论文标题
多拍阴影估计的性能分析
Performance analysis of multi-shot shadow estimation
论文作者
论文摘要
阴影估计是一种有效的方法，可以预测具有统计保证的量子状态的许多可观察结果。在多拍的情况下，一个人在相同的统一演变后以$ k $次的顺序准备的状态进行投影测量，并以$ m $ $ $ $ $的随机采样统一性重复此过程。结果，总共有$ MK $ $乘以。在这里，我们分析了这种多拍情景中影子估计的性能，其特征是估计某些可观察到的$ o $的期望值的差异。我们发现，除了影子 - norm $ \ | o \ | _ {\ mathrm {shadow}} $中，在[Huang et.Al. \ | _ {\ Mathrm {Xshadow}} $。对于随机的Pauli和Clifford测量，我们分析并显示$ \ | O \ | _ {\ Mathrm {Xshadow}} $的上限。特别是，我们找出在随机Pauli测量中可观察到的Pauli的确切方差公式。我们的工作为应用多拍阴影估计的理论指导提供了指导。
Shadow estimation is an efficient method for predicting many observables of a quantum state with a statistical guarantee. In the multi-shot scenario, one performs projective measurement on the sequentially prepared state for $K$ times after the same unitary evolution, and repeats this procedure for $M$ rounds of random sampled unitary. As a result, there are $MK$ times measurements in total. Here we analyze the performance of shadow estimation in this multi-shot scenario, which is characterized by the variance of estimating the expectation value of some observable $O$. We find that in addition to the shadow-norm $\|O \|_{\mathrm{shadow}}$ introduced in [Huang et.al.~Nat.~Phys.~2020\cite{huang2020predicting}], the variance is also related to another norm, and we denote it as the cross-shadow-norm $\|O \|_{\mathrm{Xshadow}}$. For both random Pauli and Clifford measurements, we analyze and show the upper bounds of $\|O \|_{\mathrm{Xshadow}}$. In particular, we figure out the exact variance formula for Pauli observable under random Pauli measurements. Our work gives theoretical guidance for the application of multi-shot shadow estimation.
