  We introduce the generative quantum eigensolver (GQE), a novel method for
applying classical generative models for quantum simulation. The GQE algorithm
optimizes a classical generative model to produce quantum circuits with desired
properties. Here, we develop a transformer-based implementation, which we name
the generative pre-trained transformer-based (GPT) quantum eigensolver
(GPT-QE), leveraging both pre-training on existing datasets and training
without any prior knowledge. We demonstrate the effectiveness of training and
pre-training GPT-QE in the search for ground states of electronic structure
Hamiltonians. GQE strategies can extend beyond the problem of Hamiltonian
simulation into other application areas of quantum computing.
