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Yet, applying quantum algorithms to optimization problems meets challenges related to the efficiency of 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 an approximate gradient-based quantum algorithm for hardware-efficient circuits with amplitude encoding. We show how 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<\/mml:mtext><\/mml:mrow><\/mml:math> problems with complete weighted graphs with thousands of nodes and run the algorithm on a superconducting quantum processor. We find that for unconstrained MaxCut<\/mml:mtext><\/mml:mrow><\/mml:math> problems with more than 1000 nodes, the hybrid approach combining our algorithm with a classical solver called CPLEX can find a better solution than CPLEX alone. This demonstrates that hybrid optimization is one of the leading use cases for modern quantum devices.<\/jats:p>","DOI":"10.22331\/q-2023-11-21-1186","type":"journal-article","created":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T14:25:08Z","timestamp":1700576708000},"page":"1186","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":8,"title":["NISQ-compatible approximate quantum algorithm for unconstrained and constrained discrete optimization"],"prefix":"10.22331","volume":"7","author":[{"given":"M. R.","family":"Perelshtein","sequence":"first","affiliation":[{"name":"Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland"},{"name":"QTF Centre of Excellence, Department of Applied Physics, Aalto University, P.O. Box 15100, FI-00076 AALTO, Finland"},{"name":"InstituteQ \u2013 the Finnish Quantum Institute, Aalto University, Finland"}]},{"given":"A. I.","family":"Pakhomchik","sequence":"additional","affiliation":[{"name":"Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland"}]},{"given":"Ar. 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