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 σt 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 |operatornameTr(Ei σt) - operatornameTr(Eiρ) |, the error in our prediction for the next measurement, is at least varepsilon at most operatornameO!left(n / varepsilon2 right) 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 do significantly worse than the best possible states at most operatornameO!left(√ Tnright) times 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.