Part of Advances in Neural Information Processing Systems 31 (NeurIPS 2018)
Scott Aaronson, Xinyi Chen, Elad Hazan, Satyen Kale, Ashwin Nayak
Suppose we have many copies of an unknown n-qubit state ρ. We measure some copies of ρ using a known two-outcome measurement E_1, then other copies using a measurement E_2, 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 |\trace(Eiωt)−\trace(Eiρ)|, the error in our prediction for the next measurement, is at least eps at most O(n/eps2)\ 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.