  Filter methods realize a projection from a superposed quantum state onto a
target state, which can be efficient if two states have sufficient overlap.
Here we propose a quantum Gaussian filter (QGF) with the filter operator being
a Gaussian function of the system Hamiltonian. A hybrid quantum-classical
algorithm feasible on near-term quantum computers is developed, which
implements the quantum Gaussian filter as a linear combination of Hamiltonian
evolution at various times. Remarkably, the linear combination coefficients are
determined classically and can be optimized in the postprocessing procedure.
Compared to the existing filter algorithms whose coefficients are given in
advance, our method is more flexible in practice under given quantum resources
with the help of postprocessing on classical computers. We demonstrate the
quantum Gaussian filter algorithm for the quantum Ising model with numeral
simulations under noises. We also propose an alternative full quantum approach
that implements a QGF with an ancillary continuous-variable mode.
