We develop a phase estimation method with a distinct feature: its maximal
runtime (which determines the circuit depth) is δ/ϵ, where
ϵ is the target precision, and the preconstant δ can be
arbitrarily close to 0 as the initial state approaches the target eigenstate.
The total cost of the algorithm satisfies the Heisenberg-limited scaling
O(ϵ−1). This is different from all previous
proposals, where δ≳π is required even if the initial state is
an exact eigenstate. As a result, our algorithm may significantly reduce the
circuit depth for performing phase estimation tasks on early fault-tolerant
quantum computers. The key technique is a simple subroutine called quantum
complex exponential least squares (QCELS). Our algorithm can be readily applied
to reduce the circuit depth for estimating the ground-state energy of a quantum
Hamiltonian, when the overlap between the initial state and the ground state is
large. If this initial overlap is small, we can combine our method with the
Fourier filtering method developed in [Lin, Tong, PRX Quantum 3, 010318, 2022],
and the resulting algorithm provably reduces the circuit depth in the presence
of a large relative overlap compared to ϵ. The relative overlap
condition is similar to a spectral gap assumption, but it is aware of the
information in the initial state and is therefore applicable to certain
Hamiltonians with small spectral gaps. We observe that the circuit depth can be
reduced by around two orders of magnitude in numerical experiments under
various settings