{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T18:13:34Z","timestamp":1785521614130,"version":"3.56.0"},"reference-count":63,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,21]],"date-time":"2023-08-21T00:00:00Z","timestamp":1692576000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"U.S. Department of Energy (DOE), Office of Science","award":["DE-FG02-87ER40371"],"award-info":[{"award-number":["DE-FG02-87ER40371"]}]},{"name":"U.S. Department of Energy (DOE), Office of Science","award":["DE-SC0023692"],"award-info":[{"award-number":["DE-SC0023692"]}]},{"name":"U.S. Department of Energy (DOE), Office of Science","award":["DE-SC0023707"],"award-info":[{"award-number":["DE-SC0023707"]}]},{"name":"U.S. Department of Energy (DOE), Office of Science","award":["CEX2020-001035-M"],"award-info":[{"award-number":["CEX2020-001035-M"]}]},{"name":"U.S. Department of Energy (DOE), Office of Science","award":["PID2020-119632GB-I00"],"award-info":[{"award-number":["PID2020-119632GB-I00"]}]},{"name":"U.S. Department of Energy (DOE), Office of Science","award":["ERC-2018-ADG-835105"],"award-info":[{"award-number":["ERC-2018-ADG-835105"]}]},{"name":"U.S. Department of Energy (DOE), Office of Science","award":["101109293"],"award-info":[{"award-number":["101109293"]}]},{"name":"Palmer Department Chair Endowment at Iowa State University","award":["DE-FG02-87ER40371"],"award-info":[{"award-number":["DE-FG02-87ER40371"]}]},{"name":"Palmer Department Chair Endowment at Iowa State University","award":["DE-SC0023692"],"award-info":[{"award-number":["DE-SC0023692"]}]},{"name":"Palmer Department Chair Endowment at Iowa State University","award":["DE-SC0023707"],"award-info":[{"award-number":["DE-SC0023707"]}]},{"name":"Palmer Department Chair Endowment at Iowa State University","award":["CEX2020-001035-M"],"award-info":[{"award-number":["CEX2020-001035-M"]}]},{"name":"Palmer Department Chair Endowment at Iowa State University","award":["PID2020-119632GB-I00"],"award-info":[{"award-number":["PID2020-119632GB-I00"]}]},{"name":"Palmer Department Chair Endowment at Iowa State University","award":["ERC-2018-ADG-835105"],"award-info":[{"award-number":["ERC-2018-ADG-835105"]}]},{"name":"Palmer Department Chair Endowment at Iowa State University","award":["101109293"],"award-info":[{"award-number":["101109293"]}]},{"name":"Xunta de Galicia (Centro singular de investigacion de Galicia accreditation 2019\u20132022)","award":["DE-FG02-87ER40371"],"award-info":[{"award-number":["DE-FG02-87ER40371"]}]},{"name":"Xunta de Galicia (Centro singular de investigacion de Galicia accreditation 2019\u20132022)","award":["DE-SC0023692"],"award-info":[{"award-number":["DE-SC0023692"]}]},{"name":"Xunta de Galicia (Centro singular de investigacion de Galicia accreditation 2019\u20132022)","award":["DE-SC0023707"],"award-info":[{"award-number":["DE-SC0023707"]}]},{"name":"Xunta de Galicia (Centro singular de investigacion de Galicia accreditation 2019\u20132022)","award":["CEX2020-001035-M"],"award-info":[{"award-number":["CEX2020-001035-M"]}]},{"name":"Xunta de Galicia (Centro singular de investigacion de Galicia accreditation 2019\u20132022)","award":["PID2020-119632GB-I00"],"award-info":[{"award-number":["PID2020-119632GB-I00"]}]},{"name":"Xunta de Galicia (Centro singular de investigacion de Galicia accreditation 2019\u20132022)","award":["ERC-2018-ADG-835105"],"award-info":[{"award-number":["ERC-2018-ADG-835105"]}]},{"name":"Xunta de Galicia (Centro singular de investigacion de Galicia accreditation 2019\u20132022)","award":["101109293"],"award-info":[{"award-number":["101109293"]}]},{"name":"Spanish Research State Agency","award":["DE-FG02-87ER40371"],"award-info":[{"award-number":["DE-FG02-87ER40371"]}]},{"name":"Spanish Research State Agency","award":["DE-SC0023692"],"award-info":[{"award-number":["DE-SC0023692"]}]},{"name":"Spanish Research State Agency","award":["DE-SC0023707"],"award-info":[{"award-number":["DE-SC0023707"]}]},{"name":"Spanish Research State Agency","award":["CEX2020-001035-M"],"award-info":[{"award-number":["CEX2020-001035-M"]}]},{"name":"Spanish Research State Agency","award":["PID2020-119632GB-I00"],"award-info":[{"award-number":["PID2020-119632GB-I00"]}]},{"name":"Spanish Research State Agency","award":["ERC-2018-ADG-835105"],"award-info":[{"award-number":["ERC-2018-ADG-835105"]}]},{"name":"Spanish Research State Agency","award":["101109293"],"award-info":[{"award-number":["101109293"]}]},{"name":"European Research Council","award":["DE-FG02-87ER40371"],"award-info":[{"award-number":["DE-FG02-87ER40371"]}]},{"name":"European Research Council","award":["DE-SC0023692"],"award-info":[{"award-number":["DE-SC0023692"]}]},{"name":"European Research Council","award":["DE-SC0023707"],"award-info":[{"award-number":["DE-SC0023707"]}]},{"name":"European Research Council","award":["CEX2020-001035-M"],"award-info":[{"award-number":["CEX2020-001035-M"]}]},{"name":"European Research Council","award":["PID2020-119632GB-I00"],"award-info":[{"award-number":["PID2020-119632GB-I00"]}]},{"name":"European Research Council","award":["ERC-2018-ADG-835105"],"award-info":[{"award-number":["ERC-2018-ADG-835105"]}]},{"name":"European Research Council","award":["101109293"],"award-info":[{"award-number":["101109293"]}]},{"name":"European Union\u2019s MSCA Postdoctoral Fellowships HORIZON-MSCA-2022-PF-01","award":["DE-FG02-87ER40371"],"award-info":[{"award-number":["DE-FG02-87ER40371"]}]},{"name":"European Union\u2019s MSCA Postdoctoral Fellowships HORIZON-MSCA-2022-PF-01","award":["DE-SC0023692"],"award-info":[{"award-number":["DE-SC0023692"]}]},{"name":"European Union\u2019s MSCA Postdoctoral Fellowships HORIZON-MSCA-2022-PF-01","award":["DE-SC0023707"],"award-info":[{"award-number":["DE-SC0023707"]}]},{"name":"European Union\u2019s MSCA Postdoctoral Fellowships HORIZON-MSCA-2022-PF-01","award":["CEX2020-001035-M"],"award-info":[{"award-number":["CEX2020-001035-M"]}]},{"name":"European Union\u2019s MSCA Postdoctoral Fellowships HORIZON-MSCA-2022-PF-01","award":["PID2020-119632GB-I00"],"award-info":[{"award-number":["PID2020-119632GB-I00"]}]},{"name":"European Union\u2019s MSCA Postdoctoral Fellowships HORIZON-MSCA-2022-PF-01","award":["ERC-2018-ADG-835105"],"award-info":[{"award-number":["ERC-2018-ADG-835105"]}]},{"name":"European Union\u2019s MSCA Postdoctoral Fellowships HORIZON-MSCA-2022-PF-01","award":["101109293"],"award-info":[{"award-number":["101109293"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The traveling salesman problem (TSP) is one of the most often-used NP-hard problems in computer science to study the effectiveness of computing models and hardware platforms. In this regard, it is also heavily used as a vehicle to study the feasibility of the quantum computing paradigm for this class of problems. In this paper, we tackle the TSP using the quantum approximate optimization algorithm (QAOA) approach by formulating it as an optimization problem. By adopting an improved qubit encoding strategy and a layer-wise learning optimization protocol, we present numerical results obtained from the gate-based digital quantum simulator, specifically targeting TSP instances with 3, 4, and 5 cities. We focus on the evaluations of three distinctive QAOA mixer designs, considering their performances in terms of numerical accuracy and optimization cost. Notably, we find that a well-balanced QAOA mixer design exhibits more promising potential for gate-based simulators and realistic quantum devices in the long run, an observation further supported by our noise model simulations. Furthermore, we investigate the sensitivity of the simulations to the TSP graph. Overall, our simulation results show that the digital quantum simulation of problem-inspired ansatz is a successful candidate for finding optimal TSP solutions.<\/jats:p>","DOI":"10.3390\/e25081238","type":"journal-article","created":{"date-parts":[[2023,8,21]],"date-time":"2023-08-21T08:53:31Z","timestamp":1692608011000},"page":"1238","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":37,"title":["Comparative Study of Variations in Quantum Approximate Optimization Algorithms for the Traveling Salesman Problem"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5525-0996","authenticated-orcid":false,"given":"Wenyang","family":"Qian","sequence":"first","affiliation":[{"name":"Instituto Galego de Fisica de Altas Enerxias (IGFAE), Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain"},{"name":"Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Robert A. M.","family":"Basili","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mary Mehrnoosh","family":"Eshaghian-Wilner","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Iowa State University, Ames, IA 50011, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ashfaq","family":"Khokhar","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Glenn","family":"Luecke","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Iowa State University, Ames, IA 50011, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3500-4314","authenticated-orcid":false,"given":"James P.","family":"Vary","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,21]]},"reference":[{"key":"ref_1","unstructured":"Biggs, N., Lloyd, E.K., and Wilson, R.J. (1986). Graph Theory, 1736\u20131936, Clarendon Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1126\/science.220.4598.671","article-title":"Optimization by Simulated Annealing","volume":"220","author":"Kirkpatrick","year":"1983","journal-title":"Science"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0925-2312(98)00030-7","article-title":"The self-organizing map","volume":"21","author":"Kohonen","year":"1998","journal-title":"Neurocomputing"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ambainis, A., Balodis, K., Iraids, J., Kokainis, M., Pr\u016bsis, K., and Vihrovs, J. (2019, January 6\u20139). Quantum Speedups for Exponential-Time Dynamic Programming Algorithms. Proceedings of the SODA \u201919: Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, San Diego, CA, USA.","DOI":"10.1137\/1.9781611975482.107"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/0009-2614(94)00117-0","article-title":"Quantum annealing: A new method for minimizing multidimensional functions","volume":"219","author":"Finnila","year":"1994","journal-title":"Chem. Phys. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"5355","DOI":"10.1103\/PhysRevE.58.5355","article-title":"Quantum annealing in the transverse Ising model","volume":"58","author":"Kadowaki","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_7","unstructured":"Warren, R.H. (2021). Solving combinatorial problems by two D-Wave hybrid solvers: A case study of traveling salesman problems in the TSP Library. arXiv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"760783","DOI":"10.3389\/fphy.2021.760783","article-title":"Solving the Traveling Salesman Problem on the D-Wave Quantum Computer","volume":"9","author":"Jain","year":"2021","journal-title":"Front. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Villar-Rodriguez, E., Osaba, E., and Oregi, I. (2022, January 4\u20137). Analyzing the behaviour of D\u2019WAVE quantum annealer: Fine-tuning parameterization and tests with restrictive Hamiltonian formulations. Proceedings of the 2022 IEEE Symposium Series on Computational Intelligence (SSCI), Singapore.","DOI":"10.1109\/SSCI51031.2022.10022300"},{"key":"ref_10","unstructured":"Farhi, E., Goldstone, J., and Gutmann, S. (2014). A quantum approximate optimization algorithm. arXiv."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1038\/nature23879","article-title":"Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets","volume":"549","author":"Kandala","year":"2017","journal-title":"Nature"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"043193","DOI":"10.1103\/PhysRevResearch.4.043193","article-title":"Solving hadron structures using the basis light-front quantization approach on quantum computers","volume":"4","author":"Qian","year":"2022","journal-title":"Phys. Rev. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3101724","DOI":"10.1109\/TQE.2020.3030314","article-title":"Quantum Computing for Finance: State-of-the-Art and Future Prospects","volume":"1","author":"Egger","year":"2020","journal-title":"IEEE Trans. Quantum Eng."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Preskill, J. (2018). Quantum Computing in the NISQ era and beyond. arXiv.","DOI":"10.22331\/q-2018-08-06-79"},{"key":"ref_15","first-page":"021067","article-title":"Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices","volume":"10","author":"Zhou","year":"2020","journal-title":"Phys. Rev. X"},{"key":"ref_16","unstructured":"Mesman, K., Al-Ars, Z., and M\u00f6ller, M. (2021). QPack: Quantum Approximate Optimization Algorithms as universal benchmark for quantum computers. arXiv."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1038\/s41567-020-01105-y","article-title":"Quantum approximate optimization of non-planar graph problems on a planar superconducting processor","volume":"17","author":"Harrigan","year":"2021","journal-title":"Nat. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Cook, J., Eidenbenz, S., and B\u00e4rtschi, A. (2020, January 12\u201316). The Quantum Alternating Operator Ansatz on Maximum k-Vertex Cover. Proceedings of the 2020 IEEE International Conference on Quantum Computing and Engineering (QCE), Denver, CO, USA.","DOI":"10.1109\/QCE49297.2020.00021"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"7564","DOI":"10.1109\/TITS.2022.3172241","article-title":"Solving Vehicle Routing Problem Using Quantum Approximate Optimization Algorithm","volume":"24","author":"Azad","year":"2022","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Sarkar, A., Al-Ars, Z., and Bertels, K. (2021). QuASeR: Quantum Accelerated de novo DNA sequence reconstruction. PLoS ONE, 16.","DOI":"10.1371\/journal.pone.0249850"},{"key":"ref_21","unstructured":"Fingerhuth, M., Babej, T., and Ing, C. (2018). A quantum alternating operator ansatz with hard and soft constraints for lattice protein folding. arXiv."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Khumalo, M.T., Chieza, H.A., Prag, K., and Woolway, M. (2022). An investigation of IBM Quantum Computing device performance on Combinatorial Optimisation Problems. Neural Comput. Appl., 1\u201316.","DOI":"10.1007\/s00521-022-07438-4"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Hadfield, S., Wang, Z., O\u2019Gorman, B., Rieffel, E.G., Venturelli, D., and Biswas, R. (2019). From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz. Algorithms, 12.","DOI":"10.3390\/a12020034"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Hadfield, S., Wang, Z., Rieffel, E.G., O\u2019Gorman, B., Venturelli, D., and Biswas, R. (2017, January 12\u201317). Quantum Approximate Optimization with Hard and Soft Constraints. Proceedings of the Second International Workshop on Post Moores Era Supercomputing, Denver, CO, USA.","DOI":"10.1145\/3149526.3149530"},{"key":"ref_25","unstructured":"Streif, M., and Leib, M. (2019). Comparison of QAOA with quantum and simulated annealing. arXiv."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"057701","DOI":"10.1103\/PhysRevE.70.057701","article-title":"Quantum annealing of the traveling-salesman problem","volume":"70","author":"Santoro","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"062317","DOI":"10.1103\/PhysRevA.95.062317","article-title":"Near-optimal quantum circuit for Grover\u2019s unstructured search using a transverse field","volume":"95","author":"Jiang","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/s42484-020-00036-4","article-title":"Layerwise learning for quantum neural networks","volume":"3","author":"Skolik","year":"2021","journal-title":"Quantum Mach. Intell."},{"key":"ref_29","unstructured":"ANIS, M.S., Abraham, H., Agarwal, R., Agliardi, G., Aharoni, M., Akhalwaya, I.Y., Aleksandrowicz, G., Alexander, T., and Amy, M. (2023, August 18). Qiskit: An Open-source Framework for Quantum Computing, 2021. Available online: https:\/\/zenodo.org\/record\/8190968."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5","DOI":"10.3389\/fphy.2014.00005","article-title":"Ising formulations of many NP problems","volume":"2","author":"Lucas","year":"2014","journal-title":"Front. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1145\/321043.321046","article-title":"Integer Programming Formulation of Traveling Salesman Problems","volume":"7","author":"Miller","year":"1960","journal-title":"J. ACM"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Gonzalez-Bermejo, S., Alonso-Linaje, G., and Atchade-Adelomou, P. (2022). GPS: A New TSP Formulation for Its Generalizations Type QUBO. Mathematics, 10.","DOI":"10.3390\/math10030416"},{"key":"ref_33","unstructured":"Zhu, J., Gao, Y., Wang, H., Li, T., and Wu, H. (2022). A Realizable GAS-Based Quantum Algorithm for Traveling Salesman Problem. arXiv."},{"key":"ref_34","unstructured":"Glos, A., Krawiec, A., and Zimbor\u00e1s, Z. (2020). Space-efficient binary optimization for variational computing. arXiv."},{"key":"ref_35","unstructured":"Bak\u00f3, B., Glos, A., Salehi, O., and Zimbor\u00e1s, Z. (2022). Near-Optimal Circuit Design for Variational Quantum Optimization. arXiv."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"015002","DOI":"10.1103\/RevModPhys.90.015002","article-title":"Adiabatic quantum computation","volume":"90","author":"Albash","year":"2018","journal-title":"Rev. Mod. Phys."},{"key":"ref_37","unstructured":"Blekos, K., Brand, D., Ceschini, A., Chou, C.H., Li, R.H., Pandya, K., and Summer, A. (2023). A Review on Quantum Approximate Optimization Algorithm and Its Variants. arXiv."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-022-10555-8","article-title":"Multi-angle quantum approximate optimization algorithm","volume":"12","author":"Herrman","year":"2022","journal-title":"Sci. Rep."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"013141","DOI":"10.1103\/PhysRevResearch.4.013141","article-title":"Digitized-counterdiabatic quantum approximate optimization algorithm","volume":"4","author":"Chandarana","year":"2022","journal-title":"Phys. Rev. Res."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"635","DOI":"10.22331\/q-2022-01-27-635","article-title":"Counterdiabaticity and the quantum approximate optimization algorithm","volume":"6","author":"Wurtz","year":"2022","journal-title":"Quantum"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"033029","DOI":"10.1103\/PhysRevResearch.4.033029","article-title":"Adaptive quantum approximate optimization algorithm for solving combinatorial problems on a quantum computer","volume":"4","author":"Zhu","year":"2022","journal-title":"Phys. Rev. Res."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3478519","article-title":"On the Representation of Boolean and Real Functions as Hamiltonians for Quantum Computing","volume":"2","author":"Hadfield","year":"2021","journal-title":"ACM Trans. Quantum Comput."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1900015","DOI":"10.1002\/qute.201900015","article-title":"Efficient quantum algorithms for GHZ and W states, and implementation on the IBM quantum computer","volume":"2","author":"Cruz","year":"2019","journal-title":"Adv. Quantum Technol."},{"key":"ref_44","unstructured":"Diker, F. (2016). Deterministic construction of arbitrary W states with quadratically increasing number of two-qubit gates. arXiv."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"114","DOI":"10.22331\/q-2018-12-21-114","article-title":"Exact Ising model simulation on a quantum computer","volume":"2","year":"2018","journal-title":"Quantum"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s42484-022-00069-x","article-title":"Mixer-phaser Ans\u00e4tze for quantum optimization with hard constraints","volume":"4","author":"LaRose","year":"2022","journal-title":"Quantum Mach. Intell."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"B\u00e4rtschi, A., and Eidenbenz, S. (2020, January 12\u201316). Grover Mixers for QAOA: Shifting Complexity from Mixer Design to State Preparation. Proceedings of the 2020 IEEE International Conference on Quantum Computing and Engineering (QCE), Denver, CO, USA.","DOI":"10.1109\/QCE49297.2020.00020"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"012320","DOI":"10.1103\/PhysRevA.101.012320","article-title":"XY mixers: Analytical and numerical results for the quantum alternating operator ansatz","volume":"101","author":"Wang","year":"2020","journal-title":"Phys. Rev. A"},{"key":"ref_49","unstructured":"Borgsten, C. (2021). Quantum Approximate Optimization Using SWAP Gates for Mixing. [Ph.D Thesis, Chalmers University of Technology]."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1017\/S0962492900002841","article-title":"Direct search algorithms for optimization calculations","volume":"7","author":"Powell","year":"1998","journal-title":"Acta Numer."},{"key":"ref_51","first-page":"1","article-title":"A View of Algorithms for Optimization without Derivatives","volume":"43","author":"Powell","year":"2007","journal-title":"Math. Today"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Gomez, S., and Hennart, J.P. (1994). Advances in Optimization and Numerical Analysis, Springer.","DOI":"10.1007\/978-94-015-8330-5"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1109\/9.119632","article-title":"Multivariate stochastic approximation using a simultaneous perturbation gradient approximation","volume":"37","author":"Spall","year":"1992","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_54","unstructured":"Spall, J. (1997, January 12). Accelerated second-order stochastic optimization using only function measurements. Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA, USA."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"710","DOI":"10.22331\/q-2022-05-11-710","article-title":"Approaching the theoretical limit in quantum gate decomposition","volume":"6","author":"Rakyta","year":"2022","journal-title":"Quantum"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"L030401","DOI":"10.1103\/PhysRevA.104.L030401","article-title":"Training saturation in layerwise quantum approximate optimization","volume":"104","author":"Campos","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_57","unstructured":"Atsushi, M., Yudai, S., and Shigeru, Y. (2020). Problem-specific Parameterized Quantum Circuits of the VQE Algorithm for Optimization Problems. arXiv."},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Basili, R., Qian, W., Tang, S., Castellino, A., Eshaghian-Wilner, M., Khokhar, A., Luecke, G., and Vary, J.P. (2022\u20133, January 30). Performance Evaluations of Noisy Approximate Quantum Fourier Arithmetic. Proceedings of the 2022 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW), Lyon, France.","DOI":"10.1109\/IPDPSW55747.2022.00081"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1038\/s41928-020-00498-1","article-title":"Implementation of XY entangling gates with a single calibrated pulse","volume":"3","author":"Abrams","year":"2020","journal-title":"Nat. Electron."},{"key":"ref_60","first-page":"183","article-title":"Comparing measures of sample skewness and kurtosis","volume":"47","author":"Joanes","year":"1998","journal-title":"J. R. Stat. Soc. Ser. D (Stat.)"},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Kokoska, S., and Zwillinger, D. (2000). CRC Standard Probability and Statistics Tables and Formulae, CRC Press.","DOI":"10.1201\/b16923"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"263","DOI":"10.22331\/q-2020-05-11-263","article-title":"An Adaptive Optimizer for Measurement-Frugal Variational Algorithms","volume":"4","author":"Arrasmith","year":"2020","journal-title":"Quantum"},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Qian, W., Basili, R., Eshaghian-Wilner, M., Khokhar, A., Luecke, G., and Vary, J.P. (2023, January 15\u201319). Comparative study on the variations of quantum approximate optimization algorithms to the Traveling Salesman Problem. Proceedings of the 2023 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPS), St. Petersburg, FL, USA.","DOI":"10.1109\/IPDPSW59300.2023.00094"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/8\/1238\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:38:24Z","timestamp":1760128704000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/8\/1238"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,21]]},"references-count":63,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["e25081238"],"URL":"https:\/\/doi.org\/10.3390\/e25081238","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,21]]}}}