Title: Online Learning of Quantum States
| Speaker: | Ashwin Nayak |
| Affiliation: | University of Waterloo |
| Room: | MC 5501 |
Abstract:
Suppose we have many copies of an unknown n-qubit state rho. We measure some copies of rho 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 sigma_t about the state rho, using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that Tr(E_i sigma_{i-1}) differs from Tr(E_i rho) by more than epsilon at most O(n/epsilon^2) 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 O(sqrt(Tn)) 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.
Joint work with Scott Aaronson, Xinyi Chen, Elad Hazan, and Satyen Kale.