  The discretized Poisson equation matrix (DPEM) is vital to the field of
computational fluid dynamics (CFD), and an algorithm that solves it on a
quantum computer could potentially grant exponential space and time complexity
speedups. However, the DPEM in 1D has been shown to have trouble being
represented as a decomposition in the Pauli basis. Additionally, traditional
ansatz for Variational Quantum Algorithms (VQAs) that are used to heuristically
solve linear systems (such as the DPEM) have large numbers of parameters,
making them harder to train. This research attempts to resolve these problems
by utilizing the IonQ Aria quantum computer capabilities that boast all-to-all
connectivity of qubits. We propose a decomposition of the DPEM that is based on
2- or 3-qubit entanglement gates and is shown to have $O(1)$ terms with respect
to system size, with one term having an $O(n^2)$ circuit depth and the rest
having only an $O(1)$ circuit depth (where $n$ is the number of qubits defining
the system size). To test these new improvements, we ran numerical simulations
to examine how well the VQAs performed with varying system sizes, showing that
the new setup offers an $O(n)$ scaling of the number of iterations required for
convergence, providing an exponential speedup over their classical computing
counterparts. This project shows that the future of computational fluid
dynamics may involve quantum computers to provide significant time and space
complexity speedups.
