# Continuous Variables Graph States Shaped as Complex Networks: Optimization and Manipulation
## 1 Introduction  
In the last few decades, network theory has provided a natural framework for describing complex natural, social, and technological structures [1, 2]. Recurrent types of complex networks, like the scale-free networks, have been recovered in phenomena at different scales, where the functionality of the systems seems to be closely related to their structure [3, 4]. More recently, complex networks have gained attention in the quantum realm, where several theoretical works [5, 6, 7, 8, 9, 10] show that complex structures may play a role in quantum information technologies. It is clear that, as quantum architectures are reaching larger scales, their internal arrangement starts to play a significant role in their functionality. Moreover, network structures are clearly at the base of quantum communication protocols [11], but also appear in particular kinds of multiparty entangled states that allow for measurement-based quantum computing (MBQC) protocols [12]. Recently, new records have been established in superconductor [13, 14, 15] and Rydberg [16, 17] based technologies, and extremely large entangled states have been generated in the optical domain [18, 19, 20, 21]. We can describe superconducting and Rydberg platforms as networks of interacting qubits, where quantum information is encoded in Discrete Variables (DV) states. Differently, in the optical domain, networks of entangled optical modes are deterministically generated and measurement-based quantum protocols can be implemented via a Continuous Variables (CV) encoding [22]. The control on the reconfigurability of the cited networks is gradually increasing and, in particular, in the CV case, it is possible to have totally reconfigurable networks (with all-to-all connections) mimicking complex networks structures [23]. Here, we study CV entangled states, named CV graph states or CV cluster states, arranged with shapes that are typical of real-world complex networks, in order to investigate their properties for quantum computing or quantum networking (communication) protocols. CV graphs have been introduced in the context of measurement-based quantum computing exploiting CV resources [22; 24]. While usually the name \"cluster\" is used when the graph shape allows for universal quantum computing, in this work, we will use the terms \"cluster\" state and \"graph state\" as synonyms. In CV-quantum computing, quantum information is encoded in variables that can take a continuum spectrum of values. In optics, these variables are the quadratures of the electromagnetic field (called amplitude and phase quadrature) and they correspond to the position \(q\) and momentum \(p\) of the harmonic oscillator, which represents the behavior of a single mode of the field. Ideal cluster states, which involve perfect correlations between quadratures of different optical modes, cannot be reached experimentally, as they would require an infinite amount of energy (squeezing) to be established. Thus, experimental clusters always involve a poorer degree of correlations, which cause errors and noise in computation. Given experimentally accessible resources, analytical optimization protocols have been proposed to choose state manipulations that better distribute the correlations in order to minimize errors [25], then improve the quality of the cluster. In this work, we apply the optimization procedure to complex cluster shapes and we find that denser and more regular network shapes allow for better quality [26]. As the classical internet can be described via a complex network model, it is worth considering complex network shapes in quantum communication protocols. In particular, we are interested in quantum routing, i.e., the establishment of a direct entanglement link between two arbitrary nodes of a network in order to have an exploitable channel for quantum teleportation. Quantum routing protocols have been mainly studied in the DV approach [27; 28], while here we study the particular setting of CV quantum resources.