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

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 $O(N^{2/3} \ \text{polylog } N \cdot \log(1/ε))$, 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.

Figures (8)

• Figure 1: Left: A 3D spatial grid representing a discretized domain of interest. Right: a visualization of a discretized heterogeneous elliptic PDE defined on the same 3D grid, where the cells are separated for ease of viewing. Each cell is given a spatially varying coefficient $\mathrm{k}_{i,j,k}$ (e.g., diffusivity, conductivity, or permeability) and a source term $f_{i,j,k}$. The goal is to determine the field $h_{i,j,k}$ that governs the PDE at each grid point.
• Figure 2: This figure shows the empirical scaling of the condition number (equivalent to the effective condition number of the system prior to preconditioning) for solving the groundwater flow equation. The simulation uses the 3D pitchfork fracture network described in Fig. \ref{['fig:3D_pitchfork']}. As the dimension $N$ increases, the condition number scales as $\kappa = O(N^{2/3})$, consistent with the theoretical analysis in the beginning of Sec. \ref{['sec:condition_number']}.
• Figure 3: This is an extension of the 2D pitchfork fracture pattern introduced in Golden_2022 to 3D. The 2D pitchfork fracture is started in opposite directions of the XY-plane, where each fracture scale is assigned a depth based on its length (the smaller the fracture scale, the smaller the depth). The different colors represent different cell permeability values, and in this particular example, we have $5$ fracture scales.
• Figure 4: Two dimensional cross sections of different fractal models of fine-grained fracture structure with simple arithmetic formulas. These are examples of fracture structures that could lead to efficient block-encodings for the groundwater flow matrix $G'$.
• Figure 5: For the 3D pitchfork pattern shown in Fig. \ref{['fig:3D_pitchfork']}, we observe that the number of distinct elements in the groundwater flow matrix $G'$ grows roughly linearly with the number of fracture scales $F$. This suggests that the theoretical bound of $O(F^{12})$ from eq. \ref{['eq:distinct_elements_vs_F']} may be loose for some practical examples of 3D heterogeneous Poisson problems.