Quantum computing represents one of the most significant paradigms shifts in the history of computation, promising to solve classes of problems that remain intractable for classical machines. Unlike classical computing, which is built on Boolean algebra and binary logic, quantum computing is fundamentally a mathematical discipline rooted in linear algebra, complex vector spaces, and probability theory. This article examines the mathematical structures that underpin quantum computation, including qubits, superposition, entanglement, unitary transformations, and quantum gates, and explores how these structures enable algorithms such as Shor\'s and Grover\'s to outperform their classical counterparts. The discussion further extends to quantum error correction, the challenges of decoherence, and the interdisciplinary convergence of mathematics and computer science that is shaping next-generation computing architectures. The article concludes with a reflection on open mathematical problems and the future trajectory of quantum information science.