{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T14:53:36Z","timestamp":1776956016492,"version":"3.51.4"},"reference-count":35,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,10,10]],"date-time":"2023-10-10T00:00:00Z","timestamp":1696896000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>Run-times of quantum algorithms are often studied via an asymptotic, worst-case analysis. Whilst useful, such a comparison can often fall short: it is not uncommon for algorithms with a large worst-case run-time to end up performing well on instances of practical interest. To remedy this it is necessary to resort to run-time analyses of a more empirical nature, which for sufficiently small input sizes can be performed on a quantum device or a simulation thereof. For larger input sizes, alternative approaches are required.In this paper we consider an approach that combines classical emulation with detailed complexity bounds that include all constants. We simulate quantum algorithms by running classical versions of the sub-routines, whilst simultaneously collecting information about what the run-time of the quantum routine would have been if it were run instead. To do this accurately and efficiently for very large input sizes, we describe an estimation procedure and prove that it obtains upper bounds on the true expected complexity of the quantum algorithms.We apply our method to some simple quantum speedups of classical heuristic algorithms for solving the well-studied MAX-<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><\/mml:math>-SAT optimization problem. This requires rigorous bounds (including all constants) on the expected- and worst-case complexities of two important quantum sub-routines: Grover search with an unknown number of marked items, and quantum maximum-finding. These improve upon existing results and might be of broader interest.Amongst other results, we found that the classical heuristic algorithms we studied did not offer significant quantum speedups despite the existence of a theoretical per-step speedup. This suggests that an empirical analysis such as the one we implement in this paper already yields insights beyond those that can be seen by an asymptotic analysis alone.<\/jats:p>","DOI":"10.22331\/q-2023-10-10-1133","type":"journal-article","created":{"date-parts":[[2023,10,10]],"date-time":"2023-10-10T16:16:31Z","timestamp":1696954591000},"page":"1133","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":["Quantifying Grover speed-ups beyond asymptotic analysis"],"prefix":"10.22331","volume":"7","author":[{"given":"Chris","family":"Cade","sequence":"first","affiliation":[{"name":"QuSoft & University of Amsterdam (UvA), Amsterdam, the Netherlands"},{"name":"Fermioniq, Amsterdam, the Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marten","family":"Folkertsma","sequence":"additional","affiliation":[{"name":"QuSoft & CWI, Amsterdam, the Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ido","family":"Niesen","sequence":"additional","affiliation":[{"name":"QuSoft & University of Amsterdam (UvA), Amsterdam, the Netherlands"},{"name":"Fermioniq, Amsterdam, the Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jordi","family":"Weggemans","sequence":"additional","affiliation":[{"name":"QuSoft & CWI, Amsterdam, the Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2023,10,10]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Ashish Ahuja and Sanjiv Kapoor. A quantum algorithm for finding the maximum. arXiv:quant-ph\/9911082, 1999. https:\/\/doi.org\/10.48550\/arXiv.quant-ph\/9911082.","DOI":"10.48550\/arXiv.quant-ph\/9911082"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Haifa Hamad Alkasem and Mohamed El Bachir Menai. Stochastic local search for partial MAX-SAT: an experimental evaluation. Artificial Intelligence Review, 54:2525\u20132566, 2021. https:\/\/doi.org\/10.1007\/s10462-020-09908-4.","DOI":"10.1007\/s10462-020-09908-4"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Ryan Babbush, Jarrod R McClean, Michael Newman, Craig Gidney, Sergio Boixo, and Hartmut Neven. Focus beyond quadratic speedups for error-corrected quantum advantage. PRX Quantum, 2(1):010103, 2021. https:\/\/doi.org\/10.1103\/PRXQuantum.2.010103.","DOI":"10.1103\/PRXQuantum.2.010103"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Shai Ben-David, Benny Chor, Oded Goldreich, and Michel Luby. On the theory of average case complexity. Journal of Computer and system Sciences, 44(2):193\u2013219, 1992. https:\/\/doi.org\/10.1016\/0022-0000(92)90019-F.","DOI":"10.1016\/0022-0000(92)90019-F"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Benjamin Bichsel, Maximilian Baader, Timon Gehr, and Martin Vechev. Silq: A high-level quantum language with safe uncomputation and intuitive semantics. In ACM SIGPLAN Conference on Programming Language Design and Implementation, pages 286\u2013300, 2020. https:\/\/doi.org\/10.5281\/zenodo.3764961.","DOI":"10.5281\/zenodo.3764961"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Vincent D. Blondel, Jean-Loup Guillaume, Renaud Lambiotte, and Etienne Lefebvre. Fast unfolding of communities in large networks. Journal of Statistical Mechanics: Theory and Experiment, 2008:P10008, October 2008. https:\/\/doi.org\/10.1088\/1742-5468\/2008\/10\/P10008.","DOI":"10.1088\/1742-5468\/2008\/10\/P10008"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Michel Boyer, Gilles Brassard, Peter H\u00f8yer, and Alain Tapp. Tight bounds on quantum searching. Fortschritte der Physik: Progress of Physics, 46(4-5):493\u2013505, 1998. https:\/\/doi.org\/10.1002\/(SICI)1521-3978(199806)46:4\/5<493::AID-PROP493>3.0.CO;2-P.","DOI":"10.1002\/(SICI)1521-3978(199806)46:4\/5<493::AID-PROP493>3.0.CO;2-P"},{"key":"7","doi-asserted-by":"publisher","unstructured":"Chris Cade, Marten Folkertsma, Ido Niesen, and Jordi Weggemans. Quantum algorithms for community detection and their empirical run-times. arXiv:2203.06208, 2022. https:\/\/doi.org\/10.48550\/arXiv.2203.06208.","DOI":"10.48550\/arXiv.2203.06208"},{"key":"8","doi-asserted-by":"publisher","unstructured":"Chris Cade, Lana Mineh, Ashley Montanaro, and Stasja Stanisic. Strategies for solving the Fermi-Hubbard model on near-term quantum computers. Physical Review B, 102(23):235122, 2020. https:\/\/doi.org\/10.1103\/PhysRevB.102.235122.","DOI":"10.1103\/PhysRevB.102.235122"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Shaowei Cai, Chuan Luo, and Haochen Zhang. From decimation to local search and back: A new approach to MAX-SAT. In IJCAI, pages 571\u2013577, 2017. https:\/\/doi.org\/10.24963\/ijcai.2017\/80.","DOI":"10.24963\/ijcai.2017\/80"},{"key":"10","doi-asserted-by":"publisher","unstructured":"Earl Campbell, Ankur Khurana, and Ashley Montanaro. Applying quantum algorithms to constraint satisfaction problems. Quantum, 3:167, 2019. https:\/\/doi.org\/10.22331\/q-2019-07-18-167.","DOI":"10.22331\/q-2019-07-18-167"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Pierre-Luc Dallaire-Demers, Jonathan Romero, Libor Veis, Sukin Sim, and Al\u00e1n Aspuru-Guzik. Low-depth circuit ansatz for preparing correlated fermionic states on a quantum computer. Quantum Science and Technology, 4(4):045005, 2019. https:\/\/doi.org\/10.1088\/2058-9565\/ab3951.","DOI":"10.1088\/2058-9565\/ab3951"},{"key":"12","doi-asserted-by":"publisher","unstructured":"Pasquale De Meo, Emilio Ferrara, Giacomo Fiumara, and Alessandro Provetti. Generalized louvain method for community detection in large networks. In 2011 11th international conference on intelligent systems design and applications, pages 88\u201393. IEEE, 2011. https:\/\/doi.org\/10.1109\/ISDA.2011.6121636.","DOI":"10.1109\/ISDA.2011.6121636"},{"key":"13","doi-asserted-by":"publisher","unstructured":"Christoph Durr and Peter Hoyer. A quantum algorithm for finding the minimum. arXiv:quant-ph\/9607014, 1996. https:\/\/doi.org\/10.48550\/arXiv.quant-ph\/9607014.","DOI":"10.48550\/arXiv.quant-ph\/9607014"},{"key":"14","unstructured":"Andrew Hancock, Austin Garcia, Jacob Shedenhelm, Jordan Cowen, and Calista Carey. Cirq: A python framework for creating, editing, and invoking quantum circuits. URL https:\/\/github.com\/quantumlib\/Cirq, 2019."},{"key":"15","doi-asserted-by":"publisher","unstructured":"Peter H\u00f8yer. Arbitrary phases in quantum amplitude amplification. Physical Review A, 62(5):052304, 2000. https:\/\/doi.org\/10.1103\/PhysRevA.62.052304.","DOI":"10.1103\/PhysRevA.62.052304"},{"key":"16","unstructured":"Richard M Karp and J Michael Steele. Probabilistic analysis of heuristics. The traveling salesman problem, pages 181\u2013205, 1985."},{"key":"17","doi-asserted-by":"publisher","unstructured":"Joonho Lee, Dominic W Berry, Craig Gidney, William J Huggins, Jarrod R McClean, Nathan Wiebe, and Ryan Babbush. Even more efficient quantum computations of chemistry through tensor hypercontraction. PRX Quantum, 2(3):030305, 2021. https:\/\/doi.org\/10.1103\/PRXQuantum.2.030305.","DOI":"10.1103\/PRXQuantum.2.030305"},{"key":"18","doi-asserted-by":"publisher","unstructured":"Ashley Montanaro. Quantum-walk speedup of backtracking algorithms. Theory OF Computing, 14(15):1\u201324, 2018. http:\/\/dx.doi.org\/10.4086\/toc.2018.v014a015.","DOI":"10.4086\/toc.2018.v014a015"},{"key":"19","doi-asserted-by":"publisher","unstructured":"Xinyu Que, Fabio Checconi, Fabrizio Petrini, and John A Gunnels. Scalable community detection with the louvain algorithm. In 2015 IEEE International Parallel and Distributed Processing Symposium, pages 28\u201337. IEEE, 2015. https:\/\/doi.org\/10.1109\/IPDPS.2015.59.","DOI":"10.1109\/IPDPS.2015.59"},{"key":"20","doi-asserted-by":"publisher","unstructured":"Maria Schuld and Nathan Killoran. Is quantum advantage the right goal for quantum machine learning? Prx Quantum, 3(3):030101, 2022. https:\/\/doi.org\/10.1103\/PRXQuantum.3.030101.","DOI":"10.1103\/PRXQuantum.3.030101"},{"key":"21","doi-asserted-by":"publisher","unstructured":"Daniel A Spielman and Shang-Hua Teng. Smoothed analysis: an attempt to explain the behavior of algorithms in practice. Communications of the ACM, 52(10):76\u201384, 2009. https:\/\/doi.org\/10.1145\/1562764.1562785.","DOI":"10.1145\/1562764.1562785"},{"key":"22","doi-asserted-by":"publisher","unstructured":"Damian S Steiger, Thomas H\u00e4ner, and Matthias Troyer. ProjectQ: an open source software framework for quantum computing. Quantum, 2:49, 2018. https:\/\/doi.org\/10.22331\/q-2018-01-31-49.","DOI":"10.22331\/q-2018-01-31-49"},{"key":"23","doi-asserted-by":"publisher","unstructured":"EM Stoudenmire and Xavier Waintal. Grover&apos;s algorithm offers no quantum advantage. arXiv:2303.11317, 2023. https:\/\/doi.org\/10.48550\/arXiv.2303.11317.","DOI":"10.48550\/arXiv.2303.11317"},{"key":"24","unstructured":"Thomas St\u00fctzle, Holger H. Hoos, and Andrea Roli. A review of the literature on local search algorithms for MAX-SAT. 2001."},{"key":"25","doi-asserted-by":"publisher","unstructured":"Krysta Svore, Alan Geller, Matthias Troyer, John Azariah, Christopher Granade, Bettina Heim, Vadym Kliuchnikov, Mariia Mykhailova, Andres Paz, and Martin Roetteler. Q# enabling scalable quantum computing and development with a high-level DSL. In Proceedings of the Real World Domain Specific Languages Workshop 2018, pages 1\u201310, 2018. https:\/\/doi.org\/10.1145\/3183895.3183901.","DOI":"10.1145\/3183895.3183901"},{"key":"26","doi-asserted-by":"publisher","unstructured":"V. A. Traag, L. Waltman, and N. J. van Eck. From Louvain to Leiden: guaranteeing well-connected communities. Scientific Reports, 9:5233, March 2019. https:\/\/doi.org\/10.1038\/s41598-019-41695-z.","DOI":"10.1038\/s41598-019-41695-z"},{"key":"27","doi-asserted-by":"publisher","unstructured":"Vera von Burg, Guang Hao Low, Thomas H\u00e4ner, Damian S Steiger, Markus Reiher, Martin Roetteler, and Matthias Troyer. Quantum computing enhanced computational catalysis. Physical Review Research, 3(3):033055, 2021. https:\/\/doi.org\/10.1103\/PhysRevResearch.3.033055.","DOI":"10.1103\/PhysRevResearch.3.033055"},{"key":"28","doi-asserted-by":"publisher","unstructured":"Dave Wecker, Matthew B Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms. Physical Review A, 92(4):042303, 2015. https:\/\/doi.org\/10.1103\/PhysRevA.92.042303.","DOI":"10.1103\/PhysRevA.92.042303"},{"key":"29","doi-asserted-by":"publisher","unstructured":"Jordi R Weggemans, Alexander Urech, Alexander Rausch, Robert Spreeuw, Richard Boucherie, Florian Schreck, Kareljan Schoutens, Ji\u0159\u00ed Min\u00e1\u0159, and Florian Speelman. Solving correlation clustering with QAOA and a Rydberg qudit system: a full-stack approach. Quantum, 6:687, 2022. https:\/\/doi.org\/10.22331\/q-2022-04-13-687.","DOI":"10.22331\/q-2022-04-13-687"},{"key":"30","unstructured":"James D Whitfield, Jacob Biamonte, and Al\u00e1n Aspuru-Guzik. Quantum computing resource estimate of molecular energy simulation. Science, 309:1704, 2005."},{"key":"31","doi-asserted-by":"publisher","unstructured":"Robert Wille, Rod Van Meter, and Yehuda Naveh. IBM\u2019s Qiskit tool chain: Working with and developing for real quantum computers. In 2019 Design, Automation & Test in Europe Conference & Exhibition, pages 1234\u20131240. IEEE, 2019. https:\/\/doi.org\/10.23919\/DATE.2019.8715261.","DOI":"10.23919\/DATE.2019.8715261"},{"key":"32","doi-asserted-by":"publisher","unstructured":"Margaret Wright. The interior-point revolution in optimization: history, recent developments, and lasting consequences. Bulletin of the American mathematical society, 42(1):39\u201356, 2005. https:\/\/doi.org\/10.1090\/S0273-0979-04-01040-7.","DOI":"10.1090\/S0273-0979-04-01040-7"},{"key":"33","doi-asserted-by":"publisher","unstructured":"Christof Zalka. A Grover-based quantum search of optimal order for an unknown number of marked elements. arXiv:quant-ph\/9902049, 1999. https:\/\/doi.org\/10.48550\/arXiv.quant-ph\/9902049.","DOI":"10.48550\/arXiv.quant-ph\/9902049"},{"key":"34","doi-asserted-by":"publisher","unstructured":"Leo Zhou, Sheng-Tao Wang, Soonwon Choi, Hannes Pichler, and Mikhail D Lukin. Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices. Physical Review X, 10(2):021067, 2020. https:\/\/doi.org\/10.1103\/PhysRevX.10.021067.","DOI":"10.1103\/PhysRevX.10.021067"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2023-10-10-1133\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,10,10]],"date-time":"2023-10-10T16:16:43Z","timestamp":1696954603000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2023-10-10-1133\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,10]]},"references-count":35,"URL":"https:\/\/doi.org\/10.22331\/q-2023-10-10-1133","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,10]]},"article-number":"1133"}}