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

TL;DR

A quantum-classical hybrid implementation of the Liouvillian recursion method to compute many-body Green's functions using a quantum computer, showing 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.

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 $G_{rr'}(ω)$ 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.

Figures (4)

• Figure 1: Green's functions and energy estimates obtained with the proposed algorithm. In all panels, black lines are exact values, dashed colored lines are simulation results, and full-colored lines are IBM Quebec quantum processor results. In each column, a different approximate ground-state circuit is used for taking expectation values, as identified, at the top, by the circuit's fidelity to the true ground state. The first row, (a)-(c), shows the imaginary parts (spectral weight) of the local Green's functions, $G_{00}(\omega)$, obtained with the recursion \ref{['eq:contfrac']}-\ref{['eq:fm1']}, while the second row (d)-(f) shows the first-neighbor one, $G_{01}(\omega)$, obtained with polynomial expansion of Eqs. \ref{['eq:Green_poly_hybrid']}. Simulation and hardware results are shown for iteration $k=6$, while the exact results (black lines) correspond to iteration $k=30$, at which the algorithm numerically converges. The last row, (g)-(i), compares energy estimates from the Galitskii-Migdal formula \ref{['eq:gm']} (jagged colored lines, with markers highlighting even iterations) to the expected values of the Hamiltonian for each circuit (horizontal colored lines, with standard deviation in shaded area) and the true ground state energy (black line).
• Figure 2: Convergence and computational cost of the proposed algorithm. In all panels, black lines are exact values, dashed colored lines are simulation results, and full-colored lines are hardware results. The different colors correspond to different approximate ground state circuits. Panels (a) and (b) show the raw outputs of the algorithm: the continued fraction coefficients obtained at each iteration of the recursion. Panel (c) shows the number of Pauli operators generated by the recursion as a function of iterations; it constitutes the main computational cost of the algorithm. Panel (d) shows the distance \ref{['eq_wass']}, on a logarithmic scale, between the computed Green's function and the exact Green's function, as a function of iterations.
• Figure 3: The exponential convergence of the algorithm compensates its exponential cost. The colored lines show the distance $d$ to solution, as measured by \ref{['eq_wass']}, as a function of the number of Pauli operators $p$ generated at even iteration $k$ for simulations (dashed lines) and hardware runs (full lines). The gray line highlights the polynomial scaling $d=p^{-1/4}$.
• Figure 4: Approximate ground state preparation circuits, progressively farther from the true ground state energy of $-9.9531$.