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The global race to build the world's first quantum computer has attracted enormous investment from government and industry, and it attracts a growing pool of talent. As with many cutting-edge technologies, the optimal implementation is not yet settled. This important textbook describes four of the most advanced platforms for quantum computing: nuclear magnetic resonance, quantum optics, trapped ions, and superconducting systems. The fundamental physical concepts underpinning the practical implementation of quantum computing are reviewed, followed by a balanced analysis of the strengths and weaknesses inherent to each type of hardware. The text includes more than 80 carefully designed exercises with worked solutions available to instructors, applied problems from key scenarios, and suggestions for further reading, facilitating a practical and expansive learning experience. Suitable for senior undergraduate and graduate students in physics, engineering, and computer science, Building Quantum Computers is an invaluable resource for this emerging field.