Quantum Quasi-Monte Carlo algorithm for out-of-equilibrium Green functions at long times

TL;DR

This paper extends the Quantum Quasi-Monte Carlo method to efficiently compute the full frequency-dependent Green functions in out-of-equilibrium quantum impurity models, outperforming traditional Monte Carlo methods.

Contribution

The authors develop a kernel-based model function extension for QQMC, enabling accurate Green function calculations at long times and broad frequency ranges.

Findings

QQMC achieves error scaling of ~1/N^{0.86} in best cases.

Systematic error reduction by at least two orders of magnitude over Monte Carlo.

QQMC results align with tensor network methods beyond perturbative regimes.

Abstract

We extend the recently developed Quantum Quasi-Monte Carlo (QQMC) approach to obtain the full frequency dependence of Green functions in a single calculation. QQMC is a general approach for calculating high-order perturbative expansions in power of the electron-electron interaction strength. In contrast to conventional Markov chain Monte Carlo sampling, QQMC uses low-discrepancy sequences for a more uniform sampling of the multi-dimensional integrals involved and can potentially outperform Monte Carlo by several orders of magnitudes. A core concept of QQMC is the a priori construction of a "model function" that approximates the integrand and is used to optimize the sampling distribution. In this paper, we show that the model function concept extends to a kernel approach for the computation of Green functions. We illustrate the approach on the Anderson impurity model and show that the scaling of the error with the number of integrand evaluations $N$ is $\sim 1/N^{0.86}$ in the best cases, and comparable to Monte Carlo scaling $\sim 1/N^{0.5}$ in the worst cases. We find a systematic improvement over Monte Carlo sampling by at least two orders of magnitude while using a basic form of model function. Finally, we compare QQMC results with calculations performed with the Fork Tensor Product State (FTPS) method, a recently developed tensor network approach for solving impurity problems. Applying a simple Pad\'e approximant for the series resummation, we find that QQMC matches the FTPS results beyond the perturbative regime.