  We explore Davidson methods for obtaining excitation energies and other
linear response properties within quantum self-consistent linear response
(q-sc-LR) theory. Davidson-type methods allow for obtaining only a few selected
excitation energies without explicitly constructing the electronic Hessian
since they only require the ability to perform Hessian-vector multiplications.
We apply the Davidson method to calculate the excitation energies of hydrogen
chains (up to H$_{10}$) and analyze aspects of statistical noise for computing
excitation energies on quantum simulators. Additionally, we apply Davidson
methods for computing linear response properties such as static
polarizabilities for H$_2$, LiH, H$_2$O, OH$^-$, and NH$_3$, and show that
unitary coupled cluster outperforms classical projected coupled cluster for
molecular systems with strong correlation. Finally, we formulate the Davidson
method for damped (complex) linear response, with application to the nitrogen
K-edge X-ray absorption of ammonia, and the $C_6$ coefficients of H$_2$, LiH,
H$_2$O, OH$^-$, and NH$_3$.
