We provide practical and powerful schemes for learning many properties of an unknown n-qubit quantum state using a sparing number of copies of the state. Specifically, we present a depth-modulated randomized measurement scheme that interpolates between two known classical shadows schemes based on random Pauli measurements and random Clifford measurements. These can be seen within our scheme as the special cases of zero and infinite depth, respectively. We focus on the regime where depth scales logarithmically in n and provide evidence that this retains the desirable properties of both extremal schemes whilst, in contrast to the random Clifford scheme, also being experimentally feasible. We present methods for two key tasks; estimating expectation values of certain observables from generated classical shadows and, computing upper bounds on the depth-modulated shadow norm, thus providing rigorous guarantees on the accuracy of the output estimates. We consider observables that can be written as a linear combination of poly(n) Paulis and observables that can be written as a low bond dimension matrix product operator. For the former class of observables both tasks are solved efficiently in n. For the latter class, we do not guarantee efficiency but present a method that works in practice; by variationally computing a heralded approximate inverses of a tensor network that can then be used for efficiently executing both these tasks.
Efficient Many-Body Shadow Metrology via Clifford Lensing
Indian Institute of Technology Bombay（印度理工学院孟买分校）；Indian Institute of Science Education and Research, Pune（印度科学教育研究所浦那分校）；ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（ICFO-光子科学研究院，巴塞罗那科学技术研究院）
Derandomized shallow shadows: Efficient Pauli learning with bounded-depth circuits
Harvard University（哈佛大学）；Harvard-Smithsonian Center for Astrophysics（哈佛-史密森尼天体物理中心）；University of Maryland（马里兰大学）；Joint Quantum Institute（联合量子研究所）；Joint Center for Quantum Information and Computer Science（量子信息与计算机科学联合中心）；National Institute of Standards and Technology（美国国家标准与技术研究院）；University of Innsbruck（因斯布鲁克大学）
Classical Shadows with Improved Median-of-Means Estimation
Quantum Innovation Centre (Q.InC)（量子创新中心）；Agency for Science, Technology and Research (A*STAR)（科技研究局）；Institute of High Performance Computing (IHPC)（高性能计算研究所）；Science, Mathematics and Technology Cluster（科技与数学集群）；Singapore University of Technology and Design（新加坡科技与设计大学）；Singapore Management University（新加坡管理大学）
Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadows
Harvard University（哈佛大学）；IBM Quantum（IBM量子计算部门）；IBM T.J. Watson Research Center（IBM T.J.沃森研究中心）；Berkeley Center for Quantum Information and Computation（伯克利量子信息与计算中心）；University of California San Diego（加利福尼亚大学圣迭戈分校）
Polynomial Resource Classification of Quantum Circuit Familes via Classical Shadows
Virginia Modeling, Analysis, and Simulation Center Old Dominion University（弗吉尼亚建模、分析与模拟中心 老道明大学）；Center for Secure and Intelligent Critical Systems Old Dominion University（安全与智能关键系统中心 老道明大学）
Anticoncentration in Clifford Circuits and Beyond: From Random Tensor Networks to Pseudo-Magic States
Jožef Stefan Institute（约瑟夫·斯特凡研究所）；Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, CY Cergy Paris Université（理论与建模物理实验室，法国国家科学研究中心联合研究实验室第8089号，塞吉巴黎大学）
Fudan University（复旦大学）；Peking University（北京大学）；State Key Laboratory of Surface Physics, Key Laboratory of Micro and Nano Photonic Structures (MOE), and Department of Physics, Fudan University（复旦大学表面物理国家重点实验室、微纳光子结构教育部重点实验室和物理系）；Shanghai Qi Zhi Institute（上海人工智能实验室（智源研究院））；Institute of Nanoelectronics and Quantum Computing, Fudan University（复旦大学纳米电子与量子计算研究所）；Shanghai Artificial Intelligence Laboratory（上海人工智能实验室）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）；Freie Universität Berlin（柏林自由大学）；Helmholtz-Zentrum Berlin für Materialien und Energie（亥姆霍兹柏林材料能源中心）
