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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Practical distributed quantum computing requires the development of efficient compilers, able to make quantum circuits compatible with some given hardware constraints. This problem is known to be tough, even for local computing. Here, we address it on distributed architectures. As generally assumed in this scenario, telegates represent the fundamental remote (interprocessor) operations. Each telegate consists of several tasks: i) entanglement generation and distribution, ii) local operations, and iii) classical communications. Entanglement generations and distribution is an expensive resource, as it is time-consuming and fault-prone. To mitigate its impact, we model an optimization problem that combines running-time minimization with the usage of that resource. Specifically, we provide a parametric ILP formulation, where the parameter denotes a time horizon (or time availability); the objective function count the number of used resources. To minimize the time, a binary search solves the subject ILP by iterating over the parameter. Ultimately, to enhance the solution space, we extend the formulation, by introducing a predicate that manipulates the circuit given in input and parallelizes telegates' tasks.

Practical distributed quantum computing requires the development of efficient compilers, able to make quantum circuits compatible with some given hardware constraints. This problem is known to be tough, even for local computing. Here, we address it on distributed architectures. As generally assumed in this scenario, telegates represent the fundamental remote (inter-processor) operations. Each telegate consists of several tasks: i) entanglement generation and distribution, ii) local operations, and iii) classical communications. Entanglement generations and distribution is an expensive resource, as it is time-consuming. To mitigate its impact, we model an optimization problem that combines running-time minimization with the usage of distributed entangled states. Specifically, we formulated the distributed compilation problem as a dynamic network flow. To enhance the solution space, we extend the formulation, by introducing a predicate that manipulates the circuit given in input and parallelizes telegate tasks. To evaluate our framework, we split the problem into three sub-problems, and solve it by means of an approximation routine. Experiments demonstrate that the run-time is resistant to the problem size scaling. Moreover, we apply the proposed algorithm to compile circuits under different topologies, showing that topologies with a higher ratio between edges and nodes give rise to shallower circuits