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Learning Distributions over Quantum Measurement Outcomes

Shadow tomography for quantum states provides a sample efficient approach for predicting the properties of quantum systems when the properties are restricted to expectation values of $2$-outcome POVMs. However, these shadow tomography procedures yield poor bounds if there are more than 2 outcomes per measurement. In this paper, we consider a general problem of learning properties from unknown quantum states: given an unknown $d$-dimensional quantum state $ρ$ and $M$ unknown quantum measurements $\mathcal{M}_1,...,\mathcal{M}_M$ with $K\geq 2$ outcomes, estimating the probability distribution for applying $\mathcal{M}_i$ on $ρ$ to within total variation distance $ε$. Compared to the special case when $K=2$, we need to learn unknown distributions instead of values. We develop an online shadow tomography procedure that solves this problem with high success probability requiring $\tilde{O}(K\log^2M\log d/ε^4)$ copies of $ρ$. We further prove an information-theoretic lower bound that at least $Ω(\min\{d^2,K+\log M\}/ε^2)$ copies of $ρ$ are required to solve this problem with high success probability. Our shadow tomography procedure requires sample complexity with only logarithmic dependence on $M$ and $d$ and is sample-optimal for the dependence on $K$.