High-Entanglement Capabilities for Variational Quantum Algorithms: The Poisson Equation Case

The discretized Poisson equation matrix (DPEM) in 1D has been shown to require an exponentially large number of terms when decomposed in the Pauli basis when solving numerical linear algebra problems on a quantum computer. Additionally, traditional ansatz for Variational Quantum Algorithms (VQAs) that are used to heuristically solve linear systems (such as the DPEM) have many 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). Additionally, we introduce the Globally-Entangling Ansatz which reduces the parameter space of the quantum ansatz while maintaining enough expressibility to find the solution. 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 improved scaling of the number of iterations required for convergence compared to Hardware-Efficient Ansatz.