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

Abstract

We propose quantum algorithms for projective ground-state preparation and calculations of the many-body Green's functions directly in frequency domain. The algorithms are based on the linear combination of unitary (LCU) operations and essentially only use quantum resources. To prepare the ground state, we construct the operator ${\exp}(-\tau \hat{H}^2)$ using Hubbard-Stratonovich transformation by LCU and apply it on an easy-to-prepare initial state. Our projective state preparation procedure saturates the near-optimal scaling, $\mathcal{O}(\frac{1}{\gamma\Delta} \log \frac{1}{\gamma\eta})$, of other currently known algorithms, in terms of the spectral-gap lower bound $\Delta$, the additive error $\eta$ in the state vector, and the overlap lower bound $\gamma$ between the initial state and the exact ground state. It is straightforward to combine our algorithm with spectral-gap amplification technique to achieve quadratically improved scaling $\mathcal{O}(1/\sqrt{\Delta})$ for ground-state preparation of frustration-free Hamiltonians, which we demonstrate with numerical results of the $q$-deformed XXZ chain. To compute the Green's functions, including the single-particle and other response functions, we act on the prepared ground state with the retarded resolvent operator $R(\omega + i\Gamma; \hat{H})$ in the LCU form derived from the Fourier-Laplace integral transform (FIT). Our resolvent algorithm has $\mathcal{O}(\frac{1}{\Gamma^2} \log\frac{1}{\Gamma\epsilon})$ complexity scaling for the frequency resolution $\Gamma$ of the response functions and the targeted error $\epsilon$, while classical algorithms for FIT usually have polynomial scaling over the error $\epsilon$. To illustrate the complexity scaling of our algorithms, we provide numerical results for their application to the paradigmatic Fermi-Hubbard model on a one-dimensional lattice with different numbers of sites.