Fast Transforms in Quantum Computation
Artyom M. Grigoryan, Sos S. Agaian · 2025
This chapter discusses quantum fast transforms that play a key role in the development of image processing techniques. It describes concepts of the discrete paired, Hadamard, and Fourier transforms and their circuits in quantum computation. The chapter focuses on the discrete paired transform, which is a core of the N-point discrete Hadamard and Fourier transforms. It also describes the simple quantum schemes for calculation of the paired transform and presents the method of implementation of the classical diagram for the circular convolution in the frequency domain. The chapter explains the analytical and matrix representations of the discrete Hadamard transform. It details the circuits for calculating quantum Fourier transform, which are based on the paired transform-based splitting. The N-point discrete Fourier transform is considered in the traditional form which is used in signal and image processing.