局域虚时传播子的量子张量列表示的紧致性
Compactness of quantics tensor train representations of local imaginary-time propagators
AI总结：
本文数值研究局域虚时传播子的 QTT 紧致性，采用随机极点模型和两种范数截断，证明其低温键维度饱和且压缩性优于 IR 与 DLR。
摘要：
虚时传播子的空时依赖对于基于量子场论的从头算和多体计算至关重要；已有研究表明，可利用 Quantum Tensor Trains（QTT，量子张量列）对其进行压缩 [Phys. Rev. X 13, 021015 (2023)]。然而，温度等系统参数对数据规模的影响仍未得到充分研究。本文针对单时间/频率对象和双时间/频率对象，全面数值分析了局域虚时传播子在 QTT 中的紧致性，并考虑了基于 Frobenius 范数和最大范数的截断。为研究最坏情形，我们采用随机极点模型，其中极点数量随逆温度对数增长，且系数是随机的。由这些模型生成的 Green 函数预计比物理系统所得函数更难压缩。数值分析表明，这些传播子在 QTT 中具有高度可压缩性，表现优于 intermediate representation（中间表示）和 discrete Lehmann representation（离散 Lehmann 表示）等最先进方法。对于单时间/频率对象和双时间/频率对象，键维度在低温下趋于饱和，尤其是在基于 Frobenius 范数截断时。我们为单时间/频率对象的键维度饱和提供了计数数论证，而双时间/频率对象出现这种饱和的来源仍有待阐明。本文结果强调，在实际应用中，亟需进一步研究截断方法的选择、容差水平，以及虚时表示和虚频表示之间的取舍。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
