and induced quantum networks introduced
in Sec. V could also be further explored in the context of resource states for
all-optical repeaters, new simulation tools and analysis or improvement of
entanglement management policies. Alternatively, one might consider
distribution of graph states in large scale quantum networks as in [260, 484,
485], which might facilitate new applications.

##  IX Discussion and future directions

Whereas complexity is what empirical networks seem to naturally gravitate
towards, quantumness is coy and needs to be cajoled to manifest by isolation
of the systems. The two meet in the following research lines: network-
generalized quantum problems, quantum-applied network theory, quantum-
generalized approaches for complex networks and quantum-enhanced
communications, which are currently pursued by scientists and researchers from
a variety of backgrounds. In this review we have introduced the four main
research lines, unified them under the broader context of complex quantum
networks and provided a comprehensive overview of the field.

There are multiple promising directions for further development of the field.
From the quantum systems perspective, new ways to generalize quantum phenomena
to a network scenario can be envisioned. We highlight chiral quantum walks as
an example where the underlying graph is no longer undirected and which can
have advantages over both classical and sometimes also conventional quantum
walks [202, 204, 203]. Regarding quantum correlations in a network scenario,
most work still focuses on Bell nonlocality, leaving room for other types:
steering [23], entanglement and discord. Speaking of applications of network
theory to the quantum case, a network description of a stationary state has
already been explored as a cheaper alternative to state tomography [24, 14,
222, 223] but generalization to evolving states and process tomography has
only been suggested. Using a network description to not only detect but to
discover previously unknown phenomena [25, 224] remains in its infancy.
Pivoting to quantum enhanced communication networks, both within reach trusted
node and ideal quantum repeater networks can be modeled compactly as just a
network of channels weighted by capacities. This description can be applied
without a deep understanding of the relevant physics and the results have a
clear interpretation. Introducing similar models for the NISQ stage is
undoubtedly challenging but if it could be done the field could have
contributions from researchers from a much wider variety of backgrounds and
specializations.

From the network science perspective the field of complex quantum networks
could also be transformative. For instance the new generation of quantum
computers could lead to the flourishing of quantum algorithms for classical
network inference leading to further understanding of their complexity.
Moreover new quantum technologies could be key to design complex quantum
networks in experimental set-up leading to novel quantum phenomena displaying
a rich interplay between topology and dynamics. From the dynamical point of
view directions that are particularly promising and that lie at the
classical/quantum interface are progress on quantum synchronization [122] and
on network dynamics dictated by the topological Dirac operator [27]. Finally
the full potential of networks as a powerful tool to understand quantum matter
is not yet fully explored and provides a very promising direction for
unsupervised detection of quantum phase transitions.

As we have seen, the field has already produced important contributions in
each of its research lines which have so far progressed and evolved mostly
independently with some notable exceptions. For example, Hamiltonians derived
from a graph can be both simulated with cluster or graph states [174, 175] or
be used for their adiabatic preparation [264]. The states in turn could be
used to replace conventional quantum memories [261] or for error correction in
quantum communication networks [486], but for very short distances one may
consider state transfer or transport with suitable Hamiltonians again [161,
20]. If the communication network could be prepared into a continuous variable
cluster state entanglement could then be distributed using the protocol of
[228], found to be efficient in particular in sparse complex networks.
Finally, networks with comparable complexity to the classical Internet can be
constructed even from the ground state of a two dimensional spin lattice by
taking as nodes not individual spins but regions of spins of varying sizes and
as links entangled clusters of spins shared by exactly two nodes, constituting
communication channels [91]. However we believe that there remains much more
potential in the ways the lines could further couple together. In the light of
the above the interaction between the different research lines and the
interdisciplinary collaboration between physicists and network scientists will
be key to foster new discoveries in the field and to address the new
challenges that the next generation of quantum technologies will require
science to face.

###### Acknowledgements.

JN gratefully acknowledges financial support from the Turku Collegium for
Science, Medicine and Technology as well as the Academy of Finland under
Project No. 348854.

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