{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T16:42:45Z","timestamp":1775580165051,"version":"3.50.1"},"reference-count":48,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,1,27]],"date-time":"2022-01-27T00:00:00Z","timestamp":1643241600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000185","name":"Defense Advanced Research Projects Agency","doi-asserted-by":"crossref","award":["HR001120C0068"],"award-info":[{"award-number":["HR001120C0068"]}],"id":[{"id":"10.13039\/100000185","id-type":"DOI","asserted-by":"crossref"}]},{"name":"National Science Foundation","award":["PHY-1818914"],"award-info":[{"award-number":["PHY-1818914"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>The quantum approximate optimization algorithm (QAOA) is a near-term hybrid algorithm intended to solve combinatorial optimization problems, such as MaxCut. QAOA can be made to mimic an adiabatic schedule, and in the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>p<\/mml:mi><mml:mo stretchy=\"false\">&amp;#x2192;<\/mml:mo><mml:mi mathvariant=\"normal\">&amp;#x221E;<\/mml:mi><\/mml:math>limit the final state is an exact maximal eigenstate in accordance with the adiabatic theorem. In this work, the connection between QAOA and adiabaticity is made explicit by inspecting the regime of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>p<\/mml:mi><\/mml:math>large but finite. By connecting QAOA to counterdiabatic (CD) evolution, we construct CD-QAOA angles which mimic a counterdiabatic schedule by matching Trotter \"error\" terms to approximate adiabatic gauge potentials which suppress diabatic excitations arising from finite ramp speed. In our construction, these \"error\" terms are helpful, not detrimental, to QAOA. Using this matching to link QAOA with quantum adiabatic algorithms (QAA), we show that the approximation ratio converges to one at least as<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mn>1<\/mml:mn><mml:mo>&amp;#x2212;<\/mml:mo><mml:mi>C<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>p<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>&amp;#x223C;<\/mml:mo><mml:mn>1<\/mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>\/<\/mml:mo><\/mml:mrow><mml:msup><mml:mi>p<\/mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>&amp;#x03BC;<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math>. We show that transfer of parameters between graphs, and interpolating angles for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>p<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:math>given<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>p<\/mml:mi><\/mml:math>are both natural byproducts of CD-QAOA matching. Optimization of CD-QAOA angles is equivalent to optimizing a continuous adiabatic schedule. Finally, we show that, using a property of variational adiabatic gauge potentials, QAOA is at least counterdiabatic, not just adiabatic, and has better performance than finite time adiabatic evolution. We demonstrate the method on three examples: a 2 level system, an Ising chain, and the MaxCut problem.<\/jats:p>","DOI":"10.22331\/q-2022-01-27-635","type":"journal-article","created":{"date-parts":[[2022,1,27]],"date-time":"2022-01-27T14:19:21Z","timestamp":1643293161000},"page":"635","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":87,"title":["Counterdiabaticity and the quantum approximate optimization algorithm"],"prefix":"10.22331","volume":"6","author":[{"given":"Jonathan","family":"Wurtz","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA"}]},{"given":"Peter J.","family":"Love","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA"}]}],"member":"9598","published-online":{"date-parts":[[2022,1,27]]},"reference":[{"key":"0","unstructured":"E. Farhi, J. Goldstone, and S. Gutmann, (2014), arXiv:1411.4028 [quant-ph]."},{"key":"1","doi-asserted-by":"publisher","unstructured":"J. Preskill, Quantum 2, 79 (2018).","DOI":"10.22331\/q-2018-08-06-79"},{"key":"2","unstructured":"K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, (2021), arXiv:2101.08448 [quant-ph]."},{"key":"3","doi-asserted-by":"publisher","unstructured":"A. Lucas, Frontiers in Physics 2 (2014), 10.3389\/fphy.2014.00005.","DOI":"10.3389\/fphy.2014.00005"},{"key":"4","doi-asserted-by":"publisher","unstructured":"T. Albash and D. A. Lidar, Rev. Mod. Phys. 90, 015002 (2018).","DOI":"10.1103\/RevModPhys.90.015002"},{"key":"5","doi-asserted-by":"publisher","unstructured":"M. H. S. Amin, Phys. Rev. Lett. 102, 220401 (2009).","DOI":"10.1103\/PhysRevLett.102.220401"},{"key":"6","doi-asserted-by":"publisher","unstructured":"S. H. Sack and M. Serbyn, (2021), arXiv:2101.05742 [quant-ph].","DOI":"10.22331\/q-2021-07-01-491"},{"key":"7","doi-asserted-by":"publisher","unstructured":"D. Gu\u00e9ry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Mart\u00ednez-Garaot, and J. G. Muga, Rev. Mod. Phys. 91, 045001 (2019).","DOI":"10.1103\/RevModPhys.91.045001"},{"key":"8","doi-asserted-by":"publisher","unstructured":"G. Rigolin, G. Ortiz, and V. H. Ponce, Phys. Rev. A 78, 052508 (2008).","DOI":"10.1103\/PhysRevA.78.052508"},{"key":"9","doi-asserted-by":"publisher","unstructured":"S. Bachmann, W. De Roeck, and M. Fraas, Phys. Rev. Lett. 119, 060201 (2017).","DOI":"10.1103\/PhysRevLett.119.060201"},{"key":"10","doi-asserted-by":"publisher","unstructured":"A. del Campo and W. H. Zurek, International Journal of Modern Physics A 29, 1430018 (2014).","DOI":"10.1142\/s0217751x1430018x"},{"key":"11","doi-asserted-by":"publisher","unstructured":"J. I. Latorre and R. Or\u00fas, Phys. Rev. A 69, 062302 (2004).","DOI":"10.1103\/PhysRevA.69.062302"},{"key":"12","doi-asserted-by":"publisher","unstructured":"R. Barankov and A. Polkovnikov, Physical Review Letters 101 (2008), 10.1103\/physrevlett.101.076801.","DOI":"10.1103\/physrevlett.101.076801"},{"key":"13","unstructured":"B. F. Schiffer, J. Tura, and J. I. Cirac, (2021), arXiv:2103.01226 [quant-ph]."},{"key":"14","doi-asserted-by":"publisher","unstructured":"L. Zhou, S.-T. Wang, S. Choi, H. Pichler, and M. D. Lukin, Physical Review X 10 (2020), 10.1103\/physrevx.10.021067.","DOI":"10.1103\/physrevx.10.021067"},{"key":"15","unstructured":"F. G. S. L. Brandao, M. Broughton, E. Farhi, S. Gutmann, and H. Neven, (2018), arXiv:1812.04170 [quant-ph]."},{"key":"16","doi-asserted-by":"publisher","unstructured":"Z.-C. Yang, A. Rahmani, A. Shabani, H. Neven, and C. Chamon, Physical Review X 7 (2017), 10.1103\/physrevx.7.021027.","DOI":"10.1103\/physrevx.7.021027"},{"key":"17","doi-asserted-by":"publisher","unstructured":"L. T. Brady, C. L. Baldwin, A. Bapat, Y. Kharkov, and A. V. Gorshkov, Phys. Rev. Lett. 126, 070505 (2021a).","DOI":"10.1103\/PhysRevLett.126.070505"},{"key":"18","doi-asserted-by":"publisher","unstructured":"M. Streif and M. Leib, Quantum Science and Technology 5, 034008 (2020).","DOI":"10.1088\/2058-9565\/ab8c2b"},{"key":"19","doi-asserted-by":"publisher","unstructured":"M. Born and V. Fock, Zeitschrift f\u00fcr Physik 51, 165 (1928).","DOI":"10.1007\/BF01343193"},{"key":"20","unstructured":"E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, (2000), arXiv:quant-ph\/0001106 [quant-ph]."},{"key":"21","doi-asserted-by":"publisher","unstructured":"A. T. Rezakhani, W.-J. Kuo, A. Hamma, D. A. Lidar, and P. Zanardi, Phys. Rev. Lett. 103, 080502 (2009).","DOI":"10.1103\/PhysRevLett.103.080502"},{"key":"22","doi-asserted-by":"publisher","unstructured":"C. Brif, M. D. Grace, M. Sarovar, and K. C. Young, New Journal of Physics 16, 065013 (2014).","DOI":"10.1088\/1367-2630\/16\/6\/065013"},{"key":"23","unstructured":"L. T. Brady, L. Kocia, P. Bienias, A. Bapat, Y. Kharkov, and A. V. Gorshkov, (2021b), arXiv:2107.01218 [quant-ph]."},{"key":"24","doi-asserted-by":"publisher","unstructured":"S. Bao, S. Kleer, R. Wang, and A. Rahmani, Phys. Rev. A 97, 062343 (2018).","DOI":"10.1103\/PhysRevA.97.062343"},{"key":"25","doi-asserted-by":"publisher","unstructured":"T. Chasseur, L. S. Theis, Y. R. Sanders, D. J. Egger, and F. K. Wilhelm, Phys. Rev. A 91, 043421 (2015).","DOI":"10.1103\/PhysRevA.91.043421"},{"key":"26","doi-asserted-by":"publisher","unstructured":"P. W. Claeys, M. Pandey, D. Sels, and A. Polkovnikov, Phys. Rev. Lett. 123, 090602 (2019).","DOI":"10.1103\/PhysRevLett.123.090602"},{"key":"27","doi-asserted-by":"crossref","unstructured":"S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, 2011).","DOI":"10.1017\/CBO9780511973765"},{"key":"28","doi-asserted-by":"publisher","unstructured":"H. Nishimori and K. Takada, Frontiers in ICT 4, 2 (2017).","DOI":"10.3389\/fict.2017.00002"},{"key":"29","doi-asserted-by":"publisher","unstructured":"M. W. Johnson, M. H. S. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk, E. M. Chapple, C. Enderud, J. P. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, C. J. S. Truncik, S. Uchaikin, J. Wang, B. Wilson, and G. Rose, Nature 473, 194 (2011).","DOI":"10.1038\/nature10012"},{"key":"30","doi-asserted-by":"publisher","unstructured":"S. Sugiura, P. W. Claeys, A. Dymarsky, and A. Polkovnikov, Phys. Rev. Research 3, 013102 (2021).","DOI":"10.1103\/PhysRevResearch.3.013102"},{"key":"31","doi-asserted-by":"publisher","unstructured":"M. V. Berry, Journal of Physics A: Mathematical and Theoretical 42, 365303 (2009).","DOI":"10.1088\/1751-8113\/42\/36\/365303"},{"key":"32","doi-asserted-by":"publisher","unstructured":"M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, Physics Reports 697, 1 (2017).","DOI":"10.1016\/j.physrep.2017.07.001"},{"key":"33","doi-asserted-by":"publisher","unstructured":"M. Pandey, P. W. Claeys, D. K. Campbell, A. Polkovnikov, and D. Sels, Phys. Rev. X 10, 041017 (2020).","DOI":"10.1103\/PhysRevX.10.041017"},{"key":"34","doi-asserted-by":"publisher","unstructured":"D. Sels and A. Polkovnikov, Proceedings of the National Academy of Sciences 114, E3909 (2017).","DOI":"10.1073\/pnas.1619826114"},{"key":"35","doi-asserted-by":"publisher","unstructured":"M. B. Hastings and X.-G. Wen, Phys. Rev. B 72, 045141 (2005).","DOI":"10.1103\/PhysRevB.72.045141"},{"key":"36","doi-asserted-by":"publisher","unstructured":"J. Wurtz and A. Polkovnikov, Phys. Rev. B 101, 195138 (2020).","DOI":"10.1103\/PhysRevB.101.195138"},{"key":"37","doi-asserted-by":"publisher","unstructured":"N. Hatano and M. Suzuki, Lecture Notes in Physics , 37\u201368 (2005).","DOI":"10.1007\/11526216_2"},{"key":"38","doi-asserted-by":"publisher","unstructured":"S. Blanes, F. Casas, J. Oteo, and J. Ros, Physics Reports 470, 151\u2013238 (2009).","DOI":"10.1016\/j.physrep.2008.11.001"},{"key":"39","doi-asserted-by":"publisher","unstructured":"X. Chen, I. Lizuain, A. Ruschhaupt, D. Gu\u00e9ry-Odelin, and J. G. Muga, Phys. Rev. Lett. 105, 123003 (2010).","DOI":"10.1103\/PhysRevLett.105.123003"},{"key":"40","doi-asserted-by":"publisher","unstructured":"J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005).","DOI":"10.1103\/PhysRevLett.95.245701"},{"key":"41","doi-asserted-by":"publisher","unstructured":"M. Kolodrubetz, B. K. Clark, and D. A. Huse, Physical Review Letters 109 (2012), 10.1103\/physrevlett.109.015701.","DOI":"10.1103\/physrevlett.109.015701"},{"key":"42","unstructured":"A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum, and D. Sen, (2015), arXiv:1012.0653 [cond-mat.stat-mech]."},{"key":"43","doi-asserted-by":"publisher","unstructured":"D. J. Egger, J. Mare\u010dek, and S. Woerner, Quantum 5, 479 (2021).","DOI":"10.22331\/q-2021-06-17-479"},{"key":"44","doi-asserted-by":"publisher","unstructured":"A. G. R. Day, M. Bukov, P. Weinberg, P. Mehta, and D. Sels, Phys. Rev. Lett. 122, 020601 (2019).","DOI":"10.1103\/PhysRevLett.122.020601"},{"key":"45","unstructured":"G. B. Mbeng, R. Fazio, and G. Santoro, (2019), arXiv:1906.08948 [quant-ph]."},{"key":"46","unstructured":"V. Viswanath and G. M\u00fcller, The Recursion Method (Springer US, 2008)."},{"key":"47","doi-asserted-by":"publisher","unstructured":"J. Wurtz and P. Love, Phys. Rev. A 103, 042612 (2021).","DOI":"10.1103\/PhysRevA.103.042612"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2022-01-27-635\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,1,24]],"date-time":"2023-01-24T23:28:54Z","timestamp":1674602934000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2022-01-27-635\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,27]]},"references-count":48,"URL":"https:\/\/doi.org\/10.22331\/q-2022-01-27-635","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,27]]},"article-number":"635"}}