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Chapter 41 Fast Quantum n-D Fourier and Radon Transforms

E. Rundblad, Jaakko T. Astola

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

Fast classical and quantum algorithms are introduced for a wide class of nonseparable nD discrete unitary K-transforms (DKT) KNn.

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IEEE Transactions on Quantum Engineering•Dec 13, 2021•4 citations
Fast classical and quantum algorithms are introduced for a wide class of nonseparable nD discrete unitary K-transforms (DKT) KNn {\cal K}_{N^n } KNn​ .
Fast Classical and quantum algorithms are introduced for a wide class of non-separable nD discrete unitary K- transforms(DKT)KNn. They require a number of 1D DKT Kn smaller than in the Cooley-Tukey radix-p FFT-type approach. The method utilizes a decomposition of the nDK- transform into a product of original nD discrete Radon Transform and of a family parallel/independ 1DK-transforms. If the nDK-transform has a separable kernel, that again in this case our approach leads to decrease of multiplicative complexity by factor of n compared to the tow/column separable Cooley-Tukey p-radix approach.
This paper extends the Radon transform, a classical image processing tool for fast tomography and denoising, to the quantum computing platform. A new kind of periodic discrete Radon transform (PDRT), called quantum Radon transform (QRT), is proposed. The QRT has a quantum implementation that is exponentially faster than the classical Radon transform. Based on the QRT, we design an efficient quantum image denoising algorithm. The simulation results show that QRT preserves the good denoising capability as in the classical PDRT. Also, a quantum algorithm for interpolation-based discrete Radon transform (IDRT) is proposed, which can be used for fast line detection. Both the quantum extension of IDRT and the line detection algorithm can provide polynomial speedups over the classical counterparts.
This article extends the Radon transform, a classical image-processing tool for fast tomography and denoising, to the quantum computing platform. A new kind of periodic discrete Radon transform (PDRT), called the quantum periodic discrete Radon transform (QPRT), is proposed. The quantum implementation of QPRT based on the amplitude encoding method is exponentially faster than the classical PDRT. We design an efficient quantum image denoising algorithm using QPRT. The simulation results show that QPRT preserves good denoising capability as in the classical PDRT. Also, a quantum algorithm for IDRT is proposed, which can be used for fast line detection. Both the quantum extension of IDRT and the line detection algorithm can provide polynomial speedups over the classical counterparts in certain cases.