  Quantum State Tomography is the task of determining an unknown quantum state
by making measurements on identical copies of the state. Current algorithms are
costly both on the experimental front -- requiring vast numbers of measurements
-- as well as in terms of the computational time to analyze those measurements.
In this paper, we address the problem of analysis speed and flexibility,
introducing \textit{Neural Adaptive Quantum State Tomography} (NA-QST), a
machine learning based algorithm for quantum state tomography that adapts
measurements and provides orders of magnitude faster processing while retaining
state-of-the-art reconstruction accuracy. Our algorithm is inspired by particle
swarm optimization and Bayesian particle-filter based adaptive methods, which
we extend and enhance using neural networks. The resampling step, in which a
bank of candidate solutions -- particles -- is refined, is in our case learned
directly from data, removing the computational bottleneck of standard methods.
We successfully replace the Bayesian calculation that requires computational
time of $O(\mathrm{poly}(n))$ with a learned heuristic whose time complexity
empirically scales as $O(\log(n))$ with the number of copies measured $n$,
while retaining the same reconstruction accuracy. This corresponds to a factor
of a million speedup for $10^7$ copies measured. We demonstrate that our
algorithm learns to work with basis, symmetric informationally complete (SIC),
as well as other types of POVMs. We discuss the value of measurement adaptivity
for each POVM type, demonstrating that its effect is significant only for basis
POVMs. Our algorithm can be retrained within hours on a single laptop for a
two-qubit situation, which suggests a feasible time-cost when extended to
larger systems. It can also adapt to a subset of possible states, a choice of
the type of measurement, and other experimental details.
