Online learning of quantum states

Suppose we have many copies of an unknown n-qubit state ρ. We measure some copies of ρ using a known two-outcome measurement E₁, then other copies using a measurement E₂, and so on. At each stage t, we generate a current hypothesis ωt about the state ρ, using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that |Tr (E_iω_t) - Tr (E_iρ)|, the error in our prediction for the next measurement, is at least ϵ at most O(n/ϵ²) times. Even in the 'non-realizable' setting-where there could be arbitrary noise in the measurement outcomes- we show how to output hypothesis states that incur at most O(√Tn ) excess loss over the best possible state on the first T measurements. These results generalize a 2007 theorem by Aaronson on the PAC-learnability of quantum states, to the online and regret-minimization settings. We give three different ways to prove our results-using convex optimization, quantum postselection, and sequential fat-shattering dimension-which have different advantages in terms of parameters and portability.