In-Network Computing has brought computing and communication together at every communication node in the digital world, this work lays the foundations for doing the same in the quantum world, improving communication properties in the process - a combination of quantum computing and quantum communication. Full network softwarization, in-network intelligence, and massive connectivity will create an unprecedented demand for computing resources in the digital world. Accordingly, the scientific and industrial communities have begun to explore technologies such as quantum computing and have made significant efforts to demonstrate an algorithmic advantage for various problems. However, practical quantum computers are resource-inefficient and difficult to build. This article introduces a new communication paradigm leading to the concept of quantum in-network computing in the context of entanglement-assisted communication and computing for inherent distributed resilience and sensing. We review the fundamentals of digital hardware as characterized by Turing's computability theory and demonstrate their relevance to mathematically rigorous characterizations of gate-based quantum computing. We then provide such a characterization using methods from effective analysis, leading to significant results that reveal the inherent theoretical limitations of universal gate-based quantum computers. These results support our assessment that gate-based quantum in-network computing is only possible through specialized, non-universal solutions that are seamlessly integrated with high-performance digital computing.