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We give an algorithm for approximating the quantum Fourier transform over an arbitrary Ô which requires only Ç´Ò ÐÓÓ Òµ steps where Ò ÐÓÓ Ô to achieve an approximation to within an arbitrary inverse polynomial in Ò. This improves the method of Kitaev [11] which requires time quadratic in Ò. This algorithm also leads to a general and efficient Fourier… (More)

We isolate and generalize a technique implicit in many quantum algorithms, including Shor's algorithms for factoring and discrete log. In particular, we show that the distribution sampled after a Fourier transform over Zp can be efficiently approximated by transforming over Z, for any q in a large range. Our result places no restrictions on the… (More)

The Quantum Fourier Transform and Extensions of the Abelian Hidden Subgroup Problem by The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which Shor's celebrated factoring and… (More)

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