Quantum computing is poised to redefine the algorithmic foundations of communication systems. While quantum superposition and entanglement enable quadratic or exponential speedups for specific problems, identifying use cases where these advantages yield engineering benefits is still nontrivial. This article presents the fundamentals of quantum computing in a style familiar to the communications society, outlining the current limits of fault-tolerant quantum computing and clarifying a mathematical harmony between quantum and wireless systems, which makes the topic more enticing to wireless researchers. Based on a systematic review of pioneering and state-of-the-art studies indicating speedup opportunities, we distill common design trends for the research and development of quantum-accelerated communication systems and highlight lessons learned. The key insight is that quantum algorithms, including their gate-level realizations, can benefit from the design intuition applied in communication engineering. This article aims to catalyze interdisciplinary research at the frontier of quantum information processing and future communication systems.