Learning pure quantum states almost without regret

We initiate the study of sample-optimal quantum state tomography with minimal disturbance to the samples. Can we efficiently learn a precise description of a quantum state through sequential measurements of samples while at the same time ensuring that the post-measurement states are only minimally perturbed? We quantify the accumulated cost of the protocol over T samples by an additive regret objective, which controls the cumulative expected post-measurement infidelity of the consumed copies. The challenge is to balance informative measurements with measurements that keep this accumulated cost small. Here, we answer this question for all pure states by exhibiting an adaptive protocol whose expected regret is $O({d}^{3}{\log }^{2}T)$ , while the infidelity of the online estimate after t samples scales as $\widetilde{O}(1/t)$ . We also prove a logarithmic lower bound, showing that for fixed dimension, the dependence on the number of samples is optimal up to one logarithmic factor.