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This volume explores the recent mathematical advancements made in quantum computing and technology. It begins with topics in quantum computing that include quantum algorithms and hidden subgroups, quantum search, algorithmic complexity, and quantum simulation. The next section covers quantum technology, such as mathematical tools, quantum wave functions, SQUIDs, and optical quantum computing. The section on quantum information deals with error correction, cryptography, entanglement, and communication. The final part explores quantum topology, knot theory, category algebra, and logic. The book also includes recommendations for federal funding of mathematical research on quantum computing.