  Under suitable assumptions, the quantum phase estimation (QPE) algorithm is
able to achieve Heisenberg-limited precision scaling in estimating the ground
state energy. However, QPE requires a large number of ancilla qubits and large
circuit depth, as well as the ability to perform inverse quantum Fourier
transform, making it expensive to implement on an early fault-tolerant quantum
computer. We propose an alternative method to estimate the ground state energy
of a Hamiltonian with Heisenberg-limited precision scaling, which employs a
simple quantum circuit with one ancilla qubit, and a classical post-processing
procedure. Besides the ground state energy, our algorithm also produces an
approximate cumulative distribution function of the spectral measure, which can
be used to compute other spectral properties of the Hamiltonian.
