Quantum Weighted Fractional-Order Transform

Abstract

Quantum Fourier transform (QFT) transformation plays a very important role in the design of many quantum algorithms. Fractional Fourier transform (FRFT), as an extension of the Fourier transform, is particularly important due to the design of its quantum algorithm. In this paper, a new reformulation of the weighted fractional Fourier transform (WFRFT) is proposed in order to realize quantum FRFT; however, we found that this reformulation can be applied to other transformations, and therefore, this paper presents the weighted fractional Hartley transform (WFRHT). For the universality of application, we further propose a general weighted fractional-order transform (WFRT). When designing the quantum circuits, we realized the quantum WFRFT via QFT and quantum phase estimation (QPE). Moreover, after extending our design to the WFRHT, we were able to formulate the quantum WFRHT. Finally, in accordance with the research results, we designed the quantum circuit of the general WFRT, and subsequently proposed the quantum WFRT. The research in this paper has great value as a reference for the design and application of quantum algorithms.

Sliding-Window TD-FrFT Algorithm for High-Precision Ranging of LFM Signals in the Presence of Impulse Noise

“…The echo and transmitted signals are separately transformed using nonlinear FrFTs, and the delay in the LFM signals is calculated based on the peak position offset in the fractional domain. Other researchers [27,28] have introduced a quantum weighted FrFT for application to quantum singular-value decomposition problems and quantum gradient-solving problems. Moreover, a novel random FrFT has been developed [29], inheriting excellent mathematical properties from the FrFT, which can be directly used in optical image encryption and decryption.…”