  Dynamical response functions are fundamental quantities to describe the
excited-state properties in quantum many-body systems. Quantum algorithms have
been proposed to evaluate these quantities by means of quantum phase estimation
(QPE), where the energy spectra are directly extracted from the QPE measurement
outcomes in the frequency domain. Accurate estimation of excitation energies
and transition probabilities with these QPE-based approaches is, however,
challenging because of the problem of spectral leakage (or peak broadening)
which is inherent in the QPE algorithm. To overcome this issue, in this work we
consider an extension of the QPE-based approach adopting the optimal entangled
input states, which is known to achieve the Heisenberg-limited scaling for the
estimation precision. We show that with this method the peaks in the calculated
energy spectra are more localized than those calculated by the original
QPE-based approaches, suggesting the mitigation of the spectral leakage
problem. By analyzing the probability distribution with the entangled phase
estimation, we propose a simple scheme to better estimate both the transition
energies and the corresponding transition probabilities of the peaks of
interest in the spectra. The validity of our prescription is demonstrated by
numerical simulations in various quantum many-body problems: the spectral
function of a simple electron-plasmon model in condensed-matter physics, the
dipole transitions of the H$_2$O molecule in quantum chemistry, and the
electromagnetic transitions of the $^6$Li nucleus in nuclear physics.
