论文标题
纯状态的样本 - 最佳经典阴影
Sample-optimal classical shadows for pure states
论文作者
论文摘要
我们考虑在关节和独立测量的设置中，纯状态的经典阴影任务。任务是测量未知纯状态$ρ$的几份副本，以学习一个经典描述，足以估算可观察到的期望值。具体来说，目标是在添加误差$ \ε$ $ $ \ mathrm {tr}（o^2）（o^2）\ leq b $和$ \ lvert o \ lvert o \ lvert o \ rvert = 1 $的情况下近似$ \ mathrm {tr}（oρ）$ to添加错误$ε$中的任何Hermitian可观察到的$ o $。我们的主要结果适用于关节测量设置，在该设置中，我们显示$ \tildeθ（\ sqrt {b}ε^{ - 1} +ε^{ - 2}）$ $ρ$样本是必要的，足以成功地具有很高的概率。上限是对以前最佳样本复杂性的二次改进。对于下边界，我们看到瓶颈不是我们可以学习状态的速度，而是对$ρ$的任何经典描述都可以被压缩，以进行可观察到的估计。在独立的测量设置中，我们表明$ \ mathcal o（\ sqrt {bd}ε^{ - 1} +ε^{ - 2}）$ samples就足够了。值得注意的是，这意味着对于混合状态的样本最佳状态，黄，Kueng和Preskill的随机Clifford测量算法并不是纯状态的最佳选择。有趣的是，我们的结果还使用相同的随机Clifford测量值，但采用了不同的估计器。
We consider the classical shadows task for pure states in the setting of both joint and independent measurements. The task is to measure few copies of an unknown pure state $ρ$ in order to learn a classical description which suffices to later estimate expectation values of observables. Specifically, the goal is to approximate $\mathrm{Tr}(O ρ)$ for any Hermitian observable $O$ to within additive error $ε$ provided $\mathrm{Tr}(O^2)\leq B$ and $\lVert O \rVert = 1$. Our main result applies to the joint measurement setting, where we show $\tildeΘ(\sqrt{B}ε^{-1} + ε^{-2})$ samples of $ρ$ are necessary and sufficient to succeed with high probability. The upper bound is a quadratic improvement on the previous best sample complexity known for this problem. For the lower bound, we see that the bottleneck is not how fast we can learn the state but rather how much any classical description of $ρ$ can be compressed for observable estimation. In the independent measurement setting, we show that $\mathcal O(\sqrt{Bd} ε^{-1} + ε^{-2})$ samples suffice. Notably, this implies that the random Clifford measurements algorithm of Huang, Kueng, and Preskill, which is sample-optimal for mixed states, is not optimal for pure states. Interestingly, our result also uses the same random Clifford measurements but employs a different estimator.
