解析延拓的优化最大熵算法和诊断工具
Algorithms for optimized maximum entropy and diagnostic tools for analytic continuation
摘要：
在虚时间或虚频率中得到的数值数据的解析延拓已成为量子计算物理许多分支的重要组成部分。然而，这是一个病态的过程，因此是一个很难的数值问题。基于贝叶斯推断的最大熵方法是解决该问题的最广泛使用的方法。虽然该方法是公认的，并且是最可靠和有效的方法之一，但该方法及其实施的有益发展仍然是可能的。此外，虽然有一些自由软件实现，但仍然缺乏一个文档齐全，优化，通用和用户友好的软件，专门用于该特定任务。在这里，我们分析了实现的各个方面，这些方面对准确性和速度至关重要，并提出了一种高度优化的最大熵方法。最初的算法和概念贡献包括（1）数值近似，产生的计算复杂性，几乎是独立的温度和频谱形状（例如，包括宽背景中的尖锐德鲁德峰），同时只要数据精度足够，就确保结果的定量准确性，（2）一种选择熵权$α$的鲁棒方法，该方法从该方法的简单一致性条件和信息-和噪声-可以从$χ^2$相对于$α$的行为中清楚地识别拟合状态，以及（3）几种诊断以评估结果的可靠性。基准测试不同的复杂性和一个实际的物理模拟的例子光谱功能。我们的实现，其中包括费米子，玻色子和响应函数的最典型的情况下，可作为一个开源的，用户友好的软件。
英文摘要：
Analytic continuation of numerical data obtained in imaginary time or frequency has become an essential part of many branches of quantum computational physics. It is, however, an ill-conditioned procedure and thus a hard numerical problem. The maximum-entropy approach, based on bayesian inference, is the most widely used method to tackle that problem. Although the approach is well established and among the most reliable and efficient ones, useful developments of the method and of its implementation are still possible. In addition, while a few free software implementations are available, a well-documented, optimized, general purpose and user-friendly software dedicated to that specific task is still lacking. Here we analyze all aspects of the implementation that are critical for accuracy and speed, and present a highly optimized approach to maximum-entropy. Original algorithmic and conceptual contributions include (1) numerical approximations that yield a computational complexity that is almost independent of temperature and spectrum shape (including sharp Drude peaks in broad background for example) while ensuring quantitative accuracy of the result whenever precision of the data is sufficient, (2) a robust method of choosing the entropy weight $α$ that follows from a simple consistency condition of the approach and the observation that information- and noise-fitting regimes can be identified clearly from the behavior of $χ^2$ with respect to $α$, and (3) several diagnostics to assess the reliability of the result. Benchmarks with test spectral functions of different complexity and an example with an actual physical simulation are presented. Our implementation, which covers most typical cases for fermions, bosons and response functions, is available as an open source, user friendly software.
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