  In distributed quantum computing architectures, with the network and
communications functionalities provided by the Quantum Internet, remote quantum
processing units (QPUs) can communicate and cooperate for executing
computational tasks that single NISQ devices cannot handle by themselves. To
this aim, distributed quantum computing requires a new generation of quantum
compilers, for mapping any quantum algorithm to any distributed quantum
computing architecture. With this perspective, in this paper, we first discuss
the main challenges arising with compiler design for distributed quantum
computing. Then, we analytically derive an upper bound of the overhead induced
by quantum compilation for distributed quantum computing. The derived bound
accounts for the overhead induced by the underlying computing architecture as
well as the additional overhead induced by the sub-optimal quantum compiler --
expressly designed through the paper to achieve three key features, namely,
general-purpose, efficient and effective. Finally, we validate the analytical
results and we confirm the validity of the compiler design through an extensive
performance analysis.
