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Cost and Routing of Continuous Variable Quantum Networks

We study continuous-variable graph states with regular and complex network shapes and we report for their cost as a global measure of squeezing and number of squeezed modes that are necessary to build the network. We provide an analytical formula to compute the experimental resources required to implement the graph states and we use it to show that the scaling of the squeezing cost with the size of the network strictly depends on its topology. We show that homodyne measurements along parallel paths between two nodes allow to increase the final entanglement in these nodes and we use this effect to boost the efficiency of an entanglement routing protocol. The devised routing protocol is particularly efficient in running-time for complex sparse networks.

We study continuous-variable graph states with regular and complex network shapes and we report for their cost as a global measure of squeezing and number of squeezed modes that are necessary to build the network. We provide an analytical formula to compute the experimental resources required to implement the graph states and we use it to show that the scaling of the squeezing cost with the size of the network strictly depends on its topology. We show that homodyne measurements along parallel paths between two nodes allow to increase the final entanglement in these nodes and we use this effect to boost the efficiency of an entanglement routing protocol. The devised routing protocol is particularly efficient in running-time for complex sparse networks.

I. INTRODUCTION
Quantum networks theory, i.e. the study of networks of systems with quantum properties, have applications ranging from fundamental research to real-world applications. Many physical systems, such as solid state materials and large molecules like proteins can indeed be modeled as quantum networks. The underlying structure of some quantum systems can deeply influence their properties and the theoretical study of quantum networks can help the understanding of many important phenomena. Photonics quantum networks, for instance, are essential resources for quantum information processing and notably for quantum internet applications, where quantum states of light will allow for the efficient distribution and manipulation of information [1][2][3]. Needless to say, in order to develop large scale quantum communications and build a quantum internet it is compulsory to grasp the potentialities of quantum networks and exploit all their exceptional features.
There are several ways to supply quantum communication channels among devices placed at different locations, including simple optical fibers and entangled photons. An appealing approach to study photonic quantum networks is given by continuous variables (CV) graph states, an interesting class of Gaussian quantum states that, under certain condition, can provide a substrate for universal quantum computation [4]. This type of quantum states can readily be produced in some well equipped photonic labs, however their ideal version requires an infinite amount of squeezing, which is unphysical as it would imply infinite energy.
In this work we present new methods and results to study CV graph states with regular and complex topologies. CV describes quantum states living in infinite dimensional Hilbert spaces and it is the most natural language for bosonic fields, such as photons. Moreover, generation and measurement of CV states requires only coherent control on classical laser source and weakly non-linear ma-terials, along with coherent (homodyne) detection which, differently from photon counting detectors, can be highly efficient at room temperature and easily integrated in classical communication networks. However, CV encoding in quantum networking/routing protocols has not been extensively studied yet, unlike the discrete variables (DV) case [5][6][7].
Our model aims at reproducing the existing photonic platforms with realistic experimental constraints, such as limited amount of squeezing, but without taking into account propagation losses. This work is focused on the capabilities of pure CV quantum states to act as quantum networks and the resources needed for their generation. Generation losses can be very low, so that the hypothesis of pure states is a realistic one, whereas propagation losses can be mitigated by considering local (short distances) networks, and their effect on long distances will be included in future works. At the same time we probe the capabilities of photonic platforms while the scaling of the network increases beyond the capacities allowed by the state-of-the-art technology. Since the main limitation to build these optical systems is finite squeezing, we provide in section III an analytical formula that allows to compute the amount of squeezing in each mode as a function of the spectrum of the underlying graph. We show the graph states' cost as a global measure of squeezing and number of squeezed modes that are necessary to build the network. In particular we explore the cost of different networks topologies in term of number of needed squeezers at fixed number of nodes in the networks and of the global amount of squeezing.
We then explore their potentialities to perform efficient quantum communications between two arbitrary nodes when assisted with a given class of Gaussian Local Operations and Classical Communication (GLOCC) by all the agents in the network. A typical approach of quantum networking and routing consists in distributing photonics states like single-photons, Bell pairs or Gaussian states and then use synchronous local operations that build the wanted entanglement structure between the agents [2,[8][9][10][11][12][13][14]. We rather consider the case where a preexisting CV multipartite entangled state is distributed among the players and then local operations reconfigure the entanglement connections, similarly to some protocol in the DV case [15]. The choice is motivated by the fact that multimode entangled states can be directly gener-arXiv:2108.08176v2 [quant-ph] 26 Apr 2022 ated via optical platform [16][17][18][19][20] and their shape can be easily manipulated [21,22]. We show, both analytically and numericall that in specific topologies homodyne measurements along parallel paths allow to increase the entanglement between two distant nodes.
Finally, we devise a routing protocol based on local quadrature measurements for reshaping the network in order to perform a teleportation protocol between two arbitrary nodes of the networks. The Routing protocol, which is based on wire-shortening over parallel paths among the nodes, improves the final entanglement between the two nodes in a considerable amount of cases, and it is particularly efficient in running-time for complex sparse networks.
The article is structured as it follows. In section II we discuss Gaussian quantum states, their generation by quadratic Hamiltonian and their decomposition. We then introduce the Gaussian quantum networks studied in this Article. In section III, we derive the first result of this paper, namely an equation providing the squeezing values required to experimentally build a graph state as a function of its graph spectrum. In section IV we revise a resource theory of squeezing and in section V we will adopt it in order to estimate the cost of expanding the network. Although in entangled qubits networks the resource usage is always proportional to the number of links, we show that in CV Gaussian networks the trend of the presents non-trivial scaling with the size of the network and this is strictly dependent on its topology. We present as well a few instances of the full squeezing spectra -i.e. the needed amount of squeezers with the required squeezing values -of regular and complex networks, showing that some topologies are equivalent up to a linear optical transformation.
In section VI we propose a CV architecture for the quantum internet based on the Gaussian network previously described. We simulate quantum communication protocols through the network by letting the spatially separated agents present at each node perform a homodyne measurement on their optical mode and look for the optimal measurement strategy to maximize the negativity of the entangled pair shared by the two users who want to communicate, Alice and Bob. We prove that when multiple entangled paths connect Alice to Bob the optimal measurement strategy allows to increase the negativity in the final pair. This parallel enhancement of entanglement can be used to increase the quality of quantum communications in some selected network topologies.
Lastly, in section VIII, we employ our previous findings to implement an heuristic routing protocol for distributing and boosting the entanglement between two arbitrary agents. The algorithm we provide, on the one hand, is much more efficient than directly checking all possible combinations of quadrature measurements and, on the other hand, it always provides higher negativity than the classical scheme, which is directly employing the shortest path between Alice and Bob and neglect the parallel channels.

A. Gaussian quantum states
The generation of continuous variables multimode entangled states has been demonstrated in several optical setups. In such experiments we recover networks structures as naturally appearing entanglement correlations [18], reconfigurable Gaussian interactions [22], or imprinted cluster states [16,17,20,21,23].
These quantum states produced via parametric processes and linear optical transformations are characterized by Gaussian statistical distribution of the quadratures of the involved optical modes [24]. The quadraturesq j and p j of the j th mode are canonical conjugate variables, such that [q j ,p k ] = iδ j,k , associated to the quantum harmonic oscillator describing the light mode. In this work we adopt the following relation with creation and annihilation operatorsâ † = (q − ip)/ √ 2 andâ = (q + ip)/ √ 2, such that the variance of the vacuum quadratures is normalized to 1/2.
The produced states can then be completely characterized by the first two moments of the quadratures r = Tr[ρr] and σ = Tr[ρ{(r −r), (r −r) T }], where ρ is the density matrix of the Gaussian state andr = (q 1 ...q N ,p 1 ...p N ) -we follow here qp-ordering.
Parametric processes are described by quadratic Hamil-toniansĤ I =rHr T , whose dynamics is implemented on the quadratures by S H = e ΩHt , aŝ wherer 0 are quadratures of the initial state,r are the quadratures of the final state and Ω = is a 2N × 2N skew-symmetric matrix associated to the N dimensional Hilbert space allowing to write the commutation relation of the canonical variables as Since any pure Gaussian state can be obtained by the application of a unitary generated by a quadratic Hamiltonian H to the vacuum, the most general pure Gaussian state covariance matrix is given by applying S H by congruence to the vacuum covariance matrix σ 0 = 1/2: Singular value decomposition allows one to write the symplectic transformation in the so called Bloch-Messiah decomposition [24] as a product of an orthogonal, a diagonal and an orthogonal matrices S H = O∆O , which can be interpreted as a basis rotation, a squeezing in the diagonal basis and another rotation. The mode-basis in which the covariance matrix is diagonal and each component is independently squeezed is named the supermode basis. In [18,21] where the pump and the phase matching function can be described by a Gaussian spectral profile, the supermode basis corresponds to Hermite-Gauss spectral modes. The squeezing values of ∆ can be derived from the eigenvalues of the HamiltonianĤ I , while the orthogonal matrix O can be interpreted as a measurement basis change or, equivalently, as a passive linear optical transformation. The other orthogonal matrix O is simplified in the product S H S T H and can be disregarded: The diagonal matrix ∆ contains the information on the minimum number of squeezed modes in the system and their value of squeezing, which will later be used in the chosen resource theory. If we consider a single mode field, the squeezing operation is defined as a Gaussian transformation that reduces the variance ofp by a factor 10 −s/10 , where s, measured in dB throughout this article, is called squeezing factor. Squeezing is represented by the local symplectic matrix This formalism can be used to visualize and manipulate Gaussian quantum states, that are readily available in most well-equipped photonics laboratories and, although the number of modes and their connections is still in large part limited, many efforts are employed to improve the capacities of these systems.
Targeted Gaussian quantum states, including the quantum networks of the next section, can be generated via the two following strategies: i) by tailoring Hamiltonianŝ H I of multimode parametric processes in order to get the decomposition of eq. (3) corresponding to the desired covariance matrix [19,21,[25][26][27][28]; ii) by getting a number of single-mode squeezers equal to the the number of elements with s j = 0 of ∆ in eq. (3) and producing the corresponding s j squeezed states, that are injected in a linear optic intereferometer corresponding to the orthogonal matrix O in eq. (3) [29][30][31].

B. Graph states as quantum networks
The above formalism can be employed to describe Gaussian graph states, that can be used as CV quantum networks. We at first recall that a network is mathematically described by a graph G(V, E), which is a set of vertices V (or nodes) connected by a set of edges E. Labeling the nodes of the graph in some arbitrary order, we can define a symmetric adjacency matrix A = A T whose (j, k) th entry A jk is equal to the weight of the edge linking node j to node k (with no edge corresponding to a weight of 0). Typically, the adjacency matrix is enough to completely characterize a graph, however we will see that in our case there are other degrees of freedom such as the squeezing of a node and its angle.
We can now describe the quantum networks we use in this work that are called graph-or cluster-states 1 1 While usually the name cluster is used when the graph shape [32][33][34]. Theoretically, they can be built by entangling a number of squeezed modes of light via CZ-gates, which is a Gaussian operation implementing a correlation of strength g between theq and thep of the two modes on which it acts. The corresponding symplectic matrix is The graph associated to the graph states identify edges as CZ-gates applied between nodes, that are the squeezed modes, weighted with g.
In order to simplify the many degrees of freedom present in our networks, for the moment we shall assume that all the nodes will be squeezed inp by s and all the edges have a correlation strength of g. If we apply a CZ-gate network with adjacency matrix A to a multimode squeezed vacuum σ s , with squeezing factor s we obtain a Gaussian network with covariance matrix [35] Where R = 10 s/10 . The 2N × 2N covariance matrix σ is divided in four N × N blocks, where the blocks σ qq and σ pp represent the correlations among the different nodes' q-and p-quadratures, respectively, whereas the blocks σ qp and σ pq describes the correlations between qand p-quadratures. Bear in mind that the CZ-gate operations that theoretically identify the edges of the networks are seldom realized in any laboratory being very challenging to accomplish. What is commonly done, as explained in the previous subsubsection, is the reduction of the covariance matrix of the graph state in (5) to the form of eq. (3), that is also a receipt for bulding the graph states from a certain number of squeezed modes (∆) and linear optics transformations (O).

III. INTERPLAY BETWEEN SQUEEZING AND ADJACENCY SPECTRA
Consider a N -dimensional graph with adjacency matrix A. If we apply a set of CZ-gates to the set of N modes of a vacuum state according following the edges defined by A, we end up with a Gaussian graph state with the following 2N -dimensional covariance matrix: where we assumed that the vacuum state variance is normalized to 1/2. Since A is symmetric, it is always allows for universal quantum computing, in this work we will use the terms cluster state and graph state as synonyms.
for some orthogonal matrix V , where {D i } N i=1 is the set of the real eigenvalues of A. It follows that V A 2 V T = V AV T V AV T = D 2 . Let us consider the following matrix We can verify easily that W W T = 1, hence W is an orthogonal matrix implementing a basis change that would not change the spectrum of the matrix to which it is applied. If we apply it to σ we get which is a block matrix composed of diagonal matrices. We can permute the rows and columns of the matrix to get a diagonal block matrix where Π is a permutation operator, while In this basis, each block M i represents a single mode covariance matrix of a pure unentangled Gaussian state. We can hence diagonalize each block independently. In particular, notice that det(M i ) = 1 4 , thus the eigenvalues of σ are given by These values represent the squeezed and antisqueezed variances of the uncoupled oscillators of equation (4), e.g. the uncertainty of measuring the real and imaginary part of the electromagnetic field. Equation (12) shows the interplay between the physical resources necessary to experimentally implement a CV graph state and the spectrum of the underlying graph. This implies that we can use spectral graph theory to characterize analytically the physical requirements of building Gaussian networks and thus predict which one will be easier to realize. A first crucial consequence is that different graph states whose underlying graphs are co-spectral, e.g. their adjacency matrices have the same eigenvalues, can be transformed into each other applying passive linear optics 2 . We will see later in Sec. V A that the star and diamond networks have this property, making them a relevant class of Gaussian networks for applications. The intrinsic connection between the squeezing of a Gaussian network and its topology was already put in evidence by Gu et al. [34], by proving a relation between the squeezing required to produce a CV graph state and the singular value decomposition of the associated adjacency matrix. Our result is an exact generalization of theorems 2 and 3 of Ref. [34], beyond the limit of large squeezing, i.e. in a regime that is accessible with current technology. Another crucial consequence of equation (12) is that for CV graph states the number of independent squeezed modes (squeezers) in their Bloch-Messiah decomposition corresponds to the rank rk(A) of the associated adjacency matrix .
For any non-trivial graph, we have The impossibility of rank 1 adjacency matrices comes from their null trace. If A is not full-rank, it has a nonnull kernel. A common source of such kernels vectors is when one has k linearly sets S 1 , . . . , S k of vertices sharing the same neighborhood AS 1 , where, by a slight abuse of notation, S i denotes both a set of vertices and the corresponding column vector: We have then Which is null iff i α i = 0. This gives us a nulleigenspace of dimension k − 1 .
The most obvious example of this is the star graph, where N − 1 vertices share the same neighbor, giving a null space of dimension N − 1 − 1 = N − 2, hence rk(A) = N − (N − 2) = 2. If one now looks at the complete (k, N − k)-bipartite graph, one easily finds two null-eigenspaces of dimension k − 1 and N − k − 1. This leads to rk(A) = 2. Actually, any rank 2 graph is a complete bipartite graph, as can be shown by looking at A matrices one can build from two orthogonal vectors. Its two non-null eigenvalues of a rank-2 graph are

IV. A RESOURCE THEORY OF SQUEEZING
The Gaussian bosonic states of subsection II A are of particular significance in the theory of continuous variable quantum information, in particular in their quantum optical implementations. They are in fact resources for measurement based quantum computing [32,34], quantum simulations [22], multi-party quantum communication [21,36], and quantum metrology [37,38].
Being interested in the nature of the correlations between such states, the first moments become irrelevant. In any practical realization of a quantum computation or communication protocol with Gaussian states, first moments do play a role, but these are normally managed in the post-processing and do not interfere with the dynamics of the second moments. We can thus assume that our quantum states are fully described by their covariance matrix.
We have seen in the previous section that the squeezing is the essential resource for building Gaussian entangled states. A natural question is thus: what is the squeezing cost of producing a quantum state?
A general resource theory for Gaussian states is provided in [39]. The specific case of squeezing is described in [40] where they show an operational squeezing measure for any symplectic transformations S: where λ + i (S) are the largest singular values of S, while log 10 and the factor 20 ensures that the outcome is measured in dB. Using eq. (3), we can then define a squeezing measure for covariance matrices Where we multiplied the covariance matrix σ by 2 so that the vacuum state has a vanishing cost. This definition can be generalized for arbitrary quantum states, but assumes this particularly simple form for pure Gaussian ones and it works for any number of modes. We will employ it to classify networks topologies, basing on how their squeezing cost scales with the dimension of the network. This is perfectly justified for the implementation strategies that in section II A we have defined of type i), i.e. tailored multimode down-conversion processes [19,21,[25][26][27][28], where the experimental challenge is due to the trade-off between the number of modes and the amount of squeezing in each mode, both of which are considered by the resource theory. In other kind of experimental scenarios, such as time-multiplexed regular graph states implementation [16,17], where few squeezers are used to create the network by the interference of delayed squeezed pulses, the main limitation is due to the amount of squeezing in each mode, which can be easily obtained from G if we divide by the number of squeezed modes. As a matter of fact, any resource theory should be tailored to the specific experimental scenario under consideration and eq. (18) provides a versatile tool for this purpose.

V. SQUEEZING COST FOR NETWORK GENERATION
In this section we apply the results presented above to various topologies of Gaussian networks, to study how their squeezing requirement scales with the size of the network depending on its own structure. As stated in the first section, a node in the network is a a pure continuous variables Gaussian state, that will be called a qumode, and is completely defined by its own covariance matrix. In the following, we will consider an initial set of N modes vacuum states with no squeezing (s = 0), as it can be easily proved that initial uniform squeezing only adds a constant factor to the final squeezing cost. We also assume that the CZ-gate coupling strength g = 1 for every edge of the graph, keeping the effects of nonuniform correlations for future works.

A. Regular Networks
Let us first discuss some regular network structures. We shall consider the following topologies: • The linear graph L N , with N nodes and N − 1 edges, is accomplished by connecting each node in series to the next.
L 5 : • The ring graph R N , with N nodes and N edges, is a linear graph with a closed loop.
• In the star graph topology S N , with N nodes and N − 1 edges, every peripheral node is linked to a central node, called hub.
S 5 : • The diamond graph D N , with N nodes and 2(N −2) edges, has 2 hubs, each linked to all the N − 2 central nodes of the network. It is isomorphic to the complete bipartite graph in which one of the subset has 2 nodes and the other has N − 2 nodes.  Let us first take a look at the full squeezing spectrum of these topologies for a fixed number of nodes N = 100. This is shown in figure 1, where the values were computed starting from the adjacency matrix spectrum of regular graphs [41]. We can see that, as expected, the linear and the ring graphs have a very similar spectrum, with small deviations induced by the periodicity of the latter that becomes negligible for large N . The star and the diamond networks only have two squeezed modes for all N . They are thus co-spectral up to a factor and can be transformed the one into the other with linear optics. Finally, the fully connected graph has one large eigenvalue that grows with N and N − 1 uniform eigenvalues that are constant with the size of the graph.
Let us now see how the total squeezing cost G(σ) scales with the number of nodes N for each of the network topologies presented above. The analytical expressions for all these graphs are shown in appendix A. In can see how the linear graph in blue and the ring graph in orange are superposed, sharing the same squeezing cost per node that, as shown in appendix A, is constant with N . The cost of the star and diamond grows logarithmically with N and in both cases has a simple expression and present the lowest cost among the regular graphs we studied. In all these cases, although the actual squeezing cost would be smaller for these networks, the amount of squeezing required in each mode is much larger so, depending on the experimental scenario, their implementation could be the easiest or the most challenging from an experimental point of view. In any case, these networks have interesting applications for quantum communications. In particular, the star graph can be used for secret sharing [21], while the diamond produce an effect that we called parallel enhancement of entanglement, which will be explained in the next section. Finally, the fully connected graph, has a cost that grows linearly with N and, despite having the largest number of edges and thus of squeezing increasing operations, for a large number of nodes is slightly cheaper than the linear graph. Notice that there is as well a simple relation between the adjacency matrix of a graph state and the energy difference between the state and the vacuum. In fact, assuming that the energy E of the state is the mean value of the harmonic oscillator Hamiltonian on the state where we also assumed that all the modes have the same frequency ω i = ω. From this we can write The quantity on the right hand-side is proportional to the second moment of the eigenvalues distribution and it sets a fundamental lower bound on the energy necessary to implement such states. From a topological point of view, the trace of the n-th power of the adjacency matrix equals the number of closed loops of length n on the graph [42]. Thus, for n = 2, it corresponds to the number of edges in the network and implies that each independent application of the CZ-gate adds the same amount of energy. A resource scaling that is linear with the number of edges in the network is typical of DV networks (i.e. DV graph states), where each new edge requires a new Bell pair. The scaling of the squeezing in CV networks with the size of the network, as we have seen, depends instead on the structure of the underlying graph and can be non-linear with the number of edges. We want to stress here that squeezing -and not energyis the technologically not-trivial enabling resource to be implemented for building the CV networks. If we look again at the example of the star and the diamond graph, the number of edges in the two graphs are different, hence they would have different energies. As they have the same number of squeezed mode, we can transform one into the other with passive optics, e.g. without spending energy. However, it has to be clarified that the two main squeezed modes for the diamond have larger eigenvalues than the star network. Thus, if we transform a star graph into a diamond with linear optics, it would be equivalent to a graph state obtained by the application of CZ-gate with a weaker coupling g. But again this behaviour is totally accounted by the squeezing cost, which is the relevant quantity in designing CV networks.

B. Complex Networks
So far we described graphs that are built through a deterministic algorithm, though we can also construct a graph based on statistical models [43,44]. This is the difference between regular and random networks. An exemplary standard for random networks is the Erdős-Rényi model G ER (N, p), in which each pair of the N nodes have a probability p to be linked; the network thus has N 2 p edges on average [45]. Most of the network properties observed in nature, however, simply cannot be described by regular or random graphs. For this reason, a youthful branch of scientific research is committed to the study of complex networks. In the field of network theory, complex networks are a type of graphs with non trivial topological features, that are shared by neither regular nor random graphs, but are rather akin to networks modeling real systems [46].
An important class of complex networks is characterized by the small world property. These networks exhibit the peculiarity of having a low average path length, which is the mean distance between two arbitrary nodes, and a high clustering, which is a measure of the degree to which nodes in a graph tend to cluster together. The emblematic network presenting these features is the the Watts-Strogatz model G W S (N, Q, β) [47]. In this model, we first construct a regular periodic graph with N nodes and N Q 2 edges where each node has exactly Q neighbors, then with probability β we rewire each edge with another node chosen uniformly at random while avoiding self loops and link duplications.
The second relevant class of complex networks present the typical aspect of being scale-free and having longtailed structures. Scale-free networks show a power law in the degree distribution P (k) ∝ k −γ for some γ > 0, which is self-similar at all values of k in the tail of the distribution, unlike the ER and WS models that go to zero very quickly and have no tails. This fractal like attribute is well reproduced by the Barabási-Albert model G BA (N, K), which can also emulate growth and preferential attachment in networks [48]. The graph is built sequentially by adding one node at a time and wiring it to K other nodes with a probability that is proportional to the number of links that the target node already has. While adding one node at a time, the first K nodes initially will not be linked to anything, hence this graph will result having (N − K)K edges, mostly connected to a great hub. This type of graph is the canonical example to reproduce some properties of the World Wide Web. A particular case of the Barabási-Albert network is the Tree-graph G T (N ) = G BA (N, 1).
Another notable class of complex networks is constituted by technological networks, artificial networks designed typically for distribution of some merchandise or resource, such as electricity or information. The most famous example in this category is the Internet Autonomous System (AS) G AS (N ), the physical global computer data network. In order to study this topology we will base on the work put forward in ref. [49]. All the complex networks mentioned so far are meant to emulate man-made structures, however complex topologies appear in nature in the most surprising ways. There is a large variety of biological networks, among these we will consider specifically the protein-protein interaction network model G P P (N, σ) developed in ref. [50].
CV quantum networks with complex networks shapes are particularly relevant for simulating quantum complex networks environments [22,51] and to study future quantum information/communication networks mimicking the structure of the classical communication networks. It is then worth to study the scaling of the necessary squeezing resources for their implementation. In Fig. 4 we show the squeezing cost distribution for the various topologies of complex networks by showing the squeezing cost of all the principal modes. Notice that, since complex networks are a subset of correlated random networks, one cannot find the exact eigenvalues distribution of the adjacency matrix. In some selected cases, however, the eigenvalues follow a known probability distribution f (x) [52], which can be used to derive the expectation value of the total squeezing cost as where λ (+) (x) is the largest symplectic eigenvalue of eq. (12) for a real random variable x. This gives us a straightforward recipe to compute G from the distribution f . Furthermore, it allows us do deduce the scaling of G from the distribution of the eigenvalues and especially from the width of this distribution. In fact, if the distribution does not change with N , then G scales linearly with N , whereas any growth in the width of f would imply a superlinear scaling of G. The protein and internet networks, which are scale free networks, have the additional property of having many nodes sharing the same neighborhood which implies a large kernel of their adjacency matrix and thus a slower growth of G compared to the other complex graphs, which are essentially full rank. More specifically, the low rank of the protein-protein interaction graph [50], is explained by Adjacency Spectrum the duplication process at the heart of its generation, which, by definition duplicates the neighborhood of vertices. For the internet graph of [49], in particular, many of the nodes are leafs (clients) connected to a few client providers. This results in a very low -but still linearly increasing with N -rank, explaining why its squeezing cost is the lowest of the line in Fig. 5.
In Fig. 5 we report the trend of the total squeezing cost as a function of the number of nodes for each of the above complex topologies. In this case, the networks were simulated and the values obtained were averaged over ten different samples.
From the plots we notice that the linear trend is the most common case. The spectral theory of real graphs is a much less established field and there is only a handful of results we can apply to actually make predictions. In particular, to the best of our knowledge there are only empirical results about the convergence of the spectral density for the scale-free and the small-world models. In Ref. [52], however, they show some crucial properties of the spectra of scale-free and small-world graphs, although further studies are required in order to have a deeper insight in the properties of complex quantum graph states.
In particular, it is shown that, fixing the other network's parameters, the width of their distribution is constant with N , implying as we discussed a linear trend of G.
The WS graph interpolates between a regular periodic graph and the random graph when the rewiring probability β goes from 0 to 1. In particular, for β = 1 we have approximately an ER with p = Q N −1 , although this model does not approach the actual ER graph since every node will be connected to at least K 2 other nodes. From Fig. 5 we also notice that the most expensive growth belongs to the ER topology, which is the only one among the topologies we studied whose trend is superlinear. This behaviour is actually the easiest to predict from random matrix theory: the Wigner semicircular law for the distribution of the eigenvalues of a random graph [53] gives their probability distribution in the form Fig. 3a), where R = 2 N p(1 − p). Casting it in equation (21) gives where we used the parity of f ER (x), the fact that the support is in [0, R] and that for the largest eigenvalue, lim N →∞ λ 1 /N = p, with probability 1. Figure 6 shows the comparison between the theoretical behaviour of the squeezing cost of the ER graph and the numerical experiments from the simulations. It shows a superlinear ∝ N log N increase of the squeezing cost, which is explained by the widening of the support of f due to the increasing values of R. More formally, making the variable change x = Ry, we have log 10 λ (+) (Ry) = 2 log 10 R + log 10 The second term in eq. (22) then becomes  (24) and is the dominant term in the squeezing cost. Now that we have characterized the cost of implementing Gaussian quantum networks, we will describe how to use them as a substrate to perform quantum communications.

VI. QUANTUM TELEPORTATION GAUSSIAN NETWORKS
Quantum entanglement is a paramount resource for quantum information purposes. In particular, bipartite entanglement represents the fundamental requirement that a shared quantum channel should have in order to enable a truly quantum teleportation. In the framework of Quantum Communications, the networks previously described can be seen as distributed Gaussian quantum teleportation networks [54], where each pair of nodes can employ the pre-established quantum correlations together with Local Operations and Classical Communications LOCC to teleport a Gaussian quantum state from one node to the other.
In a naïve strategy, the teleportation between two arbitrary nodes can be implemented simply by ignoring all the other nodes and exploiting the residual bipartite entanglement together with classical communications. This strategy is a direct extension of the standard teleportation protocol from two to more stations and is called non-assisted protocol [55].
Another set of strategies is based upon a cooperative behavior, where all the other nodes assist the teleportation between the chosen pair (Alice and Bob) by means of LOCC. In fact, if the external nodes perform suitable local measurements and then classically communicate their outcomes to Bob, the latter can use this additional classical information to improve the process via modified conditional displacements. These strategies are called assisted protocols and are the ones that determine what we call networking/routing protocol in this Article.
According to Gu et al. [34] quadrature measurement on a mode of a Gaussian network like the ones we considered so far can be described by two simple rules: • Vertex Removal: aq-measurement on a qumode removes it from the network, along with all the edges that connect it.
• Wire Shortening: ap-measurement on a qumode is just aq-measurement after a Fourier Transform, which corresponds to a phase rotation of π/2: S F = S R (θ = π 2 ). The node will thus be removed but the phase shift will induce correlations between the neighbouring edges. Thus, measurements in the momentum basis allow us to effectively "shorten" linear graph states.
If two nodes A and B need to teleport a quantum state, they can be helped by the other nodes in the network who will perform these operations in order to increase the Where ν − is the smallest symplectic eigenvalue of the partially transposed covariance matrix of the pair. Partial transposition is a necessary operation for the PPT criterion [56] and is easily implemented in Gaussian states by changing the sign of the momentum of one of the two subsystems. For Gaussian states, the negativity is simply connected to another measure of entanglement, the fidelity of teleporting a coherent state through that quantum channel Simple classical communication attains at most F coh = 1 2 so a bipartite system presents truly quantum correlations only if F coh > 1 2 , or equivalently ν − < 1 and N > 0. The symplectic eigenvalues ν ± of a two-mode system can be computed through the invariants of the covariance matrix [57]. More specifically, we can define the seralian ∆ = det σ A +det σ B +2 det σ AB , where σ A and σ B are the local covariance matrices of the single-mode sub-systems A and B, and σ AB represents their correlations. From this we can compute the symplectic eigenvalues as: In Fig. 7 we compare the effect of different regular topologies of quantum networks with the purpose of distributing entanglement between two of the furthest nodes inside the network.
The simplest cases are the star and complete networks. In the first case the best assisted strategy is to let everyone perform aq-measurement on their node except the central one who will make a wire shortening to link We apply two different strategies toD N : one is to measure all the central nodes in P and the other is to alternate a p and a q measurement. We can see that measuring always in p is not necessarily the optimal strategy. On the right side you can see a scheme of theD network and the two different measurement strategies.
the final pair. In the complete network A and B are already linked by an edge so it is sufficient to measure the position in all the other qumodes (notice that this strategy outperforms the non-assisted protocol). In both these cases the entanglement is constant with the number of nodes in the network as we would expect, and the wire shortening of the central node in the star graph decreases the negativity with respect to the complete graph [58]. In the linear graph all the nodes have to wire shorten from A to B. Here the negativity quickly decreases with the number of nodes. The decrease of entanglement with the wire shortening seem to be typical in all configurations except the diamond graph, where all the central nodes arep-measured.

VII. MULTI-PATH ENTANGLEMENT
The behaviour of the diamond graph is quite counterintuitive and might be expected to increase the fidelity of quantum communications. It can be shown (proof in Appendix C) that the lowest symplectic eigenvalue for this system goes like Where R = 10 s/10 is the inverse of the squeezing inp, with squeezing factor s in dB. Hence, the negativity grows logarithmically with N Rg 2 and the two modes become perfectly correlated in the limit of either infinite squeezing, infinite strength CZ-gate or infinite parallel measurements onp. This property of the Diamond network, however, is not easily generalized to all graphs that present parallel connections and the quest for the optimal measurement strategy in order to improve the final entanglement is by no means trivial. This is the case, for example, of theD graph shown in Fig. 8, generated by taking the diamond network and add a CZ-gate link between adjacent central nodes. We can see that for N > 9 measuring alwaysP in this network is not the optimal strategy, whereas a better strategy is to alternate aP andQ measurement in order to restore a (smaller) diamond network. Another important figure of merit is the entanglement per squeezing cost, shown in Fig 9 (a).
We see that the diamond is the only one that gives the a ratio of entanglement per cost of the network that becomes constant for large N . However, the linear graph is the one that links two nodes that are the furthest away from each other. Conversely, figure 9 (b) shows the negativity in the final pair divided by the number of modes in the initial state. Once again, the diamond structure is particularly convenient, yielding the highest negativity while keeping a constant number of independent squeezers.
In order to give a fair comparison between the capacity of the linear network to bridge distant nodes and that of the diamond to increase the final entanglement we need to generalize the diamond graph to a diamond chain graph, DC K,N , where K is the number of parallel branches linking the two hubs that want to perform quantum communications as in figure 10. We can then compare the entanglement concentrated using multiple path strategies to link two nodes far away from each other. We can see in figure 11 that the presence two nodes at the same distance of three lattices graphs, the square lattice Q N ×N and the two triangles T N ×N , formed from the square by adding edges on the diagonals in such a way that the distance between A and B is the same,T N ×N formed by adding edges to the diagonals so that the distance is the same as the linear graph, and the linear graph L N . In order to compare the networks with the same distance we doubled the size of theT and the L graphs.
of parallel links has indeed the desired effect, despite the quality of the final pair, which still decreases exponentially with the distance. On the other hand, notice that the parallel links can help concentrating more entanglement until the system reaches a plateau and even the additional channels will not allow to increase the negativity. Moreover, the value for effort of this networks, specifically the ratio between the entanglement of the pair after the protocol and the squeezing cost before the protocol, is maximized by the linear graph. Another important class of networks, notably for measurement based quantum computation, is constituted by grid cluster states that belong to graph shapes that allow for universal quantum computation [59]. Similarly to the diamond network, the presence of ancillary nodes between the emitter and the receiver can improve the quality of the quantum link with respect to the linear network. This, however, is not a general rule and sometimes the presence of additional links can be detrimental. This is the case of the triangular lattice, generated from the square lattice by adding a link between the nodes in the diagonal. There are two ways of generating the triangular and only one of the two,T , decreases effectively the distance between Alice and Bob. In both cases the result is detrimental, however T is slightly better thanT , while the square lattice Q seems to be the most effective. This result is shown in figure 12.

VIII. ROUTING PROTOCOL
In this section we aim at employing the abstract notions on Gaussian graphs developed so far for a specific application: the routing of entanglement. In this scenario, the highly multimode entangled Gaussian state corresponds to a distributed teleportation network, described in the previous section, where each node of the network is supplied with a mode of an electromagnetic harmonic oscillator and is linked to some other nodes in different locations through quadrature correlations, e.g. quantum entanglement. We remark that this type of communication quantum networks is inherently different from the typical qubit networks that are currently being deployed in different metropolitan areas [60]. In those cases, for example, each entanglement link is pairwise between two qubits and as a consequence each node of the network will have to receive, storage and measure as many quantum states as neighbors it has. Conversely, in a Gaussian quantum network the same qumode can be entangled with an arbitrary number of other nodes. Moreover, the production of such states, their manipulation to increase the entanglement among two nodes and their measurement to perform quantum teleportation can be achieved deterministically, unlike the discrete variables case. Nonetheless, qubits networks have been extensively studied over the last years, whereas Gaussian teleportation networks is a very recent emerging field. Our purpose is, thus, not to prove the superiority of the latter, but rather to explore its properties and the differences from the DV schemes in order to get the best of both worlds.
The results of the previous sections, highlighted some outstanding properties of Gaussian networks. The most important is the parallel enhancement of entanglement in the diamond graph. If properly used, this feature can most certainly improve the routing of entanglement in regular and complex shaped networks. On these grounds, the search for an optimal protocol that exploits all the qualities of these Gaussian networks is very desirable yet arduous, and will be subject of future investigations. Alternatively, we present a naïve entanglement routing protocol that takes into account some of these properties and we will apply it to complex topologies, to show that the enhancement of the entanglement with respect to the trivial protocol is, in principle, easily achievable. Imagine we have a distributed network of entangled harmonic oscillators, where each node is honest and can perform classical communication and local homodyne measurement, and we want to establish an entangled pair between two nodes, Alice and Bob, that want to teleport a quantum state or perform QKD. The trivial protocol -called Shortest int the following -would be to find the shortest path between them and measure in P all the qumodes along this path and in Q all the others. A careful look at the inner structure of the network, however, might help us increase the strength of the correlation. For example if at any point, two nodes on the path are linked by multiple parallel routes, we can measure these in P to exploit the parallel enhancement.
In order to show this in practice, we will test the performances of three different routing protocols (shown in figure 13) on various complex networks with the purpose of establishing a highly entangled pair [61]. We choose Alice to be one of the hubs of the graph and evaluate the efficiency of the protocol in delivering entanglement to all the other nodes. The quantum protocol that we propose to exploit the parallel enhancement of entanglement will be simply called Routing.
• Routing: it takes as input the target node, Bob; it lists all the shortest paths connecting it to Alice and measures all the nodes that are not in these paths in theQ quadrature, so that they will not influence the protocol. Among the list of paths it checks one by one those to be measured inP in order to maximize the negativity N of the final pair, while the rest will be measured inQ.
In the Routing protocol, in principle, we could have considered as well parallel paths of longer lengths that might have contributed to improve the negativity. However, in practice had the only effect of slowing down the performances while not increasing the entanglement for all the cases we considered. The effect of the parallel paths can be appreciated when comparing the negativity produced by Routing with that produced by Shortest.
• Shortest: the difference of the latter is that it only exploits one of the shortest parallel paths, directly measuring everything else in Q.
In some cases the two protocols do not give a substantial difference, either because there are not parallel routes or because these do not help increasing the entanglement, however in many instances the effects of parallel routing are significant. The last protocol we compare with is All P.
• All P : it measures all the terminal nodes with degree 1 inQ and the rest inP .
This protocol is less effective than the first two but is always the quickest, whereas Routing can be computationally very slow on regular networks, which are characterized by long distances and many parallel paths, but becomes very efficient on complex sparse networks. The following simulations can be reproduced using our python code, available at [61]. One instance of this program is given in figure 14 which shows the negativity provided by the three different protocols for each node of a G AS (N = 1000) network. At the beginning of the protocol, we pick Alice as the node, or one of the nodes, with the highest degree. The nodes are then sorted by their distance from Alice and, for the same distance, by the number of all the shortest paths connecting them to Alice. Additionally, the grey column represents the ratio of parallel paths that were useful to increase the entanglement. Notice that nodes at distance 1 cannot show a difference between the Routing and the Shortest protocols, however many nodes at distance 2 present a greater negativity than those at shorter distance after the Routing.
In figure 15 we show the graph of the network, where the nodes are again sorted by distance and number of parallel paths and the size of each node is proportional to its degree. In this figure Alice is '0' and has a thick Routing shortest All P Routing shortest All P FIG. 14: Negativity produced by the three different protocols applied to each node of the the G AS (N = 1000) network. The nodes are labeled in order of distance and of number of paths connecting to Alice. The blue, orange and green stems represent the negativity of the final pair after the Routing, Shortest and All P protocols respectively, while the dashed lines represent the mean value for all the nodes. The color of the marker indicates the distance of the node from A and the grey columns represent the ratio of paths that improved the entanglement in Routing. red contour. The node with highest negativity and all the paths that improved its entanglement are highlighted with red thick lines.
The same analysis was done in several networks with different sizes and topologies with very different results that we report in figures 16, 17, 18 and 19. A property that is not apparent in Fig. 15, is that the node with the highest enhancement of entanglement due to the multiple paths is not necessarily the one with the highest negativity in absolute. This is the case of the ER network of Fig. 16, in which the node with the highest entanglement, highlighted in green in the graph representation, is at distance 1 while the node with the highest difference in negativity between the Routing and the Shortest protocols, highlighted in red, is at distance 3. In this case, the structure of the subgraph used throughout the Routing is not a diamond chain and the intercorrelations among the parallel branches have limited the increase of the entanglement,as for theD N network in Fig. 8. In any case, in this network the nodes at greater distances are the ones that are most affected by our protocol and, although in some cases many parallel paths have been disregarded, as shown by the height of the grey column, all the nodes at distance 4 received a substantial enhancement.
The results of the simulation on the BA topology of Fig.   17 is similar to the AS, although the first only reaches a distance of 3. The nodes with the highest absolute negativity and the highest negativity difference produced by the Routing protocol coincide and are at distance 2 from Alice, whereas this time the subgraph of is a diamond with no interconnections. Also in this case distance 2 is favorable to perform quantum communications.
The WS structure of Fig. 18, on the other hand, is the worst to apply the Routing protocol. Only a few nodes, in fact, were poorly enhanced and mostly at large distances, while the negativity averaged over all the nodes for Routing and Shortest is comparable. The node 44 at distance 3 is the one that received the greatest boost from our protocol, whereas node 1 (like all the other nodes at distance 1) has the highest negativity.
Finally, the biological network of Fig. 19 produced the most interesting results. Once again, many nodes at distance 2 end up having more negativity than those at distance 1, and at this distance the nodes with the same degree have the same negativity that decreases exponentially with their degree. The nodes with highest negativity and highest difference coincide with node 139, which is linked to Alice through 33 intermediate nodes, forming a diamond network with no interconnections.

A. Conclusion
We have studied Gaussian multimode quantum networks with regular and complex topologies for quantum communication protocols. In particular: i) we have studied their cost in term of amount of squeezing and number of necessary squeezed modes to build the network; ii) we have established a multi-path routing protocol distributing entanglement between two arbitrary nodes. In details: • We have shown analytically and numerically that the cost of the networks is in general nonlinear with the number of edges and nodes and there are particular (regular and complex) graph shapes that optimize the cost and the number of squeezers over number of nodes/edges in the networks. Among regular networks the diamond and the star graph need only two squeezed nodes to be built, independently from their number of nodes. Among the complex networks shapes, the Internet Autonomous System model is the most convenient in number of needed squeezed states.
• We have studied the assisted teleportation protocol in Gaussian entangled networks, where couple of nodes are assisted in the teleportation by local measurement in all the other nodes. This naturally define a routing protocol in Gaussian networks. In particular we have consideredQ andP homodyne quadrature measurement that allow respectively for vertex-removal and wire-shortening.
• The routing is optimized by different measurement schemes in regular networks. In the linear and the diamond networks the best strategy consists in the wire shortening, but the diamond network shows the largest ratio in reached entanglement over cost.
• Inspired by the behaviour of the diamond network we have devised a routing protocol that exploits wire shortening in parallels paths and we have applied it to complex networks graphs. The protocol named Routing is compared with Shortest, where wire shortening is done only in the shortest path, and All P, which removes all the terminal nodes while it wire-shorten all the others. In most cases, the Routing improves the entanglement compared with Shortest. Also, in terms of computational complexity, the Routing is much slower than All P in regular networks, where there are long distances between nodes and several parallel paths, but it is very efficient in complex sparse networks.

B. Outlook
The devised Routing protocol is very general so that it can be applied to arbitrary networks, and it is particularly efficient for sparse not regular networks. Our simple graph exploration approach would be improved in computational efficiency by real graph-based algorithms, especially if we allow for approximate solutions. [62]. Also it would be interesting to allow for non uniform distributions of squeezing s and CZ gate strength g or more general homodyne measurements, i.e. going beyond the twoP andQ cases and considering measurements alongQ θ = cos(θ)Q + sin(θ)P . In addition, it could be interesting to examine a scenario in which the intermediate nodes are dishonest and do not cooperate to perform the routing. Moreover, in order to consider practical implementations, realistic parameters for losses and noise should be included in the model. Finally the routing protocol has been implemented to solve the particular task of creating a perfect EPR pair between two nodes, fu-ture protocols will consider general reshaping in arbitrary multiparty states.

X. ACKNOWLEDGEMENTS
We thank Dimitrios Tsintsilidas for helpful discussions. In fact, single mode squeezing and the CZ-gate both require a fixed amount of squeezing to be implemented, so we would expect G(σ) to scale linearly with the number of links and nodes. This is a direct consequence of the spectral distribution of the linear graph, which is We can use equations (12) and (18) to compute the squeezing cost exactly for any given N . For N 1, the average squeezing cost per modeḠ = G/(#squeezed modes) = G/ rk(A) can be approximated by the following integral  Fig. 20.
Similarly, we can use the eigenvalues expression of the circular graph (or its generalization with Q > 1 nearest neighbors) (A3) From this we can see why the linear and circular graph have the same scaling. In fact for Q = 1, we have D k (G C (Q1 ) = 2 cos(kπ/N ) − 1.
The spectra of the star and diamond graph have only two non-null eigenvalues [63] As a consequence, the cost of the star and diamond Gaussian networks grows logarithmically with N and in both cases has a simple expression G(σ S N ) = 20 log 10 which sum up to a cost growing essentially linearly in N .

Appendix B: Graphical Calculus
In ref. [33] it is provided a unified graphical calculus for all Gaussian pure states that is particularly suited for describing highly multimode Gaussian networks.
In this framework, a N mode Gaussian state is completely described, up to displacements, by a N × N complex valued adjacency matrix: Where the real and imaginary part of Z, V and U respectively, are related to the covariance matrix through the following unique decomposition Gaussian graph states have a particular simple graphical representation, being Where A is the weighted adjacency matrix of the graph and D is a diagonal matrix that represents momentum squeezing, i.e. for D = 10 −2s 1 the momentum variance of all modes is reduced by 2s decibels.
All symplectic operations can be reproduced in this language, however, since we already know how to represents the resource graph states, we only need to implement the quadrature measurements inx andp. We can express the state as Where t is the target mode we want to measure, W is the subgraph of the untouched modes and R their correlations with the target mode. We have the following two rules: • Z −→ Z q = W after aq measurement.
• Z −→ Z p = W − RR T t after ap measurement.
Thus, for a measurement inq we remove the node and its link from the graph, whereas for a measurement inp we apply a π/2 phase rotation and then measureq.