Abstract
Quantum networks allow us to harness networked quantum technologies and to develop a quantum internet. But how robust is a quantum network when its links and nodes start failing? We show that quantum complex networks based on typical noisy quantumrepeater nodes are prone to discontinuous phase transitions with respect to the random loss of operating links and nodes, abruptly compromising the connectivity of the network, and thus significantly limiting the reach of its operation. Furthermore, we determine the critical quantumrepeater efficiency necessary to avoid this catastrophic loss of connectivity as a function of the network topology, the network size, and the distribution of entanglement in the network. From all the network topologies tested, a scalefree network topology shows the best promise for a robust largescale quantum internet.
Introduction
Quantum networks are a paradigm of networks where the links and nodes obey the laws of quantum physics^{1,2,3}. Namely, the quantum links can be quantum correlations^{4}, quantum couplings or dynamics^{5,6}, or even quantum causal relations^{7}. Quantum nodes can be any system with quantum degrees of freedom. The nascent field of complex quantum networks^{4,8,9,10,11,12,13,14,15,16} is motivated both by the fundamental interest in understanding the nature and the properties of this object, as well as by the applied perspective of developing networked quantum technologies to fully harness their potential and their reach. The latter could be named for quantumsecure communications^{17,18}, quantumaccelerated computation^{19,20,21}, quantumenhanced sensing and metrology^{22,23,24}, and the development of a future quantum Internet^{1}. However, quantum systems and states are vulnerable to noise in general. But how does this translate to the network realm, i.e., how robust are noisy quantum networks, and how is that robustness affected by the underlying graph? And how does it compare to the robustness of classical networks, which typically evolve, to nontrivial network topologies^{25,26,27}, such as scalefree properties, topologies that are known to maintain their functionality against random failures^{28,29}?
Networks are a set of nodes and links, where each link connects a pair of nodes. This naturally includes complex networks^{25,30} such as the current classical Internet^{31}, a snapshot of which is presented in Fig. 1a. With the goal of investigating a quantum Internet, we consider quantum networks where the links correspond to entangled pairs of qubits, each lying in a different node. Now, imagine we want to realize a quantum operation, e.g., computing, communication, or metrology, between two distant nodes of a quantum network: how can they establish entanglement between them, with a certain target fidelity F_{target}, given the existing quantum correlations in the quantum network?
In our work, we consider an entanglement distribution network based on noisy quantumrepeater nodes, corresponding to the currently envisaged implementation of realistic longdistance quantum networks, distinctly from noiseless, purestate, quantum networks^{4,12}, and from networks based on quantum channels' upperbound capacities^{9,10,11,12,14,15,16}. Let us consider the general scenario where there are N_{ij} noisy Bell pairs with fidelity F_{initial} connecting nodes v_{i} and v_{j}. If necessary, these noisy Bell pairs can be purified to yield n_{ij} = N_{ij}/N_{ft} pairs exceeding a given target fidelity F_{target} (where N_{ft} is the number of initial pairs necessary to generate one F_{target} pair)^{32}. Next, entanglement swapping between link v_{i} and v_{j} and link v_{j} and v_{k} consumes those Bell pairs to create a longerrange entangled pair between nodes v_{i} and v_{k} with fidelity \( {F}_{{{{{{{{\rm{target}}}}}}}}}^{2}\)^{33}. That drop in fidelity means multiple pairs need to be available for entanglement purification to return the target fidelity F_{target} (again consuming more pairs). These entanglement swapping and purification operations continue at longer distances until we have connected the nodes/users who want to communicate in the network^{34}. A critical question that arises is the resource consumption in such an approach. Fortunately, it is well known that resources required for the firstgeneration quantumrepeater network scale polynomially with the number of links l needed to connect the source node Alice and Bob^{35}. To the leading order in this polynomial, we can define^{35},
as being the number of entangled qubit pairs in the entire chain necessary to create the connected entangled qubit pair with the desired fidelity F_{target}, and r(l) = l^{α} is the number of entangled qubit pairs necessary to create the connected entangled qubit pair with the desired fidelity F_{target} per link. Above α represents the efficiency of the protocol which of course depends heavily on the experimental apparatus used for the repeater scheme and the noise present in it but values in the range [1, 2] are not uncommon (see Supplementary Note 1, Figs. S1 and S2 for further details).
In this work we will show that largescale quantum networks based on noisy quantumrepeater nodes connected by noisy channels are prone to discontinuous phase transitions and that such transitions can be suppressed if the efficiency of the quantumrepeater protocol is above a certain threshold.
Results
The exploration of the connectivity of a quantumrepeater network requires the introduction of two types of connection between nodes, which we are going to call functional and structural connectivity. Functional connectivity in the quantum regime is the situation where a connection between the two nodes can be established with the required fidelity F_{target}. Structural connectivity on the other hand refers to the situation where a connection can be made since there is a path connecting the two nodes, but not necessary with fidelity F_{target}. We will illustrate these two concepts in Fig. 1b, c, d where the nodes v_{1} and v_{3} (and v_{3} and v_{5}) can individually establish connections with sufficient fidelity F_{target} (functionally connected), but v_{1} and v_{5}, while connected, can not (structurally connected). This means v_{1} does not belong to the same “functionally” connected component as v_{5} making it impossible to establish a connection between them with the required fidelity.
Standard Bernoulli percolation, a widely used technique to explore the robustness of classical networks^{25,36}, can not be used in these quantum scenarios due to the quality of service F_{target} requirement (except in the limit α = 0). Although in Bernoullipercolation theory, finding the largest connected component of a network is a computationally easy problem to solve^{25}, finding the largest functional component of these networks is an NPhard problem and can be related to the maximum cliqueproblem^{37} (see supplementary Note 2 for details). Our model is more manageable if one considers the case where all links operate with the same amount of purification and therefore using the same number of entangled qubit pairs in each link, n^{op}. Let us call this quantity the operational number of entangled qubit pairs. The operational number of entangled qubit pairs is a free variable that one can tune in order to maximize network connectivity. It is also associated with an operational distance \({l}^{{{{{{{{\rm{op}}}}}}}}}={\left({n}^{{{{{{{{\rm{op}}}}}}}}}\right)}^{1/\alpha }\), meaning each link can be functionally used in paths of length l^{op} or less.
Our concept of a quantum functional connection naturally suggests that we should choose the operational number of entangled qubit pairs n^{op} as large as possible in order to increase the operational distance l^{op} and therefore allow for nodes further away from each other to distribute Bell pairs with our required fidelity F_{target}. However, increasing the operational number of entangled qubit pairs reduces the probability of a given link having the required number of entangled pairs. Thus there is an important tradeoff to consider. If the number of pairs distributed between nodes can be expressed as a function g(n), then the probability that a given link has the required operational number of entangled qubit pairs or more is given by,
which indicates that for n^{op} larger than a certain value, most links are removed from the network and there is no giant component. Instead one needs to find the smallest value of n^{op}, such that the operational distance is larger or equal to the diameter of the network d, (l^{op} ≤ d). The diameter of the network is defined as the largest distance between any two nodes in the connected component^{25}, therefore any two nodes (associated with the connected component) are able to establish a functional connection (see Fig. 2). This set of nodes is termed the backbone and serves two purposes, first, it can be used as a measure of the connectivity of the network, and second for large networks, they will behave similarly to classical communication networks. Routing developed for classical networks^{38,39} should be sufficient (although not necessarily optimal) to find a good path to connect two nodes with fidelity F_{target}. In such a situation our method guarantees the existence of at least one path connecting any two nodes in the backbone, but given the possible existence of several paths connecting two nodes, if the backbone is large, one could also use a multipath routing approach to generate multiple entangled qubit pairs between them, using different routes, a strategy already proposed in relation to quantum networks^{9,10}.
One should vary the operational distances l^{op} for any change in the network (like the removal of nodes and links) in order to maximize the size of the backbone. One wants to establish the repeater network with just enough resources to reach the diameter of the largest connected network component (l^{op} = d ideally), but the question arises how the diameter of the network d and the operational distance l^{op} change when now one considers that links can fail randomly with p^{ext}. The diameter of the network will depend on the probability that a link both has sufficient pairs to create our link (p^{op}) and that there has not been any random failure in that link is given p^{ext}. The total probability that a link is both operational and not removed is therefore p = p^{op}p^{ext}, and for a given network one can write the diameter as \(\,{{\mbox{d}}}\,\left({p}^{{{{{{{{\rm{op}}}}}}}}}{p}^{{{{{{{{\rm{ext}}}}}}}}}\right)\). The operational distance on the other hand can be written solely as a function of the operational portability p^{op} by inverting Eq. (2). This leads us to the equation,
At this point \({p}_{0}^{{{{{{{{\rm{op}}}}}}}}}\) we have the minimal number of resources required to reach the diameter of the network’s largest connected component. After we find \({p}_{0}^{{{{{{{{\rm{op}}}}}}}}}\), the size of the backbone can be easily computed as the number of nodes in the largest connected component of the network composed of only links with sufficient pairs to create the link, and the link was not removed due to random failures. In the example of Fig. 2, since there are no random failures the size of the backbone is 6. Adding random failures just means that we are starting with a network with some of the links already removed. Our approach is slightly simplistic in that we have only considered links (loss of nodes can also be incorporated). We are now at the stage where we can explore actual networks.
Quantum ErdősRényi networks
There are of course many wellknown network models we could examine with the ErdősRényi model network probably being the simplest^{25,40}. In the ErdősRényi model, N nodes are randomly connected to each other using L = cN/2 links, where c is the average number of links incident to each node. This model has been well explored in complex network theory^{25,40} and as such, it is a good starting point, especially as one can compute l(p^{op}) and D(p^{op}p^{ext}) analytically^{41} when the number of nodes N → ∞ (asymptotic limit). Considering only bond percolation (loss of links) we show in Fig. 3 that the quantum backbone for a large ErdősRényi network is prone to an abrupt phase transition. We observe that the size of the quantum backbone actually drops abruptly as the probability of links not failing p^{ext} drops between a critical probability \({p}_{c}^{{{{{{{{\rm{ext}}}}}}}}}\). As it is usual in a firstorder phase transition we observe hysteresis, therefore the critical probability \({p}_{c}^{{{{{{{{\rm{ext}}}}}}}}}\) is not well defined and the phase transition might occur in a range of probabilities between \({p}_{c1}^{{{{{{{{\rm{ext}}}}}}}}} \; < \; {p}^{{{{{{{{\rm{ext}}}}}}}}} \; < \; {p}_{c2}^{{{{{{{{\rm{ext}}}}}}}}}\), with \({p}_{{{{{{{{\rm{c1}}}}}}}}}^{{{{{{{{\rm{op}}}}}}}}},({p}_{{{{{{{{\rm{c2}}}}}}}}}^{{{{{{{{\rm{op}}}}}}}}})\) corresponding to the largest, (smallest) probability of links not failing where the phase transition might occur. This hysteresis region whose span grows with the size of the network is shown in Fig. 4 (see Supplementary Note 3 for the analytical calculations). We observe that, when the average number of entangled pairs in each link of the network is larger than a critical number of qubit pairs n_{c}, the traditional percolation phase transition is recovered as shown in Fig. 4 (Supplementary Note 4 for details). The critical number of qubit pairs n_{c} increases quickly with the size of the network (see Supplementary Note 4 for details). It is useful to mention that we have used α = 1 as a conservative value (see Supplementary Note 1). As α increases the average number of qubit pairs necessary to avoid the discontinuous phase transition will also increase.
Quantum scalefree networks
These observations lead to a natural question about how general our results are—especially in terms of the network model. As such it is useful to explore the scalefree BarabásiAlbert quantum network whose classical counterpart is known to be more robust than the ErdősRényi^{25,42,43}. In scalefree networks, the degree distribution follows a power law at least asymptotically. This promotes the existence of hub nodes with a degree much larger that the average one (see Fig. 1a for an example). Such distribution is observed in a diverse types of realworld networks^{27}. The BarabásiAlbert model^{44} is a simple example that allows us to generate and explore scalefree networks, based on the preferential attachment principle. This means that when links are added to the network they disproportionately connect to nodes with higher degrees^{27}. In Fig. 5(a, b) we plot the operational distance l^{op}(p^{op}) and the diameter of connected component D(p^{op}p^{ext}) versus the probability a link is operational and not removed p = p^{op}p^{ext} for various network sizes N. Our results for the BarabásiAlbert network show that we are in the supercritical regime for much lower values of p^{ext}, which highlights its ability to distribute entanglement even when a large number of links fail. This is to be expected, the BarabásiAlbert network, and other scalefree networks, are known to be more robust than an ErdősRényi network^{43,44}. More importantly in Fig. 5c we observe no discontinuous phase transition in the N = 10^{3}–10^{5} region (unlike what occurred in the ErdősRényi situation Fig. 3, and therefore there is only one critical probability of links not failing \({p}_{{{{{{{{\rm{c1}}}}}}}}}^{{{{{{{{\rm{ext}}}}}}}}}={p}_{{{{{{{{\rm{c2}}}}}}}}}^{{{{{{{{\rm{ext}}}}}}}}}={p}_{c}^{{{{{{{{\rm{ext}}}}}}}}}\). This is exemplified in Fig. 5d by the absence of a region where both the subcritical and supercritical are stable solutions of Eq. (3). In fact, one can show that there is always a critical quantumrepeater efficiency α_{c} such that for an efficiency large than the critical one α < α_{c} the discontinuous phase transition is suppressed (our network used in Fig. 5a, b, c, d has α_{c} > 1). It is useful to explore this α_{c} parameter in a little more detail. When our resources are exponentially distributed, it is straightforward to show (supplementary Note 4 for details) that α_{c} is given by
which establishes the existence of a sufficient repeater efficiency so that most of the classical behavior is recovered. Despite the suppression of the discontinuous phase transition for α < α_{c} there are still a few differences between the various quantum cases. Unlike what one expects for a typical BarabásiAlbert network ^{25,43} the point at each of the networks breaks apart, does not change significantly with the network size. To understand this, it is useful to look at the relation between \({p}_{c}^{{{{{{{{\rm{ext}}}}}}}}}\) and the classical percolation critical probability p_{c} (which for the usual BarabásiAlbert network tends to zero as N increases). When our resources are exponentially distributed, \({p}_{c}^{{{{{{{{\rm{ext}}}}}}}}}\) (or \({p}_{c2}^{{{{{{{{\rm{ext}}}}}}}}}\) for a discontinuous phase transition) is related to the classical percolation critical value p_{c} by, (see Supplementary Note 5 and Fig. S3. for details)
with \({D}_{\max }\) being the critical network diameter, defined as the diameter of the network at the classical phase transition point \({D}_{\max }\mathop{=}\limits^{{{{{{{{\rm{def}}}}}}}}}D({p}_{c})\). This provides quite an interesting insight into this apparent change of behavior. It is well known that the classical percolation critical value p_{c} tends to zero with the network size for the BarabásiAlbert network^{43}, but so does \({D}_{\max }\) (this is what prevents the suppression of the phase transition). This means that the decrease of \({p}_{c({c}_{2})}^{{{{{{{{\rm{ext}}}}}}}}}\) can be mitigated by increasing the average degree of the network \(\left\langle n\right\rangle\) proportionally with the critical network diameter to the power of α, \({({D}_{\max })}^{\alpha }\).
Measuring the robustness of complex networks
Our exploration of the ErdősRényi and scalefree Bollobás quantum networks has highlighted how the topology of those networks plays a significant role in its robustness, meaning the ability to distribute entanglement in the presence of link failures, but how? We need to quantify this behavior using three important characterization parameters. The first two parameters are related to how many links need to be removed before the network breaks apart while the third is associated with the efficiency of the repeater protocol. These three parameters can be determined for both the ErdősRényi network and BarabásiAlbert networks. It is also useful to determine these parameters for geometric networks. In a geometric network, nodes are distributed across a geometric space, and nodes that are closer are more likely to be connected than nodes further apart. This type of model is very natural in quantum communication networks, given the fact that direct quantum links spanning large distances are difficult to generate. We consider two types of geometric networks, geometric graphs^{45}, where only nodes that are closer than a certain radius are connected to each other, and the Waxman model where the probability p_{ij} that two nodes connect to each other decays exponentially with the distance^{46} as \({p}_{ij}=\beta {e}^{\frac{{r}_{ij}}{R}}\), with r_{ij} being the distance between node i and j, and R the average connection distance, and in our work we considered β = 1. Although we can describe these networks as a function of the number of nodes and links, for the geometric networks these parameters are associated with physical dimensions^{45}. To give an example, a random geometric graph with a connecting radius of 266 km, an average degree c = 6, and a total number of 10^{3} nodes correspond to a network spanning a physical distance on the order of 10^{3} km. This conversion is explained in detail in supplementary Note 4 and displayed on the top axis of Fig. 6. With these four network topologies in mind—ErdősRényi, BarabásiAlbert, geometric random graphs and Waxman model—we plot in Fig. 6 the classical percolation critical probability p_{c}, \({D}_{\max }\), and α_{c} versus N, for values of average degree c = 6, 8, 10. Our plots clearly show that scalefree networks are more robust according to all three parameters, that it is the only network that for the selected parameters is able to avoid the discontinuous phase transition for α > 1 with N > 10^{3}. The reasons for this are as follows. The classical percolation critical probability p_{c} is the typical measure of the robustness of a network in the Bernoullipercolation model. Scalefree networks are known to be extremely robust against random failures in the Bernoullipercolation model as the hubs keep the network connected even when a large fraction of links are missing^{27}. In contrast, the geometric random graphs seem to be the less robust networks according to this parameter. The lack of links connecting distant parts of the network hinders their robustness against random failures. ErdősRényi and scalefree networks are both smallworld networks, meaning the distances between nodes in these networks grow with the logarithm of the number of nodes, and scalefree networks are something called ultrasmall networks because their network distances tend to be even lower^{27}. On the other hand, geometric network models tend not to be smallworld^{45}. As expected the critical distance \({D}_{\max }\) is lower for the scalefree network and largest for the geometric networks. Scalefree networks are also more robust than other networks in terms of the critical quantumrepeater efficiency α_{c}. This can be explained by the fact that α_{c} depends on how the diameter of the network changes when links are removed from the network, Eq. (5). It can be seen from Fig. 3, and Fig. 4 that the diameter of the scalefree network grows considerably slower than that of an ErdősRényi network, explaining why this is the case. The diameters of the Waxman and geometric random graphs show similar behavior to the ErdősRényi network when nodes are removed. Quantum networks based on capacity channel upperbounds^{15,16}, can be seen as a special case of our model when the quantumrepeater efficiency is set to α = 0, a regime in which classical percolation tools can be used, and therefore there are no discontinuous phase transitions. Our results show the importance that the quantumrepeater protocol efficiency plays in a quantum internet, and the importance of choosing the right network topology to mitigate such effects. It is possible, however, to recover the classical behaviors for quantum networks with any topology and sizes for quantum repeaters that operate with α = 0. Realistically, it is unlikely that the first generations of quantumrepeaters networks will be close to that regime, especially when local gate inefficiencies are included. Our results provided a lower, and therefore more practical, threshold for the quantum efficiency required for the construction of a robust network.
There is one more important consideration we must address here in terms of the generality of our results. This relates to our choice of an exponential resource distribution used throughout the paper. The exponential resource distribution was primarily chosen for the ease of our calculations. Other distributions, uniform, for instance, show similar network behavior and our conclusion about the robustness of the BarabásiAlbert networks remains unchanged (see Supplementary Note 4 for details). The form of the quantum repeaters, their operation, and how engineers of the future quantum Internet distribute resources throughout the network will determine what the resource distribution actually is. It is still an open question as to what the optimal resource distribution actually would be.
By introducing the quantum backbone, we derived a metric to measure the connectivity of a quantum network, and have shown how largescale quantum networks based on noisy quantumrepeater nodes connected by noisy channels are prone to discontinuous phase transitions. This abrupt behavior breaks the network into disconnected pieces, severely limiting its operational reach. We found that the discontinuous phase transitions can be suppressed if the efficiency of the quantumrepeater protocol is above a certain value that depends on the topology and size of the network, α_{c}. Furthermore, we have shown that the robustness of a quantum network can be fully characterized using three parameters, the classical percolation critical probability p_{c}, the critical network diameter \({D}_{\max }\) and the critical quantumrepeater efficiency necessary to avoid the firstorder phase transition, α_{c}. Our results capture an inherent fragility that geometric networks possess, they tend to be more fragile than other networks. On the other hand, scalefree networks are more robust than all other networks according to these three parameters, and the required quantumrepeater efficiency necessary to avoid the first=order transition is not too large. We have shown that the right network topology combined with advanced repeater architectures^{47} provide potential solutions for the realization of robust quantum networks. Nevertheless, it does not mean a scalefree network will be sure of a feasible topology for a quantum network.
Discussion
One of the key requirements to generate a scalefree network in geometric spaces is the existence of direct links spanning long distances. As it as been mentioned before, those are problematic to generate for a quantum network. A combination of short links using optical fiber, and long links using communication through satellite might be enough to generate a scalefreelike quantum network, but more research needs to be done in this regard. Our work also does not consider multipath purification protocols given the fact that they are still in their infancy. Multipath purification protocols could be an interesting way to increase the reach of our network, but further research on this type of protocol would be required before we can perform analyses. Finally, it has been shown that scalefree networks although robust against target attacks can be very fragile against target attacks^{27}. This phenomenon was also reported to be present in quantum networks^{15}. Our model does not say anything about how robust these networks are against random attacks, and it would be an interesting question to address in future work. Our results provide a guiding principle for the design and development of a robust largescale quantum Internet, at least against random failures, and provide a framework that can be generalized to study another type of failures or attacks.
Data availability
Data for a snapshot of the structure of Internet at the level of autonomous systems are available at http://wwwpersonal.umich.edu/m̃ejn/netdata/as22july06.zip.
Code availability
All codes in this work are available from the corresponding author on request.
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Acknowledgements
We thank Akshat Kumar for his feedback on Supplementary Note 2 and Marco Pezzutto for valuable discussions. We acknowledge the support from the JTF project The Nature of Quantum Networks (ID 60478). Furthermore, B.C. and Y.O. thank the support from Fundação para a Ciência e a Tecnologia (Portugal), namely through projects UIDB/EEA/50008/2020 and QuantSatPT, as well as from projects TheBlinQC and QuantHEP supported by the EU H2020 QuantERA ERANET Cofund in Quantum Technologies and by FCT (QuantERA/0001/2017 and QuantERA/0001/2019, respectively), and from the EU H2020 Quantum Flagship projects QIA (820445) and QMiCS (820505). B.C. thanks the support from FCT through project CEECINST/00117/2018/CP1495. K.N. acknowledges support from the JSPS KAKENHI Grant No. 21H04880.
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B.C., W.M., K.N., and Y.O. contributed to the development of the initial concept, the design, and analysis of the networks performance as well as the writing of the manuscript. B.C. performed the network simulations.
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Coutinho, B.C., Munro, W.J., Nemoto, K. et al. Robustness of noisy quantum networks. Commun Phys 5, 105 (2022). https://doi.org/10.1038/s42005022008667
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DOI: https://doi.org/10.1038/s42005022008667
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