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 $\rho$ in order to learn a classical description which suffices to later estimate expectation values of observables. Specifically, the goal is to approximate $\mathrm{Tr}(O \rho)$ for any Hermitian observable $O$ to within additive error $\epsilon$ provided $\mathrm{Tr}(O^2)\leq B$ and $\lVert O \rVert = 1$. Our main result applies to the joint measurement setting, where we show $\tilde{\Theta}(\sqrt{B}\epsilon^{-1} + \epsilon^{-2})$ samples of $\rho$ 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 $\rho$ can be compressed for observable estimation. In the independent measurement setting, we show that $\mathcal O(\sqrt{Bd} \epsilon^{-1} + \epsilon^{-2})$ 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.

中文翻译：

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纯状态的样本最佳经典阴影


我们在联合测量和独立测量的设置中考虑纯状态的经典阴影任务。任务是测量未知纯状态 $\rho$ 的几个副本，以便学习足以稍后估计可观测值期望值的经典描述。具体来说，目标是在 $\mathrm{Tr}(O^2)\leq B$ 的情况下，将任何 Hermitian 可观测 $O$ 的 $\mathrm{Tr}(O \rho)$ 逼近到加性误差 $\epsilon$ 内和 $\lVert O \rVert = 1$。我们的主要结果适用于联合测量设置，其中我们显示 $\tilde{\Theta}(\sqrt{B}\epsilon^{-1} + \epsilon^{-2})$ 的 $\rho$ 样本为高概率成功的必要和充分条件。上限是该问题已知的先前最佳样本复杂度的二次改进。对于下界，我们看到瓶颈不是我们学习状态的速度有多快，而是 $\rho$ 的任何经典描述可以压缩多少以进行可观察的估计。在独立测量设置中，我们表明 $\mathcal O(\sqrt{Bd} \epsilon^{-1} + \epsilon^{-2})$ 样本就足够了。值得注意的是，这意味着 Huang、Kueng 和 Preskill 的随机 Clifford 测量算法对于混合状态来说是样本最优的，但对于纯状态来说并不是最优的。有趣的是，我们的结果也使用相同的随机克利福德测量，但采用不同的估计器。