  Quantum state tomography is a powerful, but resource-intensive, general
solution for numerous quantum information processing tasks. This motivates the
design of robust tomography procedures that use relevant resources as sparingly
as possible. Important cost factors include the number of state copies and
measurement settings, as well as classical postprocessing time and memory. In
this work, we present and analyze an online tomography algorithm designed to
optimize all the aforementioned resources at the cost of a worse dependence on
accuracy. The protocol is the first to give provably optimal performance in
terms of rank and dimension for state copies, measurement settings and memory.
Classical runtime is also reduced substantially and numerical experiments
demonstrate a favorable comparison with other state-of-the-art techniques.
Further improvements are possible by executing the algorithm on a quantum
computer, giving a quantum speedup for quantum state tomography.
