{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,10]],"date-time":"2026-08-10T17:28:18Z","timestamp":1786382898350,"version":"build-2736575974"},"reference-count":58,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2026,8,10]],"date-time":"2026-08-10T00:00:00Z","timestamp":1786320000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["52225705"],"award-info":[{"award-number":["52225705"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"award":["52225705"],"award-info":[{"award-number":["52225705"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Single-path transmission optimization is a core task for resource scheduling and operation of complex networks, which requires coordinated optimization of transmission cost and flow. Classical algorithms bear heavy computational loads in high-dimensional decision spaces as networks grow. This paper constructs a hybrid quantum model integrating quantum approximate optimization algorithm (QAOA) and cubic spline interpolation. Paths, discrete flows, and trade-off coefficients are unified within a quadratic unconstrained binary optimization (QUBO) model. Least-squares fitting converts native parameters into QUBO coefficients, whose fitting errors are measured to verify robustness and penalty sensitivity, and auxiliary variables eliminate high-order terms to exponentially cut qubit consumption. QAOA narrows the feasible range via global coarse search, and cubic spline interpolation further yields precise continuous flow values. Powered by quantum superposition for parallel full-space exploration, the framework avoids repeated modeling for separate bias coefficients. Mixed integer programming (MIP) and genetic algorithm (GA) are adopted as comparative benchmarks. For the small-scale network instance, the relative error between the proposed method and the global optimum solved by MIP is less than 1%. For the large-scale case, the overall error of our approach remains within an acceptable range even when discrepancies exist between results yielded by classical algorithms.<\/jats:p>","DOI":"10.3390\/e28080900","type":"journal-article","created":{"date-parts":[[2026,8,10]],"date-time":"2026-08-10T16:35:16Z","timestamp":1786379716000},"page":"900","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Quantum Single-Path Transmission Optimization of Complex Networks"],"prefix":"10.3390","volume":"28","author":[{"given":"Zhengyi","family":"Wang","sequence":"first","affiliation":[{"name":"School of Control Science and Engineering, Shandong University, Jinan 250061, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4969-5195","authenticated-orcid":false,"given":"Feng","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Shandong University, Jinan 250061, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yunqing","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Control Science and Engineering, Shandong University, Jinan 250061, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9805-1698","authenticated-orcid":false,"given":"Xiaohui","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Shandong University, Jinan 250061, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2881-5626","authenticated-orcid":false,"given":"Jingyang","family":"Fang","sequence":"additional","affiliation":[{"name":"School of Control Science and Engineering, Shandong University, Jinan 250061, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1038\/s42254-018-0002-6","article-title":"The Statistical Physics of Real-World Networks","volume":"1","author":"Cimini","year":"2019","journal-title":"Nat. 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