  Modeling hydrological fracture networks is a hallmark challenge in
computational earth sciences. Accurately predicting critical features of
fracture systems, e.g. percolation, can require solving large linear systems
far beyond current or future high performance capabilities. Quantum computers
can theoretically bypass the memory and speed constraints faced by classical
approaches, however several technical issues must first be addressed. Chief
amongst these difficulties is that such systems are often ill-conditioned, i.e.
small changes in the system can produce large changes in the solution, which
can slow down the performance of linear solving algorithms. We test several
existing quantum techniques to improve the condition number, but find they are
insufficient. We then introduce the inverse Laplacian preconditioner, which
improves the scaling of the condition number of the system from $O(N)$ to
$O(\sqrt{N})$ and admits a quantum implementation. These results are a critical
first step in developing a quantum solver for fracture systems, both advancing
the state of hydrological modeling and providing a novel real-world application
for quantum linear systems algorithms.
