Low-depth Gaussian State Energy Estimation

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.