  We propose quantum algorithms for projective ground-state preparation and
calculations of the many-body Green's functions directly in frequency domain.
The algorithms are based on the linear combination of unitary (LCU) operations
and essentially only use quantum resources. To prepare the ground state, we
construct the operator ${\exp}(-\tau \hat{H}^2)$ using Hubbard-Stratonovich
transformation by LCU and apply it on an easy-to-prepare initial state. Our
projective state preparation procedure saturates the near-optimal scaling,
$\mathcal{O}(\frac{1}{\gamma\Delta} \log \frac{1}{\gamma\eta})$, of other
currently known algorithms, in terms of the spectral-gap lower bound $\Delta$,
the additive error $\eta$ in the state vector, and the overlap lower bound
$\gamma$ between the initial state and the exact ground state. It is
straightforward to combine our algorithm with spectral-gap amplification
technique to achieve quadratically improved scaling
$\mathcal{O}(1/\sqrt{\Delta})$ for ground-state preparation of frustration-free
Hamiltonians, which we demonstrate with numerical results of the $q$-deformed
XXZ chain. To compute the Green's functions, including the single-particle and
other response functions, we act on the prepared ground state with the retarded
resolvent operator $R(\omega + i\Gamma; \hat{H})$ in the LCU form derived from
the Fourier-Laplace integral transform (FIT). Our resolvent algorithm has
$\mathcal{O}(\frac{1}{\Gamma^2} \log\frac{1}{\Gamma\epsilon})$ complexity
scaling for the frequency resolution $\Gamma$ of the response functions and the
targeted error $\epsilon$, while classical algorithms for FIT usually have
polynomial scaling over the error $\epsilon$. To illustrate the complexity
scaling of our algorithms, we provide numerical results for their application
to the paradigmatic Fermi-Hubbard model on a one-dimensional lattice with
different numbers of sites.
