Quantum Algorithms for Ground State Preparation and Green's Function Calculation

We propose quantum algorithms for projective ground-state preparation and frequency-domain Green's function calculations. The algorithms are based on the linear combination of unitary (LCU) framework and use only quantum resources. To prepare the ground state, we apply the operator exp(-τH²) expressed using LCU on an easy-to-prepare initial state. This procedure saturates the near-optimal scaling O(log(1/(γη))/(γΔ)) of other algorithms, in terms of the spectral gap ∆, the targeted error η, and the overlap γ between the initial state and the exact ground state. Our algorithm can easily be combined with the spectral gap amplification technique to achieve better scaling O(1/√Δ) for frustration-free Hamiltonians. To compute single and multi-particle response functions, we act on the prepared ground state with the retarded resolvent operator in the LCU form derived from the Fourier-Laplace integral transform (FIT). Our resolvent algorithm has the complexity O(log(1/(Γϵ))/Γ²) for the frequency resolution Γ of the response functions and the targeted error ϵ, while classical algorithms for FIT usually have polynomial scaling over the error ϵ. To illustrate the complexity scaling of our algorithms, we provide numerical results for their application to the paradigmatic Fermi-Hubbard model.