Classical shadows (CS) offer a resource-efficient means to estimate quantum observables, circumventing the need for exhaustive state tomography. Here, we clarify and explore the connection between CS techniques and least squares (LS) and regularized least squares (RLS) methods commonly used in machine learning and data analysis. By formal identification of LS and RLS ``shadows'' completely analogous to those in CS -- namely, point estimators calculated from the empirical frequencies of single measurements -- we show that both RLS and CS can be viewed as regularizers for the underdetermined regime, replacing the pseudoinverse with invertible alternatives. Through numerical simulations, we evaluate RLS and CS from three distinct angles: the tradeoff in bias and variance, mismatch between the expected and actual measurement distributions, and the interplay between the number of measurements and number of shots per measurement. Compared to CS, RLS attains lower variance at the expense of bias, is robust to distribution mismatch, and is more sensitive to the number of shots for a fixed number of state copies -- differences that can be understood from the distinct approaches taken to regularization. Conceptually, our integration of LS, RLS, and CS under a unifying ``shadow'' umbrella aids in advancing the overall picture of CS techniques, while practically our results highlight the tradeoffs intrinsic to these measurement approaches, illuminating the circumstances under which either RLS or CS would be preferred, such as unverified randomness for the former or unbiased estimation for the latter.

翻译：经典阴影（CS）提供了一种资源高效的方式来估计量子观测量，免除了穷尽态层析的需求。本文阐明并探讨了CS技术与机器学习及数据分析中常用的最小二乘法（LS）和正则化最小二乘法（RLS）之间的联系。通过形式化识别LS和RLS的“阴影”——即完全类似于CS中根据单次测量经验频率计算得到的点估计量——我们证明RLS和CS均可视为欠定情形下的正则化器，将伪逆替换为可逆替代方案。通过数值模拟，我们从三个不同角度评估RLS和CS：偏差与方差的权衡、期望测量分布与实际测量分布的失配、以及测量次数与每次测量采样次数之间的相互作用。与CS相比，RLS以偏差为代价获得更低的方差，对分布失配具有鲁棒性，并且在固定状态副本数下对单次测量采样次数更为敏感——这些差异可通过正则化方法的不同路径来理解。在概念上，我们将LS、RLS和CS统一到“阴影”框架下，有助于推进对CS技术整体图景的认识；在实践中，我们的结果凸显了这些测量方法固有的权衡关系，阐明RLS或CS各自适用的情况，例如前者适用于未经验证的随机性，后者则适用于无偏估计。