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 \textit{Gentle Random Measurement Lemma}. We also extend the techniques used to prove this lemma to develop protocols for 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 $\{M_1, M_2, ... , M_m\}$. We call these types of problems \textit{Quantum Event Learning Problems}. In particular, 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 and which outperforms both the random measurement protocol analyzed in this paper and the protocol of Harrow, Lin, and Montanaro. However, this protocol requires a more complicated type of measurement, which we call a \textit{Blended Measurement}. When the total (summed) accepting probability of unlikely events is bounded, we show the random and blended measurement 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 call the problem of finding such a measurement \textit{Quantum Event Finding}. Finally, we show Blended Measurements also give a sample-efficient protocol for \textit{Quantum Mean Estimation}: a problem in which the goal is to estimate the average accepting probability of a set of measurements on an unknown state.