  The Poisson equation has many applications across the broad areas of science
and engineering. Most quantum algorithms for the Poisson solver presented so
far, either suffer from lack of accuracy and/or are limited to very small sizes
of the problem, and thus have no practical usage. Here we present an advanced
quantum algorithm for solving the Poisson equation with high accuracy and
dynamically tunable problem size. After converting the Poisson equation to the
linear systems through the finite difference method, we adopt the
Harrow-Hassidim-Lloyd (HHL) algorithm as the basic framework. Particularly, in
this work we present an advanced circuit that ensures the accuracy of the
solution by implementing non-truncated eigenvalues through eigenvalue
amplification as well as by increasing the accuracy of the controlled rotation
angular coefficients, which are the critical factors in the HHL algorithm. We
show that our algorithm not only increases the accuracy of the solutions, but
also composes more practical and scalable circuits by dynamically controlling
problem size in the NISQ devices. We present both simulated and experimental
solutions, and conclude that overall results on the quantum hardware are
dominated by the error in the CNOT gates.
