  The Poisson equation has wide applications in many areas of science and
engineering. Although there are some quantum algorithms that can efficiently
solve the Poisson equation, they generally require a fault-tolerant quantum
computer which is beyond the current technology. In this paper, we propose a
Variational Quantum Algorithm (VQA) to solve the Poisson equation, which can be
executed on Noise Intermediate-Scale Quantum (NISQ) devices. In detail, we
first adopt the finite difference method to transform the Poisson equation into
a linear system. Then, according to the special structure of the linear system,
we find an explicit tensor product decomposition, with only $2\log n+1$ items,
of its coefficient matrix under a specific set of simple operators, where $n$
is the dimension of the coefficient matrix. This implies that the proposed VQA
only needs $O(\log n)$ measurements, which dramatically reduce quantum
resources. Additionally, we perform quantum Bell measurements to efficiently
evaluate the expectation values of simple operators. Numerical experiments
demonstrate that our algorithm can effectively solve the Poisson equation.
