  The preparation of the ground state of a Hamiltonian $H$ with a large
spectral radius has applications in many areas such as electronic structure
theory and quantum field theory. Given an initial state with a constant overlap
with the ground state, and assuming that the Hamiltonian $H$ can be efficiently
simulated with an ideal fast-forwarding protocol, we first demonstrate that
employing a linear combination of unitaries (LCU) approach can prepare the
ground state at a cost of $\mathcal{O}(\log^2(\|H\| \Delta^{-1}))$ queries to
controlled Hamiltonian evolution. Here $\|H\|$ is the spectral radius of $H$
and $\Delta$ the spectral gap. However, traditional Quantum Signal Processing
(QSP)-based methods fail to capitalize on this efficient protocol, and its cost
scales as $\mathcal{O}(\|H\| \Delta^{-1})$. To bridge this gap, we develop a
multi-level QSP-based algorithm that exploits the fast-forwarding feature. This
novel algorithm not only matches the efficiency of the LCU approach when an
ideal fast-forwarding protocol is available, but also exceeds it with a reduced
cost that scales as $\mathcal{O}(\log(\|H\| \Delta^{-1}))$. Additionally, our
multi-level QSP method requires only $\mathcal{O}(\log(\|H\| \Delta^{-1}))$
coefficients for implementing single qubit rotations. This eliminates the need
for constructing the PREPARE oracle in LCU, which prepares a state encoding
$\mathcal{O}(\|H\| \Delta^{-1})$ coefficients regardless of whether the
Hamiltonian can be fast-forwarded.
