An Online Algorithm for Maximum-Likelihood Quantum State Tomography

We propose, to the best of our knowledge, the first online algorithm for maximum-likelihood 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 numerical 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. \textit{Int. Conf. Algorithmic Learning Theory}. 2017).