The Quantum Internet is envisioned as the final stage of the quantum revolution, opening fundamentally new communications and computing capabilities, including the distributed quantum computing. But the Quantum Internet is governed by the laws of quantum mechanics. Phenomena with no counterpart in classical networks, such as no-cloning, quantum measurement, entanglement and teleporting, impose very challenging constraints for the network design. Specifically, classical network functionalities, ranging from error-control mechanisms to overhead-control strategies, are based on the assumption that classical information can be safely read and copied. But this assumption does not hold in the Quantum Internet. As a consequence, the design of the Quantum Internet requires a major network-paradigm shift to harness the quantum mechanics specificities. The goal of this work is to shed light on the challenges and the open problems of the Quantum Internet design. To this aim, we first introduce some basic knowledge of quantum mechanics, needed to understand the differences between a classical and a quantum network. Then, we introduce quantum teleportation as the key strategy for transmitting quantum information without physically transferring the particle that stores the quantum information or violating the principles of the quantum mechanics. Finally, the key research challenges to design quantum communication networks are described.

Quantum Teleportation is the key communication functionality of the Quantum Internet, allowing the "transmission" of qubits without the physical transfer of the particle storing the qubit. Quantum teleportation is facilitated by the action of quantum entanglement, a somewhat counter-intuitive physical phenomenon with no direct counterpart in the classical word. As a consequence, the very concept of the classical communication system model has to be redesigned to account for the peculiarities of quantum teleportation. This redesign is a crucial prerequisite for constructing any effective quantum communication protocol. The aim of this manuscript is to shed light on this key concept, with the objective of allowing the reader: i) to appreciate the fundamental differences between the transmission of classical information versus the teleportation of quantum information; ii) to understand the communications functionalities underlying quantum teleportation, and to grasp the challenges in the design and practical employment of these functionalities; iii) to acknowledge that quantum information is subject to the deleterious effects of a noise process termed as quantum decoherence. This imperfection has no direct counterpart in the classical world; iv) to recognize how to contribute to the design and employment of the Quantum Internet.

To fully unleash the potentials of quantum computing, several new challenges and open problems need to be addressed. From a routing perspective, the optimal routing problem, i.e., the problem of jointly designing a routing protocol and a route metric assuring the discovery of the route providing the highest quantum communication opportunities between an arbitrary couple of quantum devices, is crucial. In this paper, the optimal routing problem is addressed for generic quantum network architectures composed by repeaters operating through single atoms in optical cavities. Specifically, we first model the entanglement generation through a stochastic framework that allows us to jointly account for the key physical-mechanisms affecting the end-to-end entanglement rate, such as decoherence time, atom-photon and photon-photon entanglement generation, entanglement swapping, and imperfect Bell-state measurement. Then, we derive the closed-form expression of the end-to-end entanglement rate for an arbitrary path and we design an efficient algorithm for entanglement rate computation. Finally, we design a routing protocol and we prove its optimality when used in conjunction with the entanglement rate as routing metric

The Quantum Internet, by enabling quantum communications among remote quantum nodes, is a network capable of supporting functionalities with no direct counterpart in the classical world. Indeed, with the network and communications functionalities provided by the Quantum Internet, remote quantum devices can communicate and cooperate for solving challenging computational tasks by adopting a distributed computing approach. The aim of this study is to provide the reader with an overview about the main challenges and open problems arising in the design of a distributed quantum computing ecosystem. For this, the authors provide a survey, following a bottom-up approach, from a communications engineering perspective. They start by introducing the Quantum Internet as the fundamental underlying infrastructure of the distributed quantum computing ecosystem. Then they go further, by elaborating on a high-level system abstraction of the distributed quantum computing ecosystem. They describe such an abstraction through a set of logical layers. Thereby, they clarify dependencies among the aforementioned layers and, at the same time, a road-map emerges.

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