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

TL;DR

This paper presents a novel entanglement-based decomposition for the discretized Poisson equation matrix in quantum algorithms, significantly reducing complexity and improving convergence in variational quantum algorithms on all-to-all connectivity quantum hardware.

Contribution

It introduces a new decomposition method using 2- or 3-qubit entanglement gates and a globally-entangling ansatz, enhancing efficiency and scalability of VQAs for solving linear systems.

Findings

Reduced number of terms in matrix decomposition to O(1) with respect to system size.

Improved convergence scaling in VQAs with the new ansatz.

Demonstrated effectiveness through numerical simulations on quantum hardware.

Abstract

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.