AI 中文总结
该研究针对含噪声的真实经典阴影协议建立完整理论，推导噪声阴影信道与方差，证明无噪声样本复杂度优势在噪声下保留，明确方差比判据及不同目标的优势情况。
AI 中文摘要
West等人提出的真实经典阴影协议将Huang-Kueng-Preskill方案中的酉（Clifford）系综替换为正交（实Clifford）系综，对于对称可观测量，其估计量方差显著更小：全局演化下的方差因子接近2，对于k局域实Pauli可观测量则为指数因子(3/2)^k。然而，真实硬件无法实现理想演化。基于Koh和Grewal的噪声经典阴影框架，我们针对正交演化后存在已知的完全正迹保持噪声信道的情况，建立了真实经典阴影协议的完整理论。我们利用正交群的Weingarten演算，从第一性原理推导了噪声全局和局域正交阴影信道，证明每个信道都是作用于输入的对称（或局域对称）分量的去极化信道，并由此推导出精确的单次射击方差闭式表达式、相关的阴影半范数、其双侧界以及对应的样本复杂度保证。我们证明，无噪声情况下的样本复杂度优势在噪声环境下仍完整保留。由于方差是精确值而非界，该比值由单个无量纲参数控制，这给出了因子2可实现性的闭式判据：当可观测量的范数剖面增长时，二阶矩和方差比均精确达到2，而噪声仅通过一个位于2/(d+2)范围内的因子进入该极限，因此该优势对噪声具有鲁棒性。对于秩一目标，该优势可证明无法实现，严格低于2；局域实Pauli优势仍为(3/2)^k。我们通过实参数和转置噪声标量处理复测量基，在适当极限下恢复酉阴影。
英文摘要
The real classical shadows protocol of West et al. replaces the unitary (Clifford) ensemble of the Huang--Kueng--Preskill scheme by the orthogonal (real Clifford) ensemble, and for symmetric observables achieves strictly smaller estimator variances: a factor approaching two for global evolution and an exponential factor $(3/2)^k$ for $k$-local real Pauli observables. Real hardware, however, never implements the ideal evolution. Building on the noisy classical shadows framework of Koh and Grewal, we give a complete theory of the real classical shadows protocol in the presence of a known completely positive trace-preserving noise channel acting after the orthogonal evolution. We derive the noisy global and local orthogonal shadow channels from first principles using the Weingarten calculus of the orthogonal group, prove that each is a depolarizing channel acting on the symmetric (respectively locally symmetric) component of its input, and derive from it the exact single-shot variance in closed form, together with the associated shadow seminorm, two-sided bounds on it, and the resulting sample-complexity guarantees. We prove that the noiseless sample-complexity advantages survive intact under noise. Because the variances are exact rather than bounded, the ratio is controlled by a single dimensionless parameter, which gives a closed-form criterion for when the factor of two is attainable: both the second-moment and the variance ratio reach it exactly when the observable's norm profile grows, and the noise enters that limit only through a factor lying within $2/(d+2)$ of two, so the advantage is uniform in the noise. For rank-one targets it is provably unattainable, saturating strictly below two. The local real-Pauli advantage remains $(3/2)^k$. We treat complex measurement bases through a reality parameter and a transposed-noise scalar, recovering unitary shadows in the appropriate limit.
Comments70 pages, 10 figures
