Quantum Computation for End-to-End Seismic Data Processing with Its Computational Advantages and Economic Sustainability
Authors: Kumar Ashutosh 
Published: October 2022

Abstract
Mathematical and computational challenges involved in seismic data processing presents an opportunity for early adoption of quantum computation methods for end-to-end seismic data processing. Existing methods of seismic data processing involve processes with exponential complexities that result in approximations as well as conversion of some of the continuous phenomena into a stochastic one. In the classical computation methods, the mentioned approximations and assumptions enable us to obtain acceptable results in commercially viable time. This paper proposes alternatives of the classical computations that exist in the quantum computation ecosystem along with the computational advantages it holds. The paper also presents potential contributions of the petroleum industry towards sustaining the quantum computation technologies.
Fundamentally seismic data processing involves solutions for systems of linear equations and its derivatives. Quantum computation ecosystem holds efficient solutions for systems of linear equations. In the frequency domain, Finite-Difference modelling reduces seismic-wave equations to systems of linear equations. In the classical computational setup the seismic acquisition involves treatment of the recorded waves as rays and has limited summation provision for recreating the natural reflection or refraction phenomena that is continuous instead of being a stochastic process. The algorithms in the quantum ecosystem allow us to consider summation of signals from all possible paths between the source and the receiver, by amplitude-probability. In addition to the systems of linear equations and their solution with corresponding methods in the quantum ecosystem the fourier transformation and partial differential equations enable us to decompose the waves and apply the physics equation to obtain the desired objective.
Quantum-algorithms facilitate exponential speed-up in seismic data processing. The PDE-constrained optimization inverts subsurface P-wave velocity. While going through the seismic data processing steps it is found that the fourier transformation algorithms are derived as a decomposition of the diagonal matrix. The key difference between the fast fourier transform and the quantum fourier transform is that the quantum fourier transformation is used as the building block of several quantum algorithms. Seismic inversion involves laws of physics and calculation that are guided by the ordinary differential equations. In the quantum computation ecosystem these algorithms for linear ordinary differential equations for linear partial differential equations have the complexity of (1/e), where ‘e’ is the tolerance. The insights brought by successful implementation of end-to-end seismic data processing with algorithms in the quantum computation domain enables us to drill most optimally located wells and hence facilitate cost saving. Even with a reduction of 10% in the total number of wells that we drill, we can possibly fund development of one quantum computer hence ensuring economic sustainability of the technology.
The novelty of the presented paper lies in the comparative analysis of the classical methods with its counterparts in the quantum ecosystem. It explains the technological and economical aspects of the technology such that extensive knowledge of quantum technology is not compulsory for grasping its contents.