Estimating energies of quantum many-body Hamiltonians via Super-resolution Filtering Spectroscopy

Determining energies of quantum many-body Hamiltonians is a paradigmatic use case of quantum computation. We present a novel quantum algorithm to determine two types of energies: the eigenenergies of a single Hamiltonian and eigenenergy differences, either within a Hamiltonian or between two Hamiltonians. Our algorithm uses an initial state with constant fidelity, Loschmidt echoes measured with one locally-controlled gate, a filter on the relevant energy window to boost the signal-to-noise ratio, and super-resolution techniques to estimate the energy as the sole frequency of a time series produced from the Loschmidt echoes.

The shot count scales quadratically with the system size, only linearly in the circuit depth, and quadratically with the inverse error. The target energy difference is retrieved in the classical post-processing, thus avoiding any gadget employed by other algorithms where the said difference is obtained at the quantum simulation stage.

Importantly, when the eigenenergy difference is small enough with respect to the magnitudes of the individual energies, we prove that our method is advantageous in the direct estimate over the subtraction of individual energies. To complement our analysis, we benchmark our algorithm with systems of chemical interest and make a comparison against standard algorithms for energy estimation.