An Online Algorithm for Maximum-Likelihood Quantum State Tomography

We propose, to the best of our knowledge, the first online algorithm to compute the maximum-likelihood estimate in quantum state tomography. Suppose the quantum state to be estimated corresponds to a $D$-by-$D$ density matrix. The per-iteration computational complexity of the algorithm is $O ( D ^ 3 )$, independent of the data size. The expected optimization error of the algorithm is $O(\sqrt{ ( 1 / T ) D \log D })$, where $T$ denotes the number of iterations. The algorithm can be viewed as a quantum extension of Soft-Bayes, a recent algorithm for online portfolio selection (Orseau et al. Soft-Bayes: Prod for mixtures of experts with log-loss. Int. Conf. Algorithmic Learning Theory. 2017).