Quantum algorithms for geologic fracture networks

Abstract

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.

Figure 1

Schematic workflow for applying classical and quantum algorithms to fracture flow problems. Discretizing fracture systems on classical computers involves reducing the computational cost by truncating the fracture network to exclude small fractures. This setting-aside of information provides a solution that does not accurately reflect all flow. Conversely, quantum computing has the potential to solve large, complete fracture systems given properties of quantum mechanics and algorithms designed to take advantage of those properties.

Figure 2

Solving 4-node and 8-node fracture systems using the SSO algorithm. Subfigure (a) presents results for a one-dimensional grid of 4 nodes, while subfigure (b) does the same for a grid of 8 nodes. Each subfigure presents the maximum, minimum, and average error in 75 runs on either IBM’s quantum simulator (solid blue line) or the ibmq_rome quantum computer (dashed red line). The error is plotted against the number of iterations, q, used. The figures also include a dotted gray line illustrating the error that would be achieved if the returned ‘solution’ from the quantum computer was the result of the maximally-mixed state. As described in the main text, this serves as a benchmark for assessing the quality of the solution SSO achieved. Subfigures (c,d) present cartoons of the problems solved: each color indicates a unique node pressure, with fixed pressures of 1 and 0 on the boundaries.

Figure 3

Solving a $6\times 8$ pitchfork fracture problem using a quantum computer. Subfigures (a) and (b) illustrate the cost and fidelity per iteration for forty sets of randomly-initialized parameters; the result with the highest fidelity is highlighted. Subfigure (c) illustrates the normalized, known, classically computed solution with overlaid permeabilities. The inner $4\times 8$ nodes are the sought-after pressure values because the top and bottom rows have fixed boundary pressures. The maroon dots illustrate low permeabilities, and the connected green squares illustrate the fracture. Subfigure (d) is the solution from quantum hardware (specifically, qubits 0, 1, 4, 7, and 10 of the ibmq_mumbai machine). This solution has fidelity 0.9911, to four figures.

Figure 4

Solving a series of larger fracture flow problems. Subfigures (a) through (d) are results achieved by running the VLS-trained circuits on the ibmq_montreal quantum computer for 128-, 512-, 2048-, and 8192-node regions. (These correspond to 7-, 9-, 11-, and 13-qubit problems, respectively.) Associated fidelities are 0.9628, 0.9353, 0.9076, and 0.8834. Subfigure (e) plots fidelities from the same series of problems (in addition to a 5-qubit problem) alongside the fidelity that would have been achieved had the quantum computer’s prepared solution degraded to the maximally-mixed state. The latter illustrates the fidelity from a result comprised solely of random ‘noise’, thus demonstrating how much better the achieved result on quantum hardware is and experimentally illustrating the noise resilience that is partially proved in Sec. XA, online.

Figure 5

Variational linear solver algorithm. As described in Fig. 1, the algorithm accepts as input an A and $\mathbf{b}$ specifying a LSP and minimizes a cost function to create a parameterized quantum circuit that solves the LSP. Specifically, the cost function optimization takes advantage of classical optimization techniques, while the cost function evaluation occurs via either quantum hardware or a classical simulator of such hardware.

Figure 6

Pitchfork and surface discretization. A cartoon of the subsurface flow situation, in which a region is discretized into $d_1\times d_2$ nodes. Pressure boundary conditions of 1 and 0 are imposed on the left- and right-hand sides of the region, respectively, and a pitchfork fracture runs through the middle of the region. Each node has a pressure that we seek as our solution.

Figure 7

Five-qubit, two-layer ansatz. The pattern for this ansatz is a ‘preliminary’ layer of $R_y$ gates followed by layers with two sets of controlled-Z and $R_y$ gates each. Each $R_y$ gate’s angle is one of the tuned parameters.

Figure 8

The qubit connectivity of IBM’s Montreal quantum computer.