Computing Green's functions and improving ground state energy estimation on quantum computers with Liouvillian recursion

TL;DR

This paper introduces a hybrid quantum-classical method using Liouvillian recursion to compute Green's functions, improving ground state energy estimates on quantum computers, demonstrated on a four-site Hubbard model with promising convergence and noise robustness.

Contribution

The paper presents a novel hybrid algorithm for Green's function computation using Liouvillian recursion, suitable for near-term quantum hardware and improving energy estimation accuracy.

Findings

Exponential convergence in the number of iterations.

Polynomial complexity in Green's-function accuracy.

Robustness to noise and ground state preparation errors.

Abstract

We present a quantum-classical hybrid implementation of the Liouvillian recursion method to compute many-body Green's functions using a quantum computer. From an approximate ground state preparation circuit, this algorithm produces the local ($r=r'$) and inter-site ($r\neq r'$) Green's functions $G_{rr'}(\omega)$ by measuring observables generated recursively. We demonstrate the approach on a superconducting quantum processor for the open-boundary four-site Hubbard model. We then use the computed Green's functions as input to the Galitskii-Migdal formula to produce better ground state energy estimation than the expectation value of the Hamiltonian for the approximate circuit. Empirical results indicate exponential convergence in the number of iterations, yielding a computational complexity polynomial in the Green's-function accuracy, as measured with the Wasserstein distance. Our results also indicate significant robustness to noise and to inaccuracies of the ground state preparation, providing evidence that Liouvillian recursion is well adapted to the constraints of near-term quantum computing.