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：对量子系统进行随机顺序的二元投影测量序列所引发的预期扰动，其上界至少存在一个测量接受的概率的平方根。我们将此界限称为"温和随机测量引理"。进一步考虑如下问题：给定未知量子态$\rho$的样本访问权限，需要估计一组测量$\{M_1, M_2, \ldots , M_m\}$的接受概率$\text{Tr}[M_i \rho]$的性质。我们将此类问题定义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解决量子OR问题，从而回答了Aaronson提出的开放问题。同时我们给出一种适用于非投影测量的量子OR协议，该协议需要一种更复杂的测量类型——我们称之为混合测量。当对测量集$\{M_1, \ldots, M_m\}$附加特定保证时，本文提出的量子OR协议还可用于寻找满足$\text{Tr}[M_i \rho]$取值较大的测量$M_i$。此外，我们提出一种基于混合测量的协议，用于估计未知态上测量集的平均接受概率。最后考虑O'Donnell与Bădescu描述的阈值搜索问题。基于量子事件发现结果，我们证明使用$O(\log^2(m) / \epsilon^2)$份$\rho$副本的随机排序（或混合）测量可解决该问题。由此获得影子层析成像算法，其样本复杂度为$\tilde{O}(\log^2(m)\log(d)/\epsilon^4)$，与当前最优样本复杂度持平。该算法无需在量子测量中注入噪声，但要求测量按随机顺序进行，因此不再保持在线特性。