标题： 统一的Matchgate经典阴影框架 显示英文标题

标题： Unified Framework for Matchgate Classical Shadows

摘要： 估计量子费米子性质是计算上困难但对电子系统研究至关重要的任务。 最近的发展开始通过引入依赖于费米子高斯单位变换（FGUs）采样的经典阴影协议来解决这一挑战：这是一类可以在费米子空间中方便地映射到匹配门电路的变换。 文献中提出的不同协议使用了正交群 $O(2n)$ 的不同子集合，FGUs 可以与这些子集合相关联。 我们提出了一种统一这些不同协议的方法，证明它们的等价性，并由此推导出一个最优的采样方案。 我们首先证明与 $SO(2n)$ 相关的 FGU 集合及其与 Clifford 群的交集的前三个矩相等，推广了已知对于 $O(2n)$的结果，并回答了之前工作中提出的问题。 在此证明的基础上，我们建立了文献中分析的 FGU 集合所产生的阴影协议之间的等价性。 最后，根据我们的结果，我们提出了一种针对匹配门电路小子集合的采样方案，该方案在门的数量方面是最优的，并且继承了之前集合的性能保证。 显示英文摘要

摘要： Estimating quantum fermionic properties is a computationally difficult yet crucial task for the study of electronic systems. Recent developments have begun to address this challenge by introducing classical shadows protocols relying on sampling of Fermionic Gaussian Unitaries (FGUs): a class of transformations in fermionic space which can be conveniently mapped to matchgates circuits. The different protocols proposed in the literature use different sub-ensembles of the orthogonal group $O(2n)$ to which FGUs can be associated. We propose an approach that unifies these different protocols, proving their equivalence, and deriving from it an optimal sampling scheme. We begin by demonstrating that the first three moments of the FGU ensemble associated with $SO(2n)$ and of its intersection with the Clifford group are equal, generalizing a result known for $O(2n)$ and addressing a question raised in previous works. Building on this proof, we establish the equivalence between the shadows protocols resulting from FGU ensembles analyzed in the literature. Finally, from our results, we propose a sampling scheme for a small sub-ensemble of matchgates circuits that is optimal in terms of number of gates and that inherits the performances guarantees of the previous ensembles.