Addressing quantum's "fine print" with efficient state preparation and information extraction for quantum algorithms and geologic fracture networks

Abstract

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.

Figure 1

A block diagram of HHL. After a separate, non-HHL algorithm has prepared the right-hand-side vector, $\mathbf{b}$, HHL applies QPE to extract the eigenvalues and eigenvectors from the matrix, A. Then, the algorithm prepares the eigenvalues for extraction from their entangled state using inverse QPE before measuring an ancilla qubit to determine if the operation was successful. A measurement of 1 indicates that $|x⟩\sim \mathbf{x}$ is stored in the b-register, while a value of 0 indicates a probabilistic circuit failure, the likelihood of which is included in the complexity of HHL via $ϵ$.

Figure 2

A schematic illustrating how information is transferred from a geologic fracture flow problem into an HHL circuit that solves for the pressure at each discretized node.

Figure 3

Subfigure (a) illustrates the swap test for two n-qubit registers, $|\varphi ⟩$ and $|\psi ⟩$, while Subfigure (b) illustrates the Hadamard test for two n-qubit states that can be efficiently prepared via the gates $U_{\varphi }$ and $U_{\psi }$.

Figure 4

A quantum circuit that prepares the state of $\mathbf{b}$ for a fracture flow problem with a pressure gradient where $n_b=8$. Note that $\mathbf{b}$ has $2^{n_b/2}$ nonzero entries.

Figure 5

On the left, a simplified two-dimensional fracture network model with two fractures intersecting and two random fluid injection/extraction wells at the red dots. The color bar represents permeability on a logarithmic scale. $\mathbf{b}$ encodes boundary conditions for a $4\times 4$ grid with $2^{n_b}=16$ cells $\Rightarrow n_b=4$. Nonzero entries in $\mathbf{b}$ correspond to well sites. On the right, a quantum circuit to prepare the state $|b⟩=\frac{1}{\sqrt{39665}}(-164|0001⟩+113|1100⟩)$ for this fracture flow problem with Neumann boundary conditions (sparse nonzero entries in $\mathbf{b}$).

Figure 6

An illustration of a two-dimensional fracture network model with fractures that intersect in a pattern of fractal-style recursion to generate a pitchfork fracture network. Random fluid injection/extraction wells are located at the red dots. The color bar represents permeability on a logarithmic scale. $\mathbf{b}$ encodes Neumann boundary conditions (sparse nonzero entries) for a $64\times 64$ grid with $2^{n_b}=4096$ cells $\Rightarrow n_b=12$. Nonzero entries in $\mathbf{b}$ correspond to well sites.

Figure 7

Total gate count (total), CNOT gate count (cx) and T gate count (t) of the quantum circuit preparing the state of a sparse vector $\mathbf{b}$. In our system, the number of qubits is fixed at $n_b=12$ while W, the number of nonzero entries in $\mathbf{b}$ corresponding to randomly-generated injection/extraction sites, increases from 1 to 25. For each value of W, gate counts from 5 random state samples are shown with $+$ symbols, along with their average and bars extending from smallest to largest value. Theoretically, the state preparation method’s total gate count scales as $O(W{n}_b)$, further broken down as $O(W{n}_b)$ CNOT gates and $O(WlogW+n_b)$ single-qubit gates.

Figure 8

A schematic of the circuit structure for extracting average pressure using the swap test. Note that only the magnitude of the average pressure is extracted; sign is not.

Figure 9

A schematic of the circuit structure for extracting average pressure using the Hadamard test. The circuit is more complicated than in Fig. 8, but it allows for extracting both the sign and magnitude of the average pressure.

Figure 10

A schematic $2\times 2$ grid illustrating a discretized region with four nodes, some combination of which holds the average pressure we seek to compute.

Figure 11

Schematic of the HHL and swap test circuits for extracting the average pressure from a full row, column, and diagonal of a discretized two-by-two region.

Figure 12

Subfigure (a) is a $4\times 4$ discretized region with a pitchfork fracture marked by yellow circles. Subfigure (b) illustrates the two circuits needed to compute the average pressure in a series of arbitrarily-selected nodes, by building r-registers as described above.