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"Progress in quantum computing: Algorithms, communication, and applications"

Final Report Summary - QPROGRESS (Progress in quantum computing: Algorithms, communication, and applications)

Computers based on quantum-mechanical effects such as superposition, interference, and entanglement, can vastly outperform our current classical computers for a number of specific problems in algorithms, cryptography, and communication. One can ask two basic questions about such computers:
(1) can we build them [experimental physicists are making good progress, but we don't have large-scaled quantum computers yet], and
(2) once they are built, what could they do?
This project made progress on the second question: finding more applications of quantum computers. It found new, more efficient quantum algorithms for problems like testing whether a given function depends on only few of its input variables, for finding ground states of certain Hamiltonians, for finding rare items in large search spaces, and for solving linear and semidefinite programs and other convex optimization problems faster. It obtained positive and negative results for learning algorithms that use quantum examples. It also discovered new quantum communication protocols that significantly outperform the best possible classical protocols, but also proved new limitations on such protocols. Finally, it has used the tools of quantum information theory to obtain new results in several non-quantum areas of mathematics, such as the analysis of various ranks of matrices (relevant for combinatorial optimization) and the theory of approximation by rational functions.