We prove the expected disturbance caused to a quantum system by a sequence of randomly ordered two-outcome projective measurements is upper bounded by the square root of the probability that at least one measurement in the sequence accepts. We call this bound the Gentle Random Measurement Lemma. We then consider problems in which we are given sample access to an unknown state $\rho$ and asked to estimate properties of the accepting probabilities $\text{Tr}[M_i \rho]$ of a set of measurements $\{M_1, M_2, \ldots , M_m\}$. We call these types of problems Quantum Event Learning Problems. Using the gentle random measurement lemma, we show randomly ordering projective measurements solves the Quantum OR problem, answering an open question of Aaronson. We also give a Quantum OR protocol which works on non-projective measurements but which requires a more complicated type of measurement, which we call a Blended Measurement. Given additional guarantees on the set of measurements $\{M_1, \ldots, M_m\}$, we show the Quantum OR protocols developed in this paper can also be used to find a measurement $M_i$ such that $\text{Tr}[M_i \rho]$ is large. We also give a blended measurement based protocol for estimating the average accepting probability of a set of measurements on an unknown state. Finally we consider the Threshold Search Problem described by O'Donnell and B\u{a}descu. By building on our Quantum Event Finding result we show that randomly ordered (or blended) measurements can be used to solve this problem using $O(\log^2(m) / \epsilon^2)$ copies of $\rho$. Consequently, we obtain an algorithm for Shadow Tomography which requires $\tilde{O}(\log^2(m)\log(d)/\epsilon^4)$ samples, matching the current best known sample complexity. This algorithm does not require injected noise in the quantum measurements, but does require measurements to be made in a random order and so is no longer online.

翻译：我们证明，在一系列随机排序的两结果投影测量中，对量子系统造成的期望扰动上限等于至少一次测量接受的概率的平方根。我们将这一界限称为“温和随机测量引理”。随后，我们考虑以下问题：给定对未知状态 $\rho$ 的样本访问权限，并要求估计一组测量 $\{M_1, M_2, \ldots , M_m\}$ 的接受概率 $\text{Tr}[M_i \rho]$ 的属性。我们将这类问题称为量子事件学习问题。利用温和随机测量引理，我们证明随机排序投影测量可以解决量子OR问题，从而回答了Aaronson的一个开放性问题。我们还给出了一种适用于非投影测量的量子OR协议，但该协议需要一种更复杂的测量类型，我们称之为“混合测量”。在额外保证测量集合 $\{M_1, \ldots, M_m\}$ 的前提下，我们表明本文开发的量子OR协议也可用于找到一个测量 $M_i$，使得 $\text{Tr}[M_i \rho]$ 较大。此外，我们给出了一种基于混合测量的协议，用于估计未知状态上一组测量平均接受概率。最后，我们考虑了O'Donnell和Bădescu描述的阈值搜索问题。基于我们的量子事件发现结果，我们证明使用随机排序（或混合）测量可以在 $O(\log^2(m) / \epsilon^2)$ 份 $\rho$ 副本下解决该问题。因此，我们获得了一种用于影子层析成像的算法，该算法需要 $\tilde{O}(\log^2(m)\log(d)/\epsilon^4)$ 个样本，与当前已知的最佳样本复杂度相当。该算法不需要在量子测量中注入噪声，但要求测量以随机顺序进行，因此不再是在线方式。