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估计量子可观测量，从而避免了全面态层析的需求。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）之间的关联。通过形式化识别出与CS完全类似的LS和RLS"阴影"——即根据单次测量的经验频率计算出的点估计量——我们证明RLS和CS均可视为欠定条件下的正则化方法，通过可逆替代方案取代伪逆。通过数值模拟，我们从三个不同角度评估了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不匹配性、以及测量次数与每次测量采择次数之间的相互作用。与CS相比，RLS以牺牲偏差为代价获得更低的方差，对分布失配具有鲁棒性，并且在固定态副本数量下对每次测量的采择次数更为敏感——这些差异可通过不同的正则化策略得到理解。从概念上讲，我们以统一的"阴影"框架整合LS、RLS和CS有助于推进对CS技术的整体认识；从实践角度而言，我们的结果突显了这些测量方法固有的权衡，揭示了在何种情况下应优先选择RLS（如未验证随机性）或CS（如无偏估计）。