{"type":"preprint","citation":{"chicago":"Medina Ramos, Raimel A, and Maksym Serbyn. “A Recursive Lower Bound on the Energy Improvement of the Quantum Approximate Optimization Algorithm.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2405.10125\">https://doi.org/10.48550/arXiv.2405.10125</a>.","apa":"Medina Ramos, R. A., &#38; Serbyn, M. (n.d.). A recursive lower bound on the energy improvement of the quantum approximate optimization algorithm. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2405.10125\">https://doi.org/10.48550/arXiv.2405.10125</a>","ista":"Medina Ramos RA, Serbyn M. A recursive lower bound on the energy improvement of the quantum approximate optimization algorithm. arXiv, 2405.10125.","short":"R.A. Medina Ramos, M. Serbyn, ArXiv (n.d.).","ieee":"R. A. Medina Ramos and M. Serbyn, “A recursive lower bound on the energy improvement of the quantum approximate optimization algorithm,” <i>arXiv</i>. .","ama":"Medina Ramos RA, Serbyn M. A recursive lower bound on the energy improvement of the quantum approximate optimization algorithm. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2405.10125\">10.48550/arXiv.2405.10125</a>","mla":"Medina Ramos, Raimel A., and Maksym Serbyn. “A Recursive Lower Bound on the Energy Improvement of the Quantum Approximate Optimization Algorithm.” <i>ArXiv</i>, 2405.10125, doi:<a href=\"https://doi.org/10.48550/arXiv.2405.10125\">10.48550/arXiv.2405.10125</a>."},"publication":"arXiv","date_created":"2024-07-10T13:12:09Z","day":"16","publication_status":"submitted","date_published":"2024-05-16T00:00:00Z","article_number":"2405.10125","external_id":{"arxiv":["2405.10125"]},"article_processing_charge":"No","_id":"17222","title":"A recursive lower bound on the energy improvement of the quantum approximate optimization algorithm","doi":"10.48550/arXiv.2405.10125","date_updated":"2024-07-17T11:14:24Z","month":"05","author":[{"id":"CE680B90-D85A-11E9-B684-C920E6697425","orcid":"0000-0002-5383-2869","last_name":"Medina Ramos","full_name":"Medina Ramos, Raimel A","first_name":"Raimel A"},{"full_name":"Serbyn, Maksym","last_name":"Serbyn","id":"47809E7E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2399-5827","first_name":"Maksym"}],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2405.10125","open_access":"1"}],"corr_author":"1","year":"2024","abstract":[{"lang":"eng","text":"The quantum approximate optimization algorithm (QAOA) uses a quantum computer\r\nto implement a variational method with $2p$ layers of alternating unitary\r\noperators, optimized by a classical computer to minimize a cost function. While\r\nrigorous performance guarantees exist for the QAOA at small depths $p$, the\r\nbehavior at large depths remains less clear, though simulations suggest\r\nexponentially fast convergence for certain problems. In this work, we gain\r\ninsights into the deep QAOA using an analytic expansion of the cost function\r\naround transition states. Transition states are constructed in a recursive\r\nmanner: from the local minima of the QAOA with $p$ layers we obtain transition\r\nstates of the QAOA with $p+1$ layers, which are stationary points characterized\r\nby a unique direction of negative curvature. We construct an analytic estimate\r\nof the negative curvature and the corresponding direction in parameter space at\r\neach transition state. The expansion of the QAOA cost function along the\r\nnegative direction to the quartic order gives a lower bound of the QAOA cost\r\nfunction improvement. We provide physical intuition behind the analytic\r\nexpressions for the local curvature and quartic expansion coefficient. Our\r\nnumerical study confirms the accuracy of our approximations and reveals that\r\nthe obtained bound and the true value of the QAOA cost function gain have a\r\ncharacteristic exponential decrease with the number of layers $p$, with the\r\nbound decreasing more rapidly. Our study establishes an analytical method for\r\nrecursively studying the QAOA that is applicable in the regime of high circuit\r\ndepth."}],"oa":1,"language":[{"iso":"eng"}],"related_material":{"record":[{"status":"for_moderation","relation":"dissertation_contains","id":"17208"}]},"status":"public","department":[{"_id":"MaSe"}],"oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"}