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Among the most remarkable successes of quantum computation are Shor's efficient quantum algorithms for the computational tasks of integer factorization and the evaluation of discrete logarithms. In this article, we review the essential ingredients of these algorithms and draw out the unifying generalization of the so-called abelian hidden subgroup problem. This involves an unexpectedly harmonious alignment of the formalism of quantum physics with the elegant mathematical theory of group representations and Fourier transforms on finite groups. Finally we consider the nonabelian hidden subgroup problem mentioning some open questions where future quantum algorithms may be expected to have a substantial impact.

R. Jozsa, "Quantum Factoring, Discrete Logarithms, and the Hidden Subgroup Problem," in Computing in Science & Engineering, vol. 3, no. , pp. 34-43, 2001.
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