  It has been shown that some non-Gaussian states are useful in quantum
metrology. In this paper, we consider an estimation problem which is not well
studied in previous researches-an estimation of displacement that follows a
Gaussian distribution when post-selection of the measurement outcome is
allowed. We derive a lower bound for the estimation error when only Gaussian
states and Gaussian operations are used. We show that this bound can be beaten
using only linear optics and simple non-Gaussian states such as single photon
states. We also obtain a lower bound for the estimation error when the maximum
photon number of the sensor state is given, using a generalized version of Van
Trees inequality. Our result further reveals the limit of methods using
Gaussian states and the role of non-Gaussianity in quantum metrology.
