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, however, 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 uncovering 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, we distill common design trends for the research and development of quantum-accelerated communication systems and highlight lessons learned. The key insight is that classical heuristics can sharpen certain quantum parameters, underscoring the complementary strengths of classical and quantum computing. This article aims to catalyze interdisciplinary research at the frontier of quantum information processing and future communication systems.