  Estimating the eigenstate properties of quantum many-body systems is a
long-standing, challenging problem for both classical and quantum computing.
For the task of eigenstate preparation, quantum signal processing (QSP) has
established near-optimal query complexity $O( \Delta^{-1} \log(\epsilon^{-1})
)$ by querying the block encoding of the Hamiltonian $H$ where $\Delta$ is the
energy gap and $\epsilon$ is the target precision. However, QSP is challenging
for both near-term noisy quantum computers and early fault-tolerant quantum
computers (FTQC), which are limited by the number of logical qubits and circuit
depth. To date, early FTQC algorithms have focused on querying the perfect time
evolution $e^{-iHt}$. It remains uncertain whether early FTQC algorithms can
maintain good asymptotic scaling at the gate level. Moreover, when considering
qubit connectivity, the circuit depth of existing FTQC algorithms may scale
suboptimally with system size. Here, we present a full-stack design of a random
sampling algorithm for estimating the eigenenergy and the observable
expectations on the eigenstates, which can achieve high precision and good
system size scaling. The gate complexity has a logarithmic dependence on
precision $ {O}(\log^{1+o(1)} (1/\epsilon))$ for generic Hamiltonians, which
cannot achieved by methods using Trottersiation to realise $e^{-iHt}$ like in
QETU. For $n$-qubit lattice Hamiltonians, our method achieves near-optimal
system size dependence with the gate complexity $O(n^{1+o(1)})$. When
restricting the qubit connectivity to a linear nearest-neighbour architecture,
The method shows advantages in circuit depth, with $O(n^{o(1)})$ for lattice
models and $O(n^{2+o(1)})$ for electronic structure problems. We compare the
resource requirements (CNOT gates, T gates and qubit numbers) by phase
estimation, QSP, and QETU, in lattice and molecular problems.
