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Rapid improvement in quantum hardware has opened the door to complexproblems, but the precise characterization of quantum systems itself remains achallenge. To address this obstacle, novel tomography schemes have beendeveloped that employ generative machine learning models, enabling quantumstate reconstruction from limited classical data. In particular,quantum-inspired Born machines provide a natural way to encode measured datainto a model of a quantum state. Born machines have shown great success inlearning from classical data; however, the full potential of a Born machine inlearning from quantum measurement has thus far been unrealized. To this end, wedevise a complex-valued basis-enhanced Born machine and show that it canreconstruct pure quantum states using projective measurements from only twoPauli measurement bases. We implement the basis-enhanced Born machine to learnthe ground states across the phase diagram of a 1D chain of Rydberg atoms,reconstructing quantum states deep in ordered phases and even at criticalpoints with quantum fidelities surpassing 99%. The model accurately predictsquantum correlations and different observables, and system sizes as large as 37qubits are considered. Quantum states across the phase diagram of a 1D XY spinchain are also successfully reconstructed using this scheme. Our method onlyrequires simple Pauli measurements with a sample complexity that scalesquadratically with system size, making it amenable to experimentalimplementation.

Reconstructing Quantum States Using Basis-Enhanced Born Machines