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Quantum algorithms are getting extremely popular due to their potential tosignificantly outperform classical algorithms. Yet, applying quantum algorithmsto optimization problems meets challenges related to the efficiency of quantumalgorithms training, the shape of their cost landscape, the accuracy of theiroutput, and their ability to scale to large-size problems. Here, we present anapproximate gradient-based quantum algorithm for hardware-efficient circuitswith amplitude encoding. We show how simple linear constraints can be directlyincorporated into the circuit without additional modification of the objectivefunction with penalty terms. We employ numerical simulations to test it onMaxCut problems with complete weighted graphs with thousands of nodes and runthe algorithm on a superconducting quantum processor. We find that forunconstrained MaxCut problems with more than 1000 nodes, the hybrid approachcombining our algorithm with a classical solver called CPLEX can find a bettersolution than CPLEX alone. This demonstrates that hybrid optimization is one ofthe leading use cases for modern quantum devices.

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