Quantum Event Learning and Gentle Random Measurements

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.