Experimental shadow tomography beyond single-copy measurements
Xu-Jie Peng, Qing Liu, Lu Liu, Ting Zhang, You Zhou, He Lu
Abstract
Shadow tomography is a promising way for quantum system characterization, enabling the extraction of multiple properties of unknown quantum states using only single-copy randomized measurements. While single-copy measurements are experimentally friendly and efficient for estimating linear functions, they show limited efficiency for nonlinear ones. In contrast, multicopy measurements enabled by joint entangling operations can remarkably reduce the sample complexity for nonlinear functions. Here, we design and realize a deterministic optical three-qubit Fredkin gate by comprehensive control over multiple degrees of freedom of a single photon, achieving controlled-swap operation with high process fidelity of $0.935\ifmmode\pm\else\textpm\fi{}0.001$. Utilizing this Fredkin gate, we demonstrate hybrid shadow tomography, in which the higher-order functions like the moments can be efficiently estimated from the two-copy swap operation in conjunction with the single-copy randomized measurement. Our demonstration indicates a significant reduction in sample complexity compared with the single-copy setting. Furthermore, we show the potential advantage of our scheme in a proof-of-principle quantum metrology experiment, wherein the accuracy of parameter estimation is enhanced with the assistance of virtual distillation. Our results suggest that hybrid shadow tomography is an efficient and powerful tool for shadow-estimation-enabled quantum information processing.