  Solving differential equations is one of the most compelling applications of
quantum computing. Most existing quantum algorithms addressing general ordinary
and partial differential equations are thought to be too expensive to execute
successfully on Noisy Intermediate-Scale Quantum (NISQ) devices. Here we
propose a compact quantum algorithm for solving one-dimensional Poisson
equation based on simple Ry rotation. The major operations are performed on
probability amplitudes. Therefore, the present algorithm avoids the need to do
phase estimation, Hamiltonian simulation and arithmetic. The solution error
comes only from the finite difference approximation of the Poisson equation.
Our quantum Poisson solver (QPS) has gate-complexity of 3n in qubits and 4n^3
in one- and two-qubit gates, where n is the logarithmic of the dimension of the
linear system of equations. In terms of solution error {\epsilon}, the
complexity is log(1/{\epsilon}) in qubits and poly(log(1/{\epsilon})) in
operations, which is consist with the best known results. The present QPS may
represent a potential application on NISQ devices.
