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International symposium on image and signal processing and analysis/ISPA ...

Fast classical and quantum fractional Walsh transforms

Valery Labunets, Ekaterina V. Labunets-Rundblad, Karen Egiazarian, Jaakko T. Astola

Published Nov 13, 2002• 14 citations
Publisher

Abstract

The fractional Fourier transform (FRFT) is one-parametric generalization of the classical Fourier transform. FRFT was introduced in the 1980s and found a lot of applications in signal processing. The time and spectral domains are both special cases of the fractional Fourier domain. They correspond to the Oth and 1st functional Fourier domains, respectively. We introduce the classical and quantum fractional Walsh transforms and develop corresponding fast algorithms.

DOI
10.1109/ispa.2001.938691

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The fractional Fourier transform (FRFT) is one-parametric generalization of the classical Fourier transform. FRFT was introduced in the 1980s and found a lot of applications in signal processing. The time and spectral domains are both special cases of the fractional Fourier domain. They correspond to the Oth and 1st functional Fourier domains, respectively. We introduce the classical and quantum fractional Walsh transforms and develop corresponding fast algorithms.
Fourier and fractional-Fourier transformations are widely used in theoretical physics. In this paper we make quantum perspectives and generalization for the fractional Fourier transformation (FrFT). By virtue of quantum mechanical representation transformation and the method of integration within normal ordered product (IWOP) of operators, we find the key point for composing FrFT, and reveal the structure of FrFT. Following this procedure, a full family of generalized fractional transformations are discovered with the usual FrFT as one special case. The eigen-functions of arbitrary GFrT are derived explicitly.
- In this paper we present a method to compute the coefficients of the fractional Fourier transform (FrFT) on a quantum computer using quantum gates of polynomial complexity of the order O(n^3). The FrFt, a generalization of the DFT, has wide applications in signal processing and is particularly useful to implement the Pseudopolar and Radon transforms. Even though the FrFT is a non-unitary operation, to develop its quantum counterpart, we develop a unitary operator called the quantum Pseudo-fraction Fourier Transform (QPFrFT) in a higher-dimensional Hilbert space, in order to computer the coefficients of the FrFT. In this process we develop a unitary operator denoted U by which is an essential step to implement the QPFrFT. We then show the application of the operator U in the problem of quantum phase estimation.
It is well known that the fractional Fourier transform (FrFT) is a non-trivial generalization of the classical Fourier transform. This transform has been extensively utilized in signal analysis and image processing. The uncertainty principle is one of the fundamental results related to the FrFT. In the present paper, we first propose the different method in proving a variant of Heisenberg uncertainty principles associated with the FrFT. This method combines the Heisenberg-type uncertainty principle pertaining to the FrFT and the additive property of the FrFT. We then provide an alternative proof of the logarithmic inequality for the FRFT and find that this result differs slightly that in the literature. Finally, we explicitly expand the variant of Heisenberg uncertainty principle to a more general case.
Quantum Fourier transform (QFT) is an important part of many quantum algorithms. However, there are few reports on quantum fractional Fourier transform (QFRFT). The main reason is that the definitions of fractional Fourier transform (FRFT) are diverse, while some definitions do not include unitarity, which leads to some studies pointing out that there is no QFRFT. In this paper, we first present a reformulation of the weighted fractional Fourier transform (WFRFT) and prove its unitarity, thereby proposing a quantum weighted fractional Fourier transform (QWFRFT). The proposal of QWFRFT provides the possibility for many quantum implementations of signal processing.