Real and Fourier space readout methods: Comparison of complexity and applications to CFD problems

TL;DR

This paper compares real and Fourier space quantum readout methods for solving PDEs, demonstrating their efficiency and potential advantages in CFD applications and quantum computing contexts.

Contribution

It introduces and evaluates new approximate real space readout methods and compares them with existing Fourier space and amplitude estimation techniques.

Findings

Fourier space readout (FSR) and ARSR are most practical for continuous functions.

QAE-based methods are suitable for future quantum devices.

Significant improvements in CFD solutions using these readout methods.

Abstract

Quantum computing is a promising technology that accelerates the partial differential equations solver for practical problems. The reconstruction of solutions (i.e., the readout of quantum states) remains a crucial problem, although numerous efficient quantum algorithms have been proposed. In this paper, we propose and compare several efficient readout methods in the real and the Fourier space. The Fourier space readout (FSR) and the proposed approximate real space readout (ARSR) methods are currently the most efficient and practical ones for the purpose of reconstructing continuous real-valued functions. In contrast, the quantum amplitude estimation (QAE) based methods (especially in the Fourier space) are favorable for mid-term/far-term quantum devices. Besides, we apply the methods for benchmark solutions in computational fluid dynamics (CFD) and demonstrate great improvements compared to the conventional sampling method for large grid numbers. Equipped with efficient readout methods, we further show that a 2D Burgers' equation can be solved efficiently without using the expensive strategy of linearization. It suggests the potential quantum advantages for some practical applications on mid-term quantum devices.