虚时响应函数的Hankel投影去噪
Denoising of Imaginary Time Response Functions with Hankel projections
AI总结：
本文提出一种基于Hankel矩阵正定投影的交替投影算法，用于对有限温度量子系统的虚时响应函数（如量子蒙特卡洛数据）进行去噪，可将噪声降低高达20倍，并建议在相关数据分析中广泛采用。
摘要：
有限温度量子系统的虚时响应函数通常通过具有随机或系统误差的方法获得。降低这些误差需要巨大的计算成本——在量子蒙特卡洛模拟中，将噪声降低两倍会导致模拟成本增加四倍。在本文中，我们将某些虚时响应函数与Fock空间上线性算子空间上的内积联系起来。然后我们证明，带噪声的数据通常不满足其相关Gramian的正定性。该Gramian具有Hankel矩阵的结构。作为一种对含噪数据进行去噪的方法，我们引入了一种交替投影算法，该算法寻找与含噪数据一致的最接近的正定Hankel矩阵。我们以连续时间量子蒙特卡洛数据的费米子格林函数为例测试了我们的方法，并展示了误差的显著改善，在实际例子中将噪声降低了高达20倍。我们认为，只要分析带有误差的有限温度虚时响应函数数据，无论是在量子蒙特卡洛、量子计算还是近似半解析方法的背景下，都应使用Hankel投影。
英文摘要：
Imaginary-time response functions of finite-temperature quantum systems are often obtained with methods that exhibit stochastic or systematic errors. Reducing these errors comes at a large computational cost -- in quantum Monte Carlo simulations, the reduction of noise by a factor of two incurs a simulation cost of a factor of four. In this paper, we relate certain imaginary-time response functions to an inner product on the space of linear operators on Fock space. We then show that data with noise typically does not respect the positive definiteness of its associated Gramian. The Gramian has the structure of a Hankel matrix. As a method for denoising noisy data, we introduce an alternating projection algorithm that finds the closest positive definite Hankel matrix consistent with noisy data. We test our methodology at the example of fermion Green's functions for continuous-time quantum Monte Carlo data and show remarkable improvements of the error, reducing noise by a factor of up to 20 in practical examples. We argue that Hankel projections should be used whenever finite-temperature imaginary-time data of response functions with errors is analyzed, be it in the context of quantum Monte Carlo, quantum computing, or in approximate semianalytic methodologies.
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