  Recent progress in quantum computing is paving the way for the realization of
early fault-tolerant quantum computers. To maximize the utility of these
devices, it is important to develop quantum algorithms that match their
capabilities and limitations. Motivated by this, recent work has developed
low-depth quantum algorithms for ground state energy estimation (GSEE), an
important subroutine in quantum chemistry and materials. We detail a new GSEE
algorithm which, like recent work, uses a number of operations scaling as
$O(1/\Delta)$ as opposed to the typical $O(1/\epsilon)$, at the cost of an
increase in the number of circuit repetitions from $O(1)$ to $O(1/\epsilon^2)$.
The relevant features of this algorithm come about from using a Gaussian
window, which exponentially reduces contamination from excited states over the
simplest GSEE algorithm based on the Quantum Fourier Transform (QFT). We adapt
this algorithm to interpolate between the low-depth and full-depth regime by
replacing $\Delta$ with anything between $\Delta$ and $\epsilon$. At the cost
of increasing the number of ancilla qubits from $1$ to $O(\log\Delta)$, our
method reduces the upper bound on the number of circuit repetitions by a factor
of four compared to previous methods.
