Hardware efficient decomposition of the Laplace operator and its application to the Helmholtz and the Poisson equation on quantum computer

With the rapid advancement of quantum computers in the past few years, there is ongoing development of algorithms aimed at solving problems that are difficult to tackle with classical computers. A pertinent instance of this is the resolution of partial differential equations (PDEs), where a current trend involves the exploration of variational quantum algorithms (VQAs) tailored to efficiently function on the noisy intermediate-scale quantum (NISQ) devices. Recently, VQAs for solving the Poisson equation have been proposed, and these algorithms require highly entangled quantum states or specific types of qubit entanglement to compute the expectation value of the Laplace operator. Implementing such requirements on NISQ devices poses a significant challenge. To overcome this problem, we propose a new method for representing the Laplace operator in the finite difference formulation. Since the quantum circuits introduced for evaluating the expectation value of the Laplace operator through proposed method do not require processes that degrade the fidelity of computation, such as swap operations or generation of highly entangled states, they can be easily implemented on NISQ devices. In the regime of quantum supremacy (the number of qubits is approximately 50), our proposed approach necessitates approximately one-third fewer CNOT operations compared to conventional methods. To assess the effectiveness of the proposed method, we conduct computations for finding the eigenvalues of the Helmholtz equation and solving the Poisson equation on cloud-based quantum hardware. We calculate the fidelity of quantum states required for each method through quantum tomography and also estimate the fidelity in the quantum supremacy regime. We believe that the proposed method can be applied to other PDEs having the Laplace operator and greatly assists in solving them.