Online Learning of Quantum States

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 omega(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 vertical bar Tr (E-i omega(t)) - Tr (E-i rho)vertical bar, the error in our prediction for the next measurement, is at least 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 incur at most O(root 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.