Fast Fourier transforms and fast Wigner and Weyl functions in large quantum systems
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2024
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| author | Lei, C. Vourdas, A. |
| author_facet | Lei, C. Vourdas, A. |
| contents | Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space $D=d^n$ 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 $D=d_0...d_{n-1}$ with $d_0,...,d_{n-1}$ 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 ${\cal O}(D^2)$ to ${\cal O}(D\log D)$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_05163 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fast Fourier transforms and fast Wigner and Weyl functions in large quantum systems Lei, C. Vourdas, A. Quantum Physics Mathematical Physics Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space $D=d^n$ 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 $D=d_0...d_{n-1}$ with $d_0,...,d_{n-1}$ 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 ${\cal O}(D^2)$ to ${\cal O}(D\log D)$. 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. |
| title | Fast Fourier transforms and fast Wigner and Weyl functions in large quantum systems |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2405.05163 |