  Learning faithful representations of quantum states is crucial to fully
characterizing the variety of many-body states created on quantum processors.
While various tomographic methods such as classical shadow and MPS tomography
have shown promise in characterizing a wide class of quantum states, they face
unique limitations in detecting topologically ordered two-dimensional states.
To address this problem, we implement and study a heuristic tomographic method
that combines variational optimization on tensor networks with randomized
measurement techniques. Using this approach, we demonstrate its ability to
learn the ground state of the surface code Hamiltonian as well as an
experimentally realizable quantum spin liquid state. In particular, we perform
numerical experiments using MPS ans\"atze and systematically investigate the
sample complexity required to achieve high fidelities for systems of sizes up
to $48$ qubits. In addition, we provide theoretical insights into the scaling
of our learning algorithm by analyzing the statistical properties of maximum
likelihood estimation. Notably, our method is sample-efficient and
experimentally friendly, only requiring snapshots of the quantum state measured
randomly in the $X$ or $Z$ bases. Using this subset of measurements, our
approach can effectively learn any real pure states represented by tensor
networks, and we rigorously prove that random-$XZ$ measurements are
tomographically complete for such states.
