1. INTRODUCTION

Quantum key distribution (QKD) promises information theoretic security [1–3]. Recent research into fiber based QKD has extended point-to-point connections over 600 km [4] and more recently over 1000 km [5] using twin-field QKD [6]. This demonstrates significant advancement in the connection distances possible over fiber based QKD; however, the PLOB bound [7] demonstrates that, in the absence of quantum repeaters, there is a limit to how far QKD can be distributed across fiber.

To connect over longer distances, satellite based QKD has been investigated as a potential solution [8]. Practical demonstrations have moved from free-space both during nighttime [9] and daytime [10] to satellite-to-ground QKD [11–13]. Several experimental results including satellite to ground connection using decoy BB84 [11], satellite based entanglement distribution [14], and quantum teleportation [15] have been demonstrated. Satellite links have been used as part of the Chinese quantum network (QN) [16]. Satellite QKD using micro-satellites has been demonstrated [12], allowing for the possibility of using smaller, cheaper CubeSats for QKD.

Practical demonstrations of satellite QKD have been matched with recent advances in theoretical work [17–20]. These models provide a theoretical framework to investigate the optical communication over turbulent atmospheric links. Results demonstrate close correlation to the results from the Micius trials [20].

With advances in satellite QKD, its use to connect a global QN is coming closer to reality. Thus, the next question to ask would be how to integrate satellite QKD into the fiber network. Work has already started looking at how to optimize these satellite distributions to best design the satellite layer of the network to a) design a method for calculating the optimal orbits to maximize the minimum connection between any two ground stations based on a given satellite number and number of different satellites [21] and b) present a method to characterize the transmission rates allowed by specific constellations of satellites with ground stations [22]. While these papers investigate how to optimize the satellite paths for a given ground station configuration, or optimize the access of satellites to ground stations, there is little mention of the underlying core networks these ground stations connect to. To the best of our knowledge, no investigation has been made into how the core networks should be configured to the ground station sites, nor have there been any investigations into how many ground stations should be used. In this paper, we will design a mixed integer linear program (MILP) model to quantify the number of satellites needed to satisfy user demand in a network, thus quantifying how efficient a given configuration of ground stations and the core networks relying on given ground stations is in its usage of satellites. We will further design a MILP model to optimize the allocation of the core network sites to ground stations such that the number of satellites in the network is minimized. The partitioning of areas into sites closely resembles the work done on classical networks in [23]. In doing so, we can investigate how core network sites should be allocated to ground stations and answer the question of how to distribute satellites and ground stations while satisfying the demands of the underlying network. We use these models to investigate securely connecting data centers in two different networks, one local European network and one global network. One important consideration in satellite QNs is the access of satellites to ground stations. During an overpass, a satellite may have physical access to multiple ground stations, but may only be able to connect with a few of them. How the satellites are allocated to ground stations during an overpass is an important consideration that can significantly affect results. To see this, consider a satellite that can only communicate with one ground station at a time. During a flyover, if there are two ground stations that are in the line-of-sight of the satellite, it must select a specific one to communicate with. This reduces the overall key rate as compared to communicating with both ground stations simultaneously. Access has been looked at from an optimization perspective. Polnik et al. [24] presented an optimal scheduling scheme for a single satellite and several ground stations in the UK given an inferred weight, while Wang et al. [25] provides greedy algorithms for the access selection problem of a given satellite constellation. Both maximize the total key rates through the network. In this work, we look at how different allocation strategies change the number of satellites needed to satisfy the network transmission requirements. We do not provide an optimization for this, but rather investigate general strategies to quantify how important appropriate allocation is to the network performance and to look at how increasing the number of sources on the satellite can affect the number of satellites needed. We note that in this paper, the number of sources on a satellite quantifies how many ground stations the satellite can communicate with simultaneously, e.g., if there are two sources, then the satellite can communicate with two ground stations at once. This increases the overall key rates able to be generated by a single satellite.

2. QUANTUM CONNECTION MODEL

In this section, we will discuss the key model used to evaluate the satellite QKD rates. Remarkable progress has been made on theoretical models for the quality of the satellite QKD links in recent years. Early investigations on the subject focused on considering the dark count rate due to stray light from outside sources in a satellite QKD channel [26]. They showed that, even with the technology available at the time, satellite QKD was possible at night and possibly even during the day. Recently, there have been more in-depth models approximating the quantum channel of satellite QKD to obtain estimates for the rates. Vasylyev et al. [17] models free-space quantum optical communication links over turbulent atmospheric links using the elliptic beam approximation, considering beam deflection, elliptic beam deformations, and extinction losses. The model evaluates the mean and covariance of beam parameters, such as offset from the center and elliptical deformation lengths and angles at the aperture site. These are used to calculate the transmittance of the channel. Comparison with experimental results shows good correlation to the model. Liorni et al. [18] generalized the method to satellite QKD links, investigating both the downlink and uplink configuration verified by comparison to the Micius demonstration [11]. The model, however, assumed uniform atmosphere below 20 km and did not consider path elongation due to refraction at low elevation angles. A more detailed model by Vasylyev et al. [19] includes this path elongation along with analysis of extinction, scintillations, turbulence, and beam wander on a satellite link. All models show that the instantaneous key rate is largely independent of the azimuthal angle of the satellite to the ground station, and thus depends only on the satellite orbital radius and the zenith angle of the satellite to the ground station. Sidhu et al. [20] demonstrate a model to evaluate the overall key rate for a given satellite overpass, evaluating the atmospheric turbulence and attenuation using the Micius demonstration, to construct representative link efficiency against azimuthal angle curves. The secret QKD key length is calculated by aggregating all blocks. Since satellite communication time is finite, a finite key analysis becomes imperative to evaluate the key rate appropriately. Methods initially described by Lim et al. [27] make finite key analysis for decoy BB84 possible.

In this work, we use results from Sidhu et al. [20] to derive the key rates using finite key analysis for the asymmetric two-decoy state BB84 protocol [28]. The satellite path is evaluated using methods described in Section 3 and the total overall keys generated during the path are obtained by the model. One important consideration is that dark counts vary between daytime and nighttime. For the regional network, we assume the best pass occurs at night. The condition of the other passes is evaluated based on whether they occur during the night or day. At night [20] ${p_{\textit{dc}}} = 5 \times {10^{- 7}}$ is used while during the day [26] ${p_{\textit{dc}}} = 2 \times {10^{- 4}}$ is used, where ${p_{\textit{dc}}}$ is the dark count probability per pulse. The model optimizes for the values of two of the intensities of the WCP, ${\mu _1}$ and ${\mu _2}$; the probabilities of Alice preparing each intensity ${p_1}$ and ${p_2}$; and the asymmetric polarization probability ${P_A}$. The other parameters used in our work are given in Table 1. It should be noted that while the model evaluates the channel efficiency due to atmospheric turbulence and attenuation due to diffraction, it does not explicitly consider the losses due to telescope and detector efficiencies. These are all considered to be included in the intrinsic system efficiency ${\eta _{\rm{int}}}$, which we set to 20 dB.

Table 1. Parameters Used to Evaluate Satellite Channel Capacities

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3. SATELLITE PATH MODEL

In this section, we will describe satellite trajectories needed for satellite QKD. Vergoossen et al. [22] designed a method to characterize the transmission rates allowed by a specific constellation of satellites and ground stations and consider how to maximize the key rates with the given satellite and ground station configuration with and without inter-satellite communication. In the model, it was shown that Sun-synchronous orbits (SSOs) typically perform better than other orbits for global distribution of ground stations, despite the potentially lower connection link time, while still outperforming most other orbits when a regional network is considered. SSOs were only outperformed by orbits where the orbital inclination is equal to the maximum latitude of any ground station. These results were further backed up by the results in [21], where an optimization algorithm for the satellite orbits was designed to maximize the minimum connection between satellite stations. The regularity and predictability of such orbits allow not only for significantly improved key rates, but also allow for consistent and predicable connectivity over the course of the day. Such consistency is vital in networks requiring reliable connections. Thus, in this paper, we focus on SSO paths. We use MATLAB’s Aerospace Toolbox [29] to determine the access times of satellites on the path with ground stations assuming SSOs. Methods described in [30] allow for the calculation of the inclination angle $i$ to the satellite orbital height for SSOs. To calculate the inclination of a SSO for a given orbital radius, the gyroscopic procession of the orbit along the equator due to the equatorial bulge of the Earth (quantified by the value of the ${J_2}$ perturbation), should equal the Earth’s mean motion about the Sun, ensuring the relative angle between the satellite orbit and the Sun is constant. This gyroscopic procession of the Earth is given by [30]

A. Satellites Access Models

Ground stations can be situated in locations such that a satellite can have access to multiple ground stations simultaneously. However, there are scenarios where satellites cannot communicate with each of the stations it has access to at a given time, which is especially prevalent in regional networks. In such cases, the ground stations will need to decide which station will talk with the satellite at the given time, meaning that each pass of the satellite will not always contribute to the key rate of the given ground station. Depending on the method used to select which ground station will communicate with the satellite during the satellite pass, the satellite rate with the ground station will be modulated differently. Past papers have considered the problem for both single [24] and satellite constellations [25] from an optimization perspective. However, an interesting question arises as to how the selection of access changes the number of satellites needed in the network to satisfy transmission requirements. In this paper, we consider four simple methods of ground station allocation and investigate how different access models can affect the number of satellites needed for the networks in this work. Here, we explain how the QKD rates are affected by each.

1. No Allocation Needed

In this scenario, we consider the satellite being fitted with a sufficient number of sources such that the satellite always has access to all ground stations in the overpass. In other words, the full capacity is calculated as-is and no modifications are needed due to satellite access restrictions, as there is always a free source with which it can downlink to a ground station. Although this model requires the most complex satellites, it does, however, provide improved rates. Therefore, it provides a benchmark for comparing other scenarios as well as allowing us to quantitatively evaluate whether the added complexity on the satellite is worth the additional cost.

2. Random Allocation for Overpass

In this scenario, the satellite has a fixed number of sources $M$. In a single given overpass of the region, the satellite can connect to any of the $N$ ground stations that have access during the overpass. However, in this scenario the selection of which ground station it connects to is random. Thus, if $M = 1$, there is a $\frac{1}{N}$ chance of a given ground station to be selected. More generally, the probability of a ground station being selected is

3. Allocation Based on Expected Transmission Requirement

In this scenario, we assume the satellite has only one source and in a single overpass of the region the satellite can connect to any of the $N$ ground stations. However, the selection of the ground station is weighted, based on the expected transmission requirement of the ground station. Let us define this expected transmission requirement as $T_i^{\rm{exp}}$. The expected transmission requirement of all ground stations in the overpass can be given by $\sum\nolimits_{i \in N} T_i^{\rm{exp}}$. We thus define

4. Allocation Based on Keys Generated During Overpass

In this scenario, we assume the satellite has only one source and in a single overpass of the region the satellite can connect to any of the $N$ ground stations. However, the selection of the ground station is weighted based on the key rate generated with the ground station during the overpass. It is reasonable that the satellite should prioritize high key rate connections over lower key rate connections. In this case, we assume

4. METHOD

In this section, we will formulate MILP models to quantify the number of satellites needed in a single configuration of core network sites, ground stations, and satellites, and to determine the optimal distribution of core sites to ground stations, such that we minimize the number of satellites. In this work, we assume there are no inter-satellite links, although the results would not change if inter-satellite links between satellites along the same path are considered. This is because the overall rates between a satellite path and a ground station do not change with inter-satellite links along the same path, since the total access time is still the same. However, inter-satellite links will generally improve the latency of the keys generated between two ground stations, since two ground stations will not need to wait for a given satellite to pass over both of them, but instead the ground stations need simply to wait for one of the inter-communicating satellites to pass over either of them. We note that the models can easily be extended to include inter-satellite links between satellites in different paths. We further assume that each core site can be configured only to a single ground station and that core sites on different ground stations need to be connected via satellite QKD, while core sites on the same ground station are connected via fiber. We use CPLEX V20.1 [31] to run the models. We present the code used in this investigation in the repositories [32,33] along with instructions on how to use it to reproduce the results in this paper.

A. Model for Metric Analysis

As specified in Section 3, we consider well-defined satellite paths with repeatable surface ground-tracks for their simplicity. We will define the set of these paths as $P$, where each element in the set has a distinct ground-track, but all satellites in the path have the same ground-track. All satellites in this path will have well-defined connectivity to the ground stations, and thus a QKD rate can be calculated based on the methods described in Sections 2 and 3, ${R_{\textit{ip}}}: I \in S,p \in P$, where $S$ is the set of ground station sites.

The MILP for the metric analysis is given by

Table 2. Input Parameter Definitions for the TF-QKD MILP Model

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Table 3. Model Variable Definitions for the TF-QKD MILP Model

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B. Model for Network Segregation

Model Eq. (7) allows the evaluation of a specific configuration of ground station and core site. In this section, we will describe a more general model that can be used to determine the optimal configuration to minimize the number of satellites in the network. Assuming ${\delta _{\textit{ls}}}$ can be 1 for multiple ground station sites for each core site, we would like to determine how to distribute the core sites to ensure the number of satellites is minimized. The model can be written as

As an aside, we will discuss how to adapt model Eq. (10) to slight variations of the initial assumptions. First, our initial assumption is that each core site can only be configured to a single ground station. It is possible to generalize the model to allow for the core site to be configured to any number of ground stations. This may be of interest for rigidity purposes. If we desire the core site to connect to ${\Lambda _l}$ ground stations, we can change Constraint (vi) in model Eq. (10) to

Finally, we shall discuss how to estimate $T_i^{\rm{exp}}$. In this model, there is no well-defined transmission requirement between the satellites a priori. However, to evaluate the rate for the model in Section 3.A.3, an estimate of this is required. We will assume a uniform a priori allocation of core sites to their available ground stations. Thus, for a given site the probability that the site is associated with a ground station is given by

C. Including Core Network Costs

It should be noted that, while model Eq. (10) attempts to minimize the overall number of satellites in the network, the overall goal of building the network is to minimize the overall cost. Some connections via the core network might be more expensive than others. Ultimately, we would like to consider how this cost would affect the results. If we define the cost of connecting a pair of core nodes using the core network $i,j \in L$ as $C_{\textit{ij}}^{\rm{core}}$ and the cost of connecting a core node to a satellite node as $C_{\textit{is}}^{\rm{core}}$, then if we further define the set of variables ${\alpha _{\textit{ijs}}} \in \{0,1\} \forall i,j \in L,s \in S$ as whether the core nodes $i,j$ are both on ground station site $s$ or not, we can define the cost of the network as

Table 4. All Possible Values of ${\alpha _{\textit{ijs}}}$^a

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It should be noted that $C_{\textit{ij}}^{\rm{core}}$, $C_{\textit{is}}^{\rm{core}}$ are normalized relative to the cost of the satellite. Different satellite types have different costs. CubeSats are significantly cheaper than other models and thus the relative core network cost will be higher for the CubeSats than for other satellites. In our model, we will use the costs given in Table 5, assuming a uniform core fiber network cost per km.

Table 5. Cost Parameters Used in the Investigation

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D. Selection of M

The selection of the value of $M$ should be considered carefully in terms of potential problems caused by floating point errors. CPLEX allows the tolerance of an integer to vary up to ${\epsilon _{\rm{tol}}} = 1 \times {10^{- 10}}$. As such, small variations in ${\chi _{\textit{in}}}$ are possible up to ${\epsilon _{\rm{tol}}}$. In Constraint (xi), ${M_{\textit{in}}}$ is set to 0 if the core site $i$ is associated with the ground station $n$, since we would like to include this contribution in ${T_{\textit{nim}}}$. However, since in practice ${\chi _{\textit{in}}} = [{1 - {\epsilon _{\rm{tol}}},1 + {\epsilon _{\rm{tol}}}}]$, ${M_{\textit{in}}}$ can be as large as ${\epsilon _{\rm{tol}}}M$, even when we expect ${M_{\textit{in}}} = 0$. Thus, too large a value of $M$ will lead to significant errors in practice. In reality, we only require

5. RESULTS AND DISCUSSION

In this section, we will use our models to evaluate the use of satellites to secure data center traffic. Currently, extrapolation from the Cisco Global Cloud Index [35] predicted that there will be 30 ZB/year of global data center traffic across 8000 global data centers worldwide in 2023. If we assume the traffic is uniform across data centers, and that 512 AES key bits are needed per TB of encrypted data, far below the birthday bound, this means that ${\sim}0.02\;{\rm bits}/{\rm s}$ per data center pair is needed to encrypt the current yearly data center traffic. This value could be higher or lower, depending on how much of the data traffic needs to be secured and whether the data traffic is segmented into multiple traffic streams with different customer policies. In such cases, individual fixed length keys may be refreshed more frequently than every 1 TB of data. In this work, we focus on securing data center traffic in two networks. One is a local European network with up to 18 ground station sites and 45 major data center hubs, which are considered core sites connecting 1227 data centers. The second is a global network looking at 32 major data center hubs around the world, connected via up to 18 ground stations, connecting 3553 data centers. These networks have different properties that are interesting to investigate. The European network has a more uniform distribution of data centers, and, by extension, more uniform traffic requirements as compared to the uneven distribution in the global network. The European network has many ground stations that are geographically close, and this causes access issues for satellites as compared to the global network, where ground stations are more spread out geographically. The locations of the ground stations and core sites for these graphs are depicted in Fig. 1, and the data center core sites were selected based on data from [36].

 

Fig. 1. Locations of the ground stations and core sites for the (a) European and (b) global networks considered in this paper.

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A. Satellite Allocation

In this section, we will investigate the variation in the number of satellites needed to support the transmission requirements for each of the different allocation strategies described in Section 3.A in the networks. We assume that core sites are connected to their nearest, in terms of geographical distance, ground station, and use model Eq. (7) to evaluate the number of satellites needed for different transmission requirements. For all investigations, we use six different satellite paths, $p \in P$, with RAAN equally spaced between 80° and 130°.

Figure 2 presents the number of satellites needed as a function of the key rate for the different allocation strategies. Figure 2(a) is for the European network, while Fig. 2(b) is for the global network. Since the ground station sites are geographically close in the European network, it is particularly susceptible to the effects of overloaded satellites discussed in Section 3.A. As a result, the relative variation of the number of satellites needed for different allocation strategies is significantly higher in the European network, as depicted in Fig. 2(a). In both networks, with a single source, the number of satellites needed is smallest when the allocation is based on the expected transmission requirement. In fact, the European QN requires 40% fewer satellites to provide the same key rate, as compared to a random allocation strategy, and 37% fewer than an allocation based on maximizing the keys generated. In contrast, the global network has fewer ground stations that are geographically close. However, the ground stations with the highest key requirements, Albany and London, compete with two to three other ground stations in the network. As a result, the satellite allocation strategy used is still important for satellites with only one source. Allocation of satellites based on transmission requirements reduces the number of satellites needed by 30%, compared to allocation based on maximizing keys generated and by 8.6% compared to random allocation. We see in the global network that random allocation is better than allocation based on maximizing keys. This is because overall key maximization does not achieve the goal of maximizing keys to the ground stations that most require them, but, instead, maximizes the overall keys between all ground stations. This may result in a beneficial case of increasing the number of keys to the ground stations that require them, as in the European network, or it may reduce the keys on ground stations that need them, since the overall key rate is increased by allocation to different ground stations. Since the number of satellites needed depends on the worse ratio of the secure key rate needed between the ground station and satellite $\sum\nolimits_{k \in K} x_{i,j}^k + x_{j,i}^k$ and the capacity between the ground station and satellite ${R_{\textit{ij}}}$, reducing ${R_{\textit{ij}}}$ on the worse ground station will increase the total number of satellites needed instead as seen in the global network. Providing more sources on each satellite significantly reduces the number of satellites needed in both cases. In the European network, providing two sources per satellite exactly halves the number of satellites needed compared to just one for a random allocation strategy. Further, providing 18 sources per satellite, which removes the need to allocate the satellites, reduces the number of satellites needed by 84%, compared to the optimum strategy with just one source. In the global network, providing two sources per satellite reduces the number of satellites needed by 41%, compared to a single source with the random allocation strategy, but significant further improvements upon increasing the number of sources are not observed. This is because, in the global network, ground stations are far apart, and only up to a maximum of three can be seen by a satellite at any given time.

 

Fig. 2. Plot of the number of satellites needed as a function of the key rate to fully connect the network for different allocation procedures, assuming 0.02 bits/s per data center pair is required for a bit-per-second scale factor of 1. Results for (a) the European and (b) the global network.

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Based on these results, fewer than 40 satellites are needed to secure all data center traffic currently between the major data center hubs in Europe, even if each satellite has only one source. It is possible to reduce this to just six satellites if each satellite has multiple sources. Fewer than 160 satellites are needed to secure major global data center hubs using satellites with just a single source, while this can be reduced to as few as 70 if there are sufficient sources on each satellite.

B. Configuration of Core Sites to Ground Stations

In this section, we will investigate how the overall cost of the network varies with the number of ground stations. The overall cost of the network is given by the sum of the satellite cost and the cost of the core network. We start by looking at how the overall cost varies when core sites are connected to the geographically closest ground station. Figure 3 plots the variation of the overall network cost for the European QN. We assume 0.02 bits/s per data center pair for all results in this section. Figure 3(a) shows the overall cost, including the cost of the core network and the satellites for the CubeSat case. Since CubeSats are relatively cheap, the cost of the core network is significant. As can be seen, the core network cost decreases as the number of ground stations increases, because the distance between core sites that need to be connected to each other and the distance between core sites and ground stations decrease with a larger number of ground stations. The satellite cost barely contributes to the overall cost of the network in this case, so it is beneficial to have a large number of ground station sites to minimize the core network cost.

 

Fig. 3. Overall cost of the European network for different numbers of ground stations and different satellite types for the case where core sites are connected to the geographically closest ground station. Results for (a) CubeSat and (b) satellite costs.

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Fig. 4. Overall cost of the European network for different numbers of ground stations and different satellite types. The optimal results given by model Eq. (10) are compared to the results given by model Eq. (7) for the configuration of core sites connected to their geographically closest ground station. Results for (a) CubeSat and (b) satellite costs.

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Figure 3(b) shows the overall cost for the satellite case. In this scenario, satellite cost contributes significantly to the overall network cost. It can be seen that, in this case, the satellite cost increases with increasing number of ground stations, as more traffic is needed through the satellite network and the satellite key rates, ${R_{\textit{ij}}}$, are lower due to the increased number of ground stations sharing the same satellites. Thus, there is competition between the decreasing cost of the core network and the increasing cost of the satellites, with more ground stations when these costs are comparable. As can be seen in the graph, when connecting to the shortest distance ground station, a minimum cost is obtained when there are just eight ground stations in the network.

In the previous studies, we assumed that core sites are connected to the nearest ground station. This may not necessarily be the best configuration. We compare the results obtained in the previous studies with the results from the full network optimization, using model Eq. (10), which are presented in Fig. 4 for the European QN and Fig. 5 for the global QN. We allocate satellites based on expected transmission requirements to obtain these results. The optimization is stopped when the solution reaches within 1% of optimum, as this is sufficient for most purposes, and model Eq. (10) takes a long time to reach the true optimum for the graphs used in this work, which explains the slight increase in cost of the optimization to the shortest path solution for three ground stations shown in Fig. 4.

For the European network, it can be seen that the optimal configuration can result in significant cost reductions, compared to the use of shortest distance connections. In the CubeSat case, there is a reduction in cost of the core network by increasing the number of satellites compared to the shortest distance configuration. Since the CubeSats are very cheap compared to the core network cost, this can result in significant cost benefits of up to 42% in the case of five ground stations. In the CubeSat scenario, spreading out the core site allocation, so that fewer core sites need to be connected together by the fiber network, is important to reduce the overall network cost, even if this means that more satellites are needed. In the satellite scenario, the core network cost is comparable to the satellite cost. It is, thus, harder to justify distributing core sites to minimize the core network cost at the expense of requiring more satellites. In fact, the optimal solutions in this case have, for the most part, fewer satellites than the shortest connection solutions. It may sound like a good strategy to distribute the core sites to minimize the number of satellites. However, we see that the solution for three ground stations has fewer satellites than that for more ground stations, while also being a more expensive overall solution. A balance must be found between satellite cost and core network cost. The satellite cost depends not on the total key rate through the satellites, but on the maximum value for any ground station of the ratio of the secure key rate needed between the ground station and satellite $\sum\nolimits_{k \in K} x_{i,j}^k + x_{j,i}^k$ and the capacity between the ground station and satellite ${R_{\textit{ij}}}$. It is not necessary to minimize the overall key flow through the satellite network to minimize the number of satellites. The optimal strategy would thus be to keep this ratio small while simultaneously reducing the core network cost by appropriate allocation of the core sites.

 

Fig. 5. Overall cost of the global network for different numbers of ground stations and different satellite types. The optimal results given by model Eq. (10) are compared to the results given by model Eq. (7) for the configuration of core sites connected to their geographically closest ground station. Results for (a) CubeSat and (b) satellite costs.

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A similar result can be seen in the global network in the case of CubeSats. More ground stations mean less core network is needed, and this reduces the overall network cost. We assume that core sites can only connect to ground stations in the same country as the core site or to ground stations that are in Europe if the core site is in Europe. This means there is less flexibility in the global network to allocate core sites to different ground stations compared to the European network, and the savings that can be expected are smaller on average than those in the local European network. Cost savings of up to 21% can be expected in the CubeSat case shown in Fig. 5(a), while, in the 18-node graph, the optimal configuration of core sites to ground stations has a cost that is 38% less than the nearest neighbor connection in the satellite case shown in Fig. 5(b).

The addition of the ground station in Nashville in the global network significantly increases the number of satellites needed in the shortest distance network, which can be seen in Fig. 5(b) moving from 16 to 17 ground stations. This is because there are now two ground stations in the region, and there is a clash in ground station access of satellites in the region around Albany. This reduces the key rates to the individual ground stations, and this has an effect on the total number of satellites required, since the East Coast US has the most data centers, and thus requires the most satellites. In contrast, changes to less traffic-demanding regions result in smaller changes to overall cost. This demonstrates the positives and negatives of adding additional ground stations close to the points with the highest key requirement. The positive change in this case is that this allows a redistribution of the core sites to ease the traffic requirement on a single ground station. This also means the core network cost is reduced. However, the need to share the satellite means there is a reduced average key rate to each individual ground station. If the core sites are not allocated to the ground stations appropriately in light of this reduced rate, then the cost can be much higher. The optimal result allocates the sites appropriately, and thus the addition of this site does not lead to a significant cost increase in this case.

Finally, we see that, in the global network, the number of satellites does not increase as the number of ground stations increases, in contrast to our findings in the European network. It was previously noted that this increase in the number of satellites required is due to the overall increased traffic rate distributed across the satellite section of the network and to the decreased key rates due to sharing. In the European network, most regions had similar numbers of data centers; however, in the global network the distribution of data centers is heavily skewed towards the US and Europe. These two regions are the bottlenecks in the network that in turn determine the number of satellites needed. Changes to the ground station configuration in other regions does not affect the number of satellites needed, while changes to the US and European ground stations will have a large effect. Increasing the number of ground stations in Europe and the US will allow the core sites to be distributed more evenly across different ground stations, thus reducing the maximum key rate on a single ground station. However, these ground stations may also need to share satellites, which reduces the overall key rates to each individual ground station, and the overall key traffic transmitted through the satellite network is increased. This shows that additional regional ground stations could have either a positive or a negative effect on the overall network cost in bottleneck regions. Therefore, an in-depth analysis, such as the one presented here, is needed to determine if the addition of a ground station is beneficial. It may be better to add a ground station in a region where there will be no clash between the two ground stations to offload traffic on a single ground station, although this will increase the overall core network cost.

6. CONCLUSIONS

In this paper, we considered the design of satellite QKD quantum networks. We constructed two MILP models, a simpler model to quantify the number of satellites needed to satisfy the desired connectivity of a given configuration of core sites to ground stations, and a more complicated model to investigate an optimal allocation strategy of core sites to satellite ground stations, ensuring that the cost of both the core network and the satellite constellation is minimized. We used these models to investigate two different networks, a regional network covering Europe, and a global network to connect major data center hubs. These networks have different properties that are interesting to investigate. The European network has a more uniform distribution of data centers, and by extension more uniform traffic requirements, compared to the uneven distribution in the global network. The European network has many ground stations geographically close, and this causes access issues for satellites, as compared to the global network, where ground stations are more spread out geographically. We also considered a series of satellite-to-ground station allocation strategies to deal with the situation when satellites do not have enough sources to connect with all ground stations they have access to.

We showed that allocation strategies can have a significant effect on the number of satellites needed to satisfy the network transmission requirements: by over 40% in the European network and up to 30% in the global network. This effect is more pronounced in the European network due to the geographical closeness of the ground stations in the network. However, it is still important, even when ground stations are more dispersed, as in the global network. Allocating the satellites based on the expected transmission requirements is the optimum strategy of those investigated in this paper. We further show that adding more sources per satellite can significantly reduce the number of satellites needed.

We compared the results of a fixed network configuration, where core sites are allocated to the nearest ground station, to the optimal configuration for both the European and global networks. Such investigations showed that significant cost savings of up to 40% could be expected from an optimal configuration, thus demonstrating that appropriate core site allocation is vital to ensuring minimum cost of the network. We see that, in the limit where the core network cost is much higher than the satellite cost, appropriate allocation takes the form of minimizing the total core network distance, which may involve connecting core sites to ground stations that are not their geographically closest sites, as this could lead to the need to connect to multiple other core sites in the network, hence driving up costs. In this limit, it is sometimes better to connect a core site to an unused ground station and connect the core site to the network using satellite QKD reducing the core network cost. In the limit where the core network cost is negligible compared to the satellite cost, it is important to ensure the ratio of transmission requirement rate to satellite to ground station key rate on any given ground station is minimized. Optimum solutions in intermediate cost regimes involve a fine balance between the satellite and core site costs. We also investigate how the addition of a ground station can affect the cost of the network. The additions can either reduce the cost of the network, since it allows for a redistribution of the core sites to ease traffic requirements on individual ground stations, or can increase the network cost, since the need to share each satellite with more ground stations results in a reduced average key rate to individual ground stations. In the European network, since the distribution of traffic is more uniform, ground station additions will affect the cost of the network, and a reallocation of the core sites is needed to reduce overall costs. In the global network, the non-uniformity of the traffic means that only changes to ground stations in areas with high traffic demand will significantly affect the network cost. When this happens, a reallocation of core sites is imperative to maintain lower costs. Changes to areas with lower traffic demands will have a lesser effect on the overall network costs.

In this work, we have not considered resilience due to ground station or satellite dropout. However, we note that ground station resilience can be integrated into the model by using the modification Eq. (12). Satellite outage can occur due to solar flares or space debris collisions. Further work is required to identify the best strategies to increase network resilience at efficient cost. Possible strategies include adding reserve satellites to each path or alternatively, an unused path could be used for resilience satellites. The best path for this can be obtained using the models described in this paper.