Transformer neural networks and quantum simulators: a hybrid approach for simulating strongly correlated systems

Owing to their great expressivity and versatility, neural networks have gained attention for simulating large two-dimensional quantum many-body systems. However, their expressivity comes with the cost of a challenging optimization due to the in general rugged and complicated loss landscape. Here, we present a hybrid optimization scheme for neural quantum states (NQS), involving a data-driven pretraining with numerical or experimental data and a second, Hamiltonian-driven optimization stage. By using both projective measurements from the computational basis as well as expectation values from other measurement configurations such as spin-spin correlations, our pretraining gives access to the sign structure of the state, yielding improved and faster convergence that is robust w.r.t. experimental imperfections and limited datasets. We apply the hybrid scheme to the ground state search for the 2D transverse field Ising model and dipolar XY model on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>6</mml:mn><mml:mo>&amp;#x00D7;</mml:mo><mml:mn>6</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>10</mml:mn><mml:mo>&amp;#x00D7;</mml:mo><mml:mn>10</mml:mn></mml:math> square lattices with a patched transformer wave function, using numerical data as well as experimental data from a programmable Rydberg quantum simulator [Chen et al., Nature 616 (2023)], and show that the information from a second measurement basis highly improves the performance. Our work paves the way for a reliable and efficient optimization of neural quantum states.