Yes / Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space
with d an odd integer, and is inspired by the Cooley-Tukey formalism. The ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (together with some other transforms) in quantum systems with d-dimensional Hilbert space. Limitations of the method are discussed. In some special cases, the n Fourier transforms can be performed in parallel. The second method is for systems with dimension of the Hilbert space
with
odd integers coprime to each other. It is inspired by the Good formalism, which in turn is based on the Chinese reminder theorem. In this case also the ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (that involve some constants related to the number theory that describes the formalism). The ‘small Fourier transforms’ can be performed in a classical computer or in a quantum computer (in which case we have the additional well known advantages of quantum Fourier transform circuits). In the case that the small Fourier transforms are performed with a classical computer, complexity arguments for both methods show the reduction in computational time from
to
. The second method is also used for the fast calculation of Wigner and Weyl functions, in quantum systems with large finite dimension of the Hilbert space.
| Identifer | oai:union.ndltd.org:BRADFORD/oai:bradscholars.brad.ac.uk:10454/19917 |
| Date | 05 July 2024 |
| Creators | Lei, Ci, Vourdas, Apostolos |
| Source Sets | Bradford Scholars |
| Language | English, English |
| Detected Language | English |
| Type | Article, Published version |
| Rights | © The Author(s) 2024. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/., CC-BY |
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