Explaining the Design of the Quantum Fourier Transform

Zhongwei Wang, Xirui Gou, Ruiqing Fu +1 more

Quantum Fourier transform (QFT) plays an eminent role in quantum computation.It creates a superposition of different quantum states, allowing simultaneous calculation, which would take many steps were the same program implemented on a classical computer.The computational speed of a quantum computer is thus boosted dramatically.In this article, we explain the application of QFT in Shor's algorithm, which was proposed by Peter Shor to factor large integers on quantum computers.Specifically, the principle and design of quantum Fourier transform are explained.We stress the subtle distinction between QFT and its inverse.Since former articles did not emphasize it, we hope it could be a supplement to former articles.Our next work is to verify the existing articles on executing Shor's algorithm by conducting two experiments on IBMQ of factoring N = 15 in two ways (a = 7 and a = 11).We find that the effect of quantum entanglement might be crucial to the speed boost of factoring large integers in Shor's algorithm.

Shinil Cho

Fourier transform has been applied to applications of quantum computation. In this chapter, we explain how quantum Fourier transform (QFT) is defined to apply to quantum computation problems such as Shor’s algorithm and quantum walks.

Weng-Long Chang, Athanasios V. Vasilakos

Eduardo Bayro‐Corrachono, Zuleima Vazquez‐Flores

There is a growing interest in quantum image processing (QIP) that rose from the desire to exploit the properties of quantum computing to improve the performance of classical techniques and their applications. Since the introduction of a quaternion by Hamilton in 1843, quaternions had been used in a lot of applications. We show how the quantum states and quantum operators work in the field of quaternions and how we can use it to extend the quantum complex Fourier transform (QCFT) to the quantum quaternion Fourier transform (QQFT). We propose for the first time the novel algorithm for the QQFT. We explain the impossibility of the use of convolution in quantum image processing; we design filters in the quaternion Fourier spectrum and the space domain an edge detector and a quantum median filter.

Ivan V. Murashko, Constantine C. Korikov