The quantum algorithms of Deutsch, Simon and Shor are described in a way which highlights their dependence on the Fourier transform. The general construction of the Fourier transform on an Abelian group is outlined and this provides a unified way of understanding the efficacy of the algorithms. Finally we describe an efficient quantum factoring algorithm based on a general formalism of Kitaev and contrast its structure to the ingredients of Shor' algorithm.

arXiv:quant-ph/9707033Proceedings of the Royal Society A Mathematical Physical and Engineering SciencesJan 8, 1998128 citations

arXiv:quant-ph/9812060

Sampling Fourier Transforms on Different Domains

Lisa Hales, Sean Hallgren

Published Dec 21, 1998• 1 citation

arXiv HTML PDF

Abstract

{\mathbb Z}_p

can be efficiently approximated by transforming over

{\mathbb Z}_q

for any q in a large range. Our result places no restrictions on the superposition to be transformed, generalizing the result implicit in Shor which applies only to periodic superpositions. In addition, our proof easily generalizes to multi-dimensional transforms for any constant number of dimensions.

DOI: 10.48550/arxiv.quant-ph/9812060

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can be efficiently approximated by transforming over

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In this work, we introduce a definition of the Discrete Fourier Transform (DFT) on Euclidean lattices in

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