# Quantum Complex networks

## Quantum Complex Networks

At different scales,from molecular systems to technological infrastructures, physical systems group in structures which are neither simply regular or random, but can be represented by **networks **with **complex **shape. Proteins in metabolic structures and the World Wide Web, for example, share the same kind of statistical distribution of connections of their constituents. In addition, the individual elements of natural samples, like atoms or electrons, are **quantum **objects. Hence replicating complex networks in a scalable quantum platform is a formidable opportunity to learn more about the intrinsic quantumness of real world and for the efficient exploitation of quantum-complex structures in future technologies. Future trusted large-scale communications and efficient big data handling, in fact, will depend on at least one of the two aspects -quantum or complex- of scalable systems, or on an appropriate combination of the two.

*Growing networks constructed by addition of connected complexes of dimension 2, i.e. triangles **( Z. Wu et al. Scientific Reports 5, 10073 (2015)). The network is composed of 100 nodes, single links are added with hopping probability p=0.1, any link can belong to a large number (m) of triangles. The network shows community structures*

We investigate **theory** and **experimental implementation** of quantum complex networks in the framework of continuous variables encoding of quantum resources.

Multimode quantum resources based on parametric process, femtosecond laser along with mode-selective and multimode homodyne measurements, allows for the implementation of networks with **reconfigurable****all-to-all **coupling and topology, with sufficient size and diversity to be relevant in the context of complex networks. We can deterministically generate complex graphs of entanglement correlations between the involved fields, and establish mapping between the resource and complex networks of physical interactions (Nokkala et al, New J. of Phys. 2018)

**Co**ntinuous Variables **Q**uantum **Co**mplex **N**etworks are the topic of the Consolidator ERC grant **COQCOoN **recently awarded to Valentina Parigi. Contact us (valentina.parigi@lkb.upmc.fr) for more information.

Some of the topic involved in the project:

**Simulation of structured quantum environment****Resilience of quantum communications/information protocols****Network structures in quantum states**

# Complex Networks

__Complex networks: __Networks are physical structures represented by graph: they are composed by (nodes) connected via links (edges), which depict physical interactions or correlations. They are often classified via the probability distributions of their edges per node (= degree), which reflects the way they grow and which is strongly connected with their global behavior (dynamics) and functionalities (resilience under errors and failures) [Alb02]

*Left: Barabasi -Albert (B.-A.) graph of 200 nodes. The more connected a node is, the more likely it is to receive new links. B.-A. networks are characterized by power law distributions of links per node, which are typical of many real-world networks. Right: Watt-Strogatz graph of 200 nodes: it is built from a regular ring lattice, where each node is symmetrically connected to a fixed portion of the neighbours, then connections are rewired it with a given probability.*

In the last twenty years network theory has provided a deep insight of **complex **systems, assembling theoretical tools able to the describe dynamical behaviour of biological, social and technological structures. During the recent years a new area applying network theory and complex networks to quantum physical systems has emerged [Bian15,Bia17]

##### References

- [Nian15] BianconiG 2015 Interdisciplinary and physics challenges of network theory Europhys. Lett. 11156001
- [Bia17] Biamonte J, Faccin Mand De Domenico M 2017 Complex networks :fromclassical to quantum arXiv:1702.08459
- [Alb02] R. Albert, A. L. Barabasi, Reviews of Modern Physics, 74, 47 (2002)

# Research projets

##### Publications

- J. Nokkala, F. Arzani, F. Galve, R. Zambrini, S. Maniscalco, J. Piilo, N. Treps, V. Parigi, Reconfigurable optical implementation of quantum complex networks, New J. Phys.
**20,**053024 (2018) - F. Sansavini, V. Parigi, Continuous Variables Graph States as Complex Networks: Optimization and Manipulation, Entropy, 22,
**26**(2019)

### Simulation of structured quantum environment

In the framework of open quantum system, complex networks of coupled harmonic oscillators can be used as a model of finite environment for a quantum harmonic oscillator. The coupled evolution of quantum system plus quantum environment and the properties of the system can be manipulated by structuring the complex environment, i.e. by shaping its topology [Nok17]. The spectral density of the environment, which gives the density of environmental modes as well as coupling strengths to them as a function of frequency, can be probed via an additional oscillator. Both the dynamics complex networks of coupled harmonic oscillators and the probing scheme can be mapped to multimode parametric processes [Nok18]. In the picture: Right side: spectral density of two complex environments having the shape of two real-world complex networks (*Scannel et al*. – Cat cerebral hemisphere connections and *Lusseau et al*. – A social network of dolphins-) Blue curve: analitycal calculation of the spectral density, purple curve: theoretical derivation of the spectral density from probing , yellow curve: simulation of probing technique based on experimental parameters.

### Network structures in quantum states

Based on the degree (=number of links per nodes) distributions, collective behavior of networks can be characterized by network measurements as, e.g., Density, Heterogeneity or Clustering. They play an important role in classical networks behavior as demonstrated for example in [Gao16] . The same quantities can be defined for networks of quantum correlations in multi-party quantum systems and they have been demonstrated to be able to capture quantum phase transition in Ising spin chains [Val17] We currently work on classification of multipartite continuous variables quantum states (Gaussian and non- Gaussian) via networks measurements.

### Resilience of quantum communications/information protocols

Internet is a physical complex network characterized by a scale-free structure which has been investigated to be resilient to external attacks [Goh02] Quantum communication will be delivered in networks, and part of quantum information protocols too. We investigate which complex shapes of networks will be the most effective in quantum technologies.

In particular we are interested in CV cluster states, a particular kind of multipartite entangled states, shaped as complex networks. In the Continuos Variable regime it is easy to produce large entangled network. You may ask, given finite realistic resources, what are the best networks we can build in term of errors and computational noise for quantum information protocols. Our graphs are built from a fixed number of (finitely) squeezed modes and linear optics transformations, which are optimize to give the best nullifiers. We found that DENSER and LESS RANDOM graphs give the best results!

**Quantum routing**. Imagine that a graph structure is shared between Alice and Bob and they want to reshape the entanglement connections to create a teleportation channel between two chosen nodes. They also want to perform the protocol only via linear optics operations on their nodes. It turns out that, even for simple structures like grids, results are not trivial: given a fixed amount of squeezing for the initial nodes (10 dB in this case) it is not always possible to create quantum channels between given nodes of the two parties. Moreover, it seems that grids with node number that is twice an odd number have solutions, while it is not the case for grids with node number that is twice an even number.

To read more: F. Sansavini and V. Parigi Entropy 22 (1), 26 (2019)

##### References

- [Nok17] J. Nokkala, S. Maniscalco and J.Piilo Open Systems & Information Dynamics Vol. 24, No. 04, 1740018 (2017)
- [Nok18] J. Nokkala, F. Arzani, F. Galve, R. Zambrini, S. Maniscalco, J. Piilo, N. Treps, V. Parigi, Reconfigurable optical implementation of quantum complex networks, New J. Phys.
**20,**053024 (2018) - [Gao16] J.Gao, B. Barzel, A.L. Barabási, Nature 530, 307 (2016)
- [Val17] M. A. Valdez, D. Jaschke, D. L. Vargas, and L. D. Carr Physical Review Letters 119, 225301 (2017)).
- [Goh02] K-Il Goh et al. PNAS 99, 12583, (2002).