{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,4]],"date-time":"2025-12-04T10:09:09Z","timestamp":1764842949651,"version":"3.37.3"},"reference-count":23,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2024,11,25]],"date-time":"2024-11-25T00:00:00Z","timestamp":1732492800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/researchcouncilofnorwayrcn","name":"Research Council of Norway","doi-asserted-by":"publisher","award":["332023"],"award-info":[{"award-number":["332023"]}],"id":[{"id":"10.13039\/researchcouncilofnorwayrcn","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/researchcouncilofnorwayrcn","name":"Research Council of Norway","doi-asserted-by":"publisher","award":["324944"],"award-info":[{"award-number":["324944"]}],"id":[{"id":"10.13039\/researchcouncilofnorwayrcn","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"We present a novel formalism to both understand and construct mixers that preserve a given subspace. The method connects and utilizes the stabilizer formalism that is used in error correcting codes. This can be useful in the setting when the quantum approximate optimization algorithm (QAOA), a popular meta-heuristic for solving combinatorial optimization problems, is applied in the setting where the constraints of the problem lead to a feasible subspace that is large but easy to specify. The proposed method gives a systematic way to construct mixers that are resource efficient in the number of controlled not gates and can be understood as a generalization of the well-known X and XY mixers and a relaxation of the Grover mixer: Given a basis of any subspace, a resource efficient mixer can be constructed that preserves the subspace. The numerical examples provided show a dramatic reduction of CX gates when compared to previous results. We call our approach logical X-Mixer or logical X QAOA (LX-QAOA<\/mml:mtext><\/mml:mrow><\/mml:math>), since it can be understood as dividing the subspace into code spaces of stabilizers S and consecutively applying logical rotational X gates associated with these code spaces. Overall, we hope that this new perspective can lead to further insight into the development of quantum algorithms.<\/jats:p>","DOI":"10.22331\/q-2024-11-25-1535","type":"journal-article","created":{"date-parts":[[2024,11,25]],"date-time":"2024-11-25T13:01:27Z","timestamp":1732539687000},"page":"1535","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["LX-mixers for QAOA: Optimal mixers restricted to subspaces and the stabilizer formalism"],"prefix":"10.22331","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3558-503X","authenticated-orcid":false,"given":"Franz G.","family":"Fuchs","sequence":"first","affiliation":[{"name":"SINTEF AS, Department of Mathematics and Cybernetics, Oslo, Norway"},{"name":"Department of Mathematics, University of Oslo, Norway"}]},{"given":"Ruben Pariente","family":"Bassa","sequence":"additional","affiliation":[{"name":"SINTEF AS, Department of Mathematics and Cybernetics, Oslo, Norway"}]}],"member":"9598","published-online":{"date-parts":[[2024,11,25]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Andreas B\u00e4rtschiand Stephan Eidenbenz ``Grover mixers for QAOA: Shifting complexity from mixer design to state preparation'' 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) 72-82 (2020).","DOI":"10.1109\/QCE49297.2020.00020"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Sebastian Brandhofer, Daniel Braun, Vanessa Dehn, Gerhard Hellstern, Matthias H\u00fcls, Yanjun Ji, Ilia Polian, Amandeep Singh Bhatia, and Thomas Wellens, ``Benchmarking the performance of portfolio optimization with QAOA'' Quantum Information Processing 22, 1\u201327 (2023).","DOI":"10.1007\/s11128-022-03766-5"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Sergey Bravyi, Alexander Kliesch, Robert Koenig, and Eugene Tang, ``Obstacles to Variational Quantum Optimization from Symmetry Protection'' Phys. 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