Fig. 1: Scheme for our generator-enhanced optimization (GEO) strategy.

The GEO framework leverages generative models to utilize previous samples coming from any quantum or classical solver. The trained quantum or classical generator is responsible for proposing candidate solutions which might be out of reach for conventional solvers. This seed data set (step 0) consists of observation bitstrings \({\{{{{{{{{{\boldsymbol{x}}}}}}}}}^{(i)}\}}_{{{{{{{{\rm{seed}}}}}}}}}\) and their respective costs \({\{{\sigma }^{(i)}\}}_{{{{{{{{\rm{seed}}}}}}}}}\). To give more weight to samples with low cost, the seed samples and their costs are used to construct a softmax function which serves as a surrogate to the cost function but in probabilistic domain. This softmax surrogate also serves as a prior distribution from which the training set samples are withdrawn to train the generative model (steps 1–3). As shown in the figure between steps 1 and 2, training samples from the softmax surrogate are biased favoring those with low cost value. For the work presented here, we implemented a tensor-network (TN)-based generative model. Therefore, we refer to this quantum-inspired instantiation of GEO as TN-GEO. Other families of generative models from classical, quantum, or hybrid quantum-classical can be explored as expounded in the main text. The quantum-inspired generator corresponds to a tensor-network Born machine (TNBM) model which is used to capture the main features in the training data, and to propose new solution candidates which are subsequently post-selected before their costs \({\{{\sigma }^{(i)}\}}_{{{{{{{{\rm{new}}}}}}}}}\) are evaluated (steps 4-6). The new set is merged with the seed data set (step 7) to form an updated seed data set (step 8) which is to be used in the next iteration of the algorithm. More algorithmic details for the two TN-GEO strategies proposed here, as a booster or as a stand-alone solver, can be found in the main text and in Supplementary Note 1.F and 1.G.