  Quantum state tomography (QST) is the art of reconstructing an unknown
quantum state through measurements. It is a key primitive for developing
quantum technologies. Neural network quantum state tomography (NNQST), which
aims to reconstruct the quantum state via a neural network ansatz, is often
implemented via a basis-dependent cross-entropy loss function. State-of-the-art
implementations of NNQST are often restricted to characterizing a particular
subclass of states, to avoid an exponential growth in the number of required
measurement settings. To provide a more broadly applicable method for efficient
state reconstruction, we present "neural-shadow quantum state tomography"
(NSQST)-an alternative neural network-based QST protocol that uses infidelity
as the loss function. The infidelity is estimated using the classical shadows
of the target state. Infidelity is a natural choice for training loss,
benefiting from the proven measurement sample efficiency of the classical
shadow formalism. Furthermore, NSQST is robust against various types of noise
without any error mitigation. We numerically demonstrate the advantage of NSQST
over NNQST at learning the relative phases of three target quantum states of
practical interest, as well as the advantage over direct shadow estimation.
NSQST greatly extends the practical reach of NNQST and provides a novel route
to effective quantum state tomography.
