  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 a
linear system through the finite difference method, we adopt the 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. Consequently, we are able to drastically
reduce the relative error in the solution while achieving higher success
probability as the amplification level is increased. 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 NISQ
devices. We present both simulated and experimental results and discuss the
sources of errors. Finally, we conclude that though overall results on the
existing NISQ hardware are dominated by the error in the CNOT gates, this work
opens a path to realizing a multidimensional Poisson solver on near-term
quantum hardware.
