Fourier Transforms and Quantum Computation

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.

S. S. Zhou, T. Loke, Josh Izaac +1 more

Weng-Long Chang, Athanasios V. Vasilakos

Vinayak V. Dixit, Sisi Jian

Drive cycles in vehicle systems are important determinants for energy consumption, emissions, and safety. Estimating the frequency of the drive cycle quickly is important for control applications related to fuel efficiency, emission reduction and improving safety. Quantum computing has established the computational efficiency that can be gained. A drive cycle frequency estimation algorithm based on the quantum Fourier transform is exponentially faster than the classical Fourier transform. The algorithm is applied on real world data set. We evaluate the method using a quantum computing simulator, demonstrating remarkable consistency with the results from the classical Fourier transform. Current quantum computers are noisy, a simple method is proposed to mitigate the impact of the noise. The method is evaluated on a 15 qubit IBM-q quantum computer. The proposed method for a noisy quantum computer is still faster than the classical Fourier transform.

Chien‐Cheng Tseng, Su‐Ling Lee

The graph Fourier transform (GFT) is a vital tool for processing irregular data collected across diverse structured systems, including sensor, social, transportation, and brain networks. Given its wide utility in graph signal applications like compression, prediction, classification, denoising, detection, and estimation, developing efficient methods for computing the GFT over graphs and networks is a crucial task. This paper presents a quantum computation method for calculating the GFT of a graph signal, executed through five main steps: (1) The graph signal is first normalized to a unit norm during preprocessing. (2) A quantum state preparation circuit loads this unit-norm data into a quantum register, generating a superposition state. (3) The quantum circuit representing the unitary GFT matrix is obtained using the IBM Qiskit transpiler, which is then applied to transform the vertex-domain data into the spectral domain. (4) Repeated quantum measurements are performed to generate a histogram that encodes the spectral-domain data. (5) Finally, the histogram's probability data is scaled by the original input signal's norm to yield the complete GFT result. The effectiveness of this proposed quantum method is demonstrated through the GFT computation of temperature data from a sensor network.