NISQ-compatible approximate quantum algorithm for unconstrained and constrained discrete optimization

Quantum algorithms are gaining extreme popularity due to their potential to significantly outperform classical algorithms. Yet, practical applications of quantum algorithms to optimization problems meet challenges related to the efficiency of the existing quantum algorithms training, the shape of their cost landscape, the accuracy of their output, and their ability to scale to large-size problems. Here, we present a gradient-based quantum algorithm for hardware efficient circuits with amplitude encoding. We show that simple linear constraints can be directly incorporated into the circuit without additional modification of the objective function with penalty terms. We employ numerical simulations to test it on MaxCut problems with complete weighted graphs with thousands of nodes and run the algorithm on a superconducting quantum processor. We find that being applied to unconstrained MaxCut problems with more than 1000 nodes, the hybrid approach combining our algorithm with a classical solver called CPLEX realizes a better solution than the CPLEX alone. This demonstrates that hybrid optimization is one of the leading use cases for modern quantum devices.