Compactness of quantics tensor train representations of local imaginary-time propagators

TL;DR

This paper demonstrates that local imaginary-time propagators in quantum systems can be highly compressed using Quantum Tensor Trains, outperforming existing methods, with bond dimensions saturating at low temperatures, indicating potential for efficient quantum simulations.

Contribution

The study provides a comprehensive numerical analysis of the compactness of local imaginary-time propagators in QTT, revealing their high compressibility and saturation behavior at low temperatures, and compares this with state-of-the-art approaches.

Findings

Propagators are highly compressible in QTT.

Bond dimensions saturate at low temperatures.

QTT outperforms intermediate and Lehmann representations.

Abstract

Space-time dependence of imaginary-time propagators, vital for \textit{ab initio} and many-body calculations based on quantum field theories, has been revealed to be compressible using Quantum Tensor Trains (QTTs) [Phys. Rev. X {\bf 13}, 021015 (2023)]. However, the impact of system parameters, like temperature, on data size remains underexplored. This paper provides a comprehensive numerical analysis of the compactness of local imaginary-time propagators in QTT for one-time/-frequency objects and two-time/-frequency objects, considering truncation in terms of the Frobenius and maximum norms. To study worst-case scenarios, we employ random pole models, where the number of poles grows logarithmically with the inverse temperature and coefficients are random. The Green's functions generated by these models are expected to be more difficult to compress than those from physical systems. The numerical analysis reveals that these propagators are highly compressible in QTT, outperforming the state-of-the-art approaches such as intermediate representation and discrete Lehmann representation. For one-time/-frequency objects and two-time/-frequency objects, the bond dimensions saturate at low temperatures, especially for truncation in terms of the Frobenius norm. We provide counting-number arguments for the saturation of bond dimensions for the one-time/-frequency objects, while the origin of this saturation for two-time/-frequency objects remains to be clarified. This paper's findings highlight the critical need for further research on the selection of truncation methods, tolerance levels, and the choice between imaginary-time and imaginary-frequency representations in practical applications.