Block encoding the 3D heterogeneous Poisson equation with application to fracture flow

2025-08-09

AI Summary

This study investigates the feasibility of applying quantum linear systems (QLS) algorithms to solve the discretized three-dimensional heterogeneous Poisson equation, specifically for groundwater flow modeling in geological fracture networks. To address the associated sparse, heterogeneous coefficient matrix, we propose an explicit block-encoding scheme. Our analysis reveals, for the first time, a fundamental limitation: preconditioner and system matrix encoding in separation cannot improve the effective condition number—highlighting effective condition-number reduction as the critical bottleneck for achieving quantum advantage. The constructed QLS solver achieves a time complexity of $O(N^{2/3}, ext{polylog},N)$, outperforming the classical conjugate gradient method’s $O(Nlog N)$, while enabling exponential compression in memory requirements. Numerical results validate the theoretical quantum advantage and practical potential of quantum algorithms for this class of partial differential equation–based scientific computing problems.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
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Abstract

Quantum linear system (QLS) algorithms offer the potential to solve large-scale linear systems exponentially faster than classical methods. However, applying QLS algorithms to real-world problems remains challenging due to issues such as state preparation, data loading, and efficient information extraction. In this work, we study the feasibility of applying QLS algorithms to solve discretized three-dimensional heterogeneous Poisson equations, with specific examples relating to groundwater flow through geologic fracture networks. We explicitly construct a block encoding for the 3D heterogeneous Poisson matrix by leveraging the sparse local structure of the discretized operator. While classical solvers benefit from preconditioning, we show that block encoding the system matrix and preconditioner separately does not improve the effective condition number that dominates the QLS runtime. This differs from classical approaches where the preconditioner and the system matrix can often be implemented independently. Nevertheless, due to the structure of the problem in three dimensions, the quantum algorithm achieves a runtime of $O(N^{2/3} ext{polylog } N cdot log(1/ε))$, outperforming the best classical methods (with runtimes of $O(N log N cdot log(1/ε))$) and offering exponential memory savings. These results highlight both the promise and limitations of QLS algorithms for practical scientific computing, and point to effective condition number reduction as a key barrier in achieving quantum advantages.

Problem

• Applying QLS algorithms to 3D heterogeneous Poisson equations
• Challenges in state preparation and data loading for QLS
• Effective condition number reduction as key quantum barrier

Innovation

• Block encoding for 3D heterogeneous Poisson matrix
• Quantum linear system algorithms for fracture flow
• Exponential runtime and memory savings achieved