Solving large systems of equations is a challenge for modeling natural phenomena, such as simulating subsurface flow. To avoid systems that are intractable on current computers, it is often necessary to neglect information at small scales, an approach known as coarse-graining. For many practical applications, such as flow in porous, homogenous materials, coarse-graining offers a sufficiently-accurate approximation of the solution. Unfortunately, fractured systems cannot be accurately coarse-grained, as critical network topology exists at the smallest scales, including topology that can push the network across a percolation threshold. Therefore, new techniques are necessary to accurately model important fracture systems. Quantum algorithms for solving linear systems offer a theoretically-exponential improvement over their classical counterparts, and in this work we introduce two quantum algorithms for fractured flow. The first algorithm, designed for future quantum computers which operate without error, has enormous potential, but we demonstrate that current hardware is too noisy for adequate performance. The second algorithm, designed to be noise resilient, already performs well for problems of small to medium size (order 10-1000 nodes), which we demonstrate experimentally and explain theoretically. We expect further improvements by leveraging quantum error mitigation and preconditioning.
Solving large systems of equations is a challenge for modeling natural phenomena, such as simulating subsurface flow. To avoid systems that are intractable on current computers, it is often necessary to neglect information at small scales, an approach known as coarse-graining. For many practical applications, such as flow in porous, homogenous materials, coarse-graining offers a sufficiently-accurate approximation of the solution. Unfortunately, fractured systems cannot be accurately coarse-grained, as critical network topology exists at the smallest scales, including topology that can push the network across a percolation threshold. Therefore, new techniques are necessary to accurately model important fracture systems. Quantum algorithms for solving linear systems offer a theoretically-exponential improvement over their classical counterparts, and in this work we introduce two quantum algorithms for fractured flow. The first algorithm, designed for future quantum computers which operate without error, has enormous potential, but we demonstrate that current hardware is too noisy for adequate performance. The second algorithm, designed to be noise resilient, already performs well for problems of small to medium size (order 10-1000 nodes), which we demonstrate experimentally and explain theoretically. We expect further improvements by leveraging quantum error mitigation and preconditioning.
Abstract Uncertainty quantification plays a crucial role in the modeling of subsurface flow. For instance, uncertainties in the properties of geologic fracture networks significantly impact flow, requiring numerous simulations to accurately estimate quantities of interest. However, each simulation is computationally expensive because it requires solving a large linear system to capture features that involve both small and large fractures. An example is in percolation, where the interaction of many small fractures (which cumulatively can have a large surface area) with the rock matrix must be modeled precisely. Quantum computing is an emerging tool with the potential to address this issue. Quantum algorithms offer a significant speedup in solving linear systems, achieving efficiencies that are challenging to match with classical approaches. These classical approaches include direct solvers, such as LU decomposition, and iterative methods, notably preconditioned conjugate gradient, commonly used in subsurface modeling to solve large sparse systems. However, applying quantum algorithms to geologic fracture flow requires careful attention to algorithmic and problem‐specific constraints to fully realize this quantum advantage. In this work we describe a quantum algorithm for generalized Monte Carlo applications with a quadratic speedup over the classical approaches which can be combined with the quantum speedup, currently under investigation, for solving quantum linear systems for subsurface flow. We show that for quantum algorithms the computational cost of estimating a quantity of interest for a statistical ensemble of networks is roughly the same as that of a single realization, essentially implying that one can get uncertainty quantification for free.
Accurate modelling of fluid flow in porous media is crucial for applications including hydrology, geological CO₂ storage, hydrogen storage, geothermal energy, and hydrocarbon recovery. The inherent heterogeneity of subsurface formations, characterized by spatial permeability contrasts, fractures, and vugs, leads to large and often ill-conditioned linear systems when discretized at high resolution. Although upscaling techniques reduce computational cost, they may obscure fine-scale features that significantly influence flow behavior, particularly in the presence of small but highly conductive fractures. This study investigates the Variational Quantum Linear Solver (VQLS) as an alternative framework for solving discretized Darcy flow equations in one-, two-, and three-dimensional (1D, 2D, and 3D) homogeneous and heterogeneous systems. VQLS approximates the solution to linear systems of the form Ax = b by preparing a quantum state proportional to the classical solution vector. The framework incorporates a hybrid quantum-classical cost formulation, which strengthens optimization in ill-conditioned systems, and a curvature-aware Adaptive Dynamic Ansatz, which enables principled circuit depth selection without manual tuning. Five experimental modes are evaluated, including fixed-depth training, hybrid cost formulation, adaptive ansatz growth, depth sweep analysis, and hybrid weight sweep. All experiments are conducted using noiseless statevector simulation. The results examine convergence behavior, solution accuracy, and algorithmic limitations in heterogeneous media, providing insight into the practical potential and current constraints of variational quantum solvers for porous media flow modelling.
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 [Formula: see text] 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.
Quantum algorithms provide an exponential speedup for solving certain classes of linear systems, including those that model geologic fracture flow. However, this revolutionary gain in efficiency does not come without difficulty. Quantum algorithms require that problems satisfy not only algorithm-specific constraints, but also application-specific ones. Otherwise, the quantum advantage carefully attained through algorithmic ingenuity can be entirely negated. Previous work addressing quantum algorithms for geologic fracture flow has illustrated core algorithmic approaches while incrementally removing assumptions. This work addresses two further requirements for solving geologic fracture flow systems with quantum algorithms: efficient system state preparation and efficient information extraction. Our approach to addressing each is consistent with an overall exponential speed-up.
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 (3D) 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 run-time. 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 run-time of O ( N 2 / 3 polylog N ⋅ log ( 1 / ϵ ) ) , outperforming the best classical methods (with run times of O ( N log N ⋅ 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.
Quantum computing not only holds the potential to solve long-standing problems in quantum physics, but also to offer speed-ups across a broad spectrum of other fields. Access to a computational space that incorporates quantum effects, such as superposition and entanglement, enables the derivation of promising quantum algorithms for important tasks, including preparing the ground state of a quantum system or predicting its evolution over time. Successfully tackling these tasks promises insights into significant theoretical and technological questions, such as superconductivity and the design of new materials. The aim of quantum algorithms is to use a series of quantum operations in a quantum circuit to solve a problem beyond the reach of classical computers. However, the noise and limited scale of current quantum computers restricts these circuits to moderate sizes and depths. As a result, many prominent algorithms are currently infeasible to run for problem sizes of practical interest. In response, recent research focused on variational quantum algorithms, which allow the selection of circuits that act within a quantum device's capabilities. Yet, these algorithms can require the execution of a large number of circuits, leading to prohibitively long computation times. This doctoral thesis develops two main techniques to reduce these quantum computational resource requirements, with the goal of scaling up application sizes on current quantum processors. The first approach is based on stochastic approximations of computationally costly quantities, such as quantum circuit gradients or the quantum geometric tensor (QGT). The second method takes a different perspective on the QGT, leading to a potentially more efficient description of time evolution on current quantum computers. Both techniques rely on maintaining available information and only computing necessary corrections, instead of re-computing possibly redundant data. The main focus of application for our algorithms is the simulation of quantum systems, broadly defined as including the preparation of ground and thermal states, and the real- and imaginary-time propagation of a system. The developed subroutines, however, can further be utilized in the fields of optimization or machine learning. Our algorithms are benchmarked on a range of representative models, such as Ising or Heisenberg spin models, both in numerical simulations and experiments on the hardware. In combination with error mitigation techniques, the latter is scaled up to 27 qubits; into a regime that variational quantum algorithms are challenging to scale to on noisy quantum computers without our algorithms.