Reconstructing a quantum state with a variational autoencoder

Quantum state tomography (QST) is an important and challenging task in the field of quantum information, which has attracted a lot of attentions in recent years. Machine learning models can provide a classical representation of the quantum state after trained on the measurement outcomes, which are part of effective techniques to solve QST problem. In this work, we use a variational autoencoder (VAE) to learn the measurement distribution of two quantum states generated by MPS circuits. We first consider the Greenberger-Horne-Zeilinger (GHZ) state which can be generated by a simple MPS circuit. Simulation results show that a VAE can reconstruct 3- to 8-qubit GHZ states with a high fidelity, i.e., 0.99, and is robust to depolarizing noise. The minimum number (N-s(*)) of training samples required to reconstruct the GHZ state up to 0.99 fidelity scales approximately linearly with the number of qubits (N). However, for the quantum state generated by a complex MPS circuit, N-s(*) increases exponentially with N, especially for the quantum state with high entanglement entropy.