{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T13:06:53Z","timestamp":1779887213442,"version":"3.53.1"},"reference-count":28,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,29]],"date-time":"2022-12-29T00:00:00Z","timestamp":1672272000000},"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":["11974205"],"award-info":[{"award-number":["11974205"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11774197"],"award-info":[{"award-number":["11774197"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2017YFA0303700"],"award-info":[{"award-number":["2017YFA0303700"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["11974205"],"award-info":[{"award-number":["11974205"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["11774197"],"award-info":[{"award-number":["11774197"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2017YFA0303700"],"award-info":[{"award-number":["2017YFA0303700"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>When faced with a quantum-solving problem for partial differential equations, people usually transform such problems into Hamiltonian simulation problems or quantum-solving problems for linear equation systems. In this paper, we propose a third approach to solving partial differential equations that differs from the two approaches. By using the duality quantum algorithm, we construct a quantum-solving algorithm for solving the first-order wave equation, which represents a typical class of partial differential equations. Numerical results of the quantum circuit have high precision consistency with the theoretical d\u2019Alembert solution. Then the routine is applied to the wave equation with either a dissipation or dispersion term. As shown by complexity analysis for all these cases of the wave equation, our algorithm has a quadratic acceleration for each iteration compared to the classical algorithm.<\/jats:p>","DOI":"10.3390\/e25010062","type":"journal-article","created":{"date-parts":[[2022,12,30]],"date-time":"2022-12-30T03:26:11Z","timestamp":1672370771000},"page":"62","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Quantum-Solving Algorithm for d\u2019Alembert Solutions of the Wave Equation"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6745-2883","authenticated-orcid":false,"given":"Yuanye","family":"Zhu","sequence":"first","affiliation":[{"name":"Center on Frontiers of Computing Studies and School of Computer Science, Peking University, Beijing 100871, China"},{"name":"State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.1126\/science.273.5278.1073","article-title":"Universal Quantum Simulators","volume":"273","author":"Lloyd","year":"1996","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Aharonov, D., and Ta-Shma, A. (2003, January 9\u201311). Adiabatic quantum state generation and statistical zero knowledge. Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, San Diego, CA, USA.","DOI":"10.1145\/780542.780546"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"090502","DOI":"10.1103\/PhysRevLett.114.090502","article-title":"Simulating Hamiltonian dynamics with a truncated Taylor series","volume":"114","author":"Berry","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"163","DOI":"10.22331\/q-2019-07-12-163","article-title":"Hamiltonian Simulation by Qubitization","volume":"3","author":"Low","year":"2019","journal-title":"Quantum"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"150502","DOI":"10.1103\/PhysRevLett.103.150502","article-title":"Quantum algorithm for linear systems of equations","volume":"103","author":"Harrow","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","unstructured":"Leyton, S.K., and Osborne, T.J. (2008). A quantum algorithm to solve nonlinear differential equations. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"105301","DOI":"10.1088\/1751-8113\/47\/10\/105301","article-title":"High-order quantum algorithm for solving linear differential equations","volume":"47","author":"Berry","year":"2014","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"012323","DOI":"10.1103\/PhysRevA.99.012323","article-title":"Quantum algorithm for simulating the wave equation","volume":"99","author":"Costa","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_9","unstructured":"Gonzalez-Conde, J., Rodr\u00edguez-Rozas, \u00c1., Solano, E., and Sanz, M. (2021). Pricing financial derivatives with exponential quantum speedup. arXiv."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1007\/s00220-017-3002-y","article-title":"Quantum Algorithm for Linear Differential Equations with Exponentially Improved Dependence on Precision","volume":"356","author":"Berry","year":"2017","journal-title":"Commun. Math. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1427","DOI":"10.1007\/s00220-020-03699-z","article-title":"Quantum Spectral Methods for Differential Equations","volume":"375","author":"Childs","year":"2020","journal-title":"Commun. Math. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"013021","DOI":"10.1088\/1367-2630\/15\/1\/013021","article-title":"Quantum algorithm and circuit design solving the Poisson equation","volume":"15","author":"Cao","year":"2013","journal-title":"New J. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1088\/0253-6102\/45\/5\/013","article-title":"General Quantum Interference Principle and Duality Computer","volume":"45","year":"2006","journal-title":"Commun. Theor. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s11704-008-0021-z","article-title":"Duality quantum computing","volume":"2","author":"Long","year":"2008","journal-title":"Front. Comput. Sci. China"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1088\/0253-6102\/51\/1\/13","article-title":"Allowable Generalized Quantum Gates","volume":"51","author":"Yang","year":"2009","journal-title":"Commun. Theor. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/s11128-006-0040-3","article-title":"Mathematical Theory of Duality Quantum Computers","volume":"6","author":"Gudder","year":"2007","journal-title":"Quantum Inf. Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"022307","DOI":"10.1103\/PhysRevA.64.022307","article-title":"Grover algorithm with zero theoretical failure rate","volume":"64","author":"Long","year":"2001","journal-title":"Phys. Rev. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1088\/0253-6102\/32\/3\/335","article-title":"Arbitrary phase rotation of the marked state cannot be used for Grover\u2019s quantum search algorithm","volume":"32","author":"Guilu","year":"1999","journal-title":"Commun. Theor. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/S0375-9601(99)00631-3","article-title":"Phase matching in quantum searching","volume":"262","author":"Long","year":"1999","journal-title":"Phys. Lett. A"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Zhu, Y., Wang, Z., Yan, B., and Wei, S. (2021). Robust Quantum Search with Uncertain Number of Target States. Entropy, 23.","DOI":"10.3390\/e23121649"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Berry, D.W., Childs, A.M., and Kothari, R. (2015, January 17\u201320). Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters. Proceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, Berkeley, CA, USA.","DOI":"10.1109\/FOCS.2015.54"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"30727","DOI":"10.1038\/srep30727","article-title":"Duality quantum algorithm efficiently simulates open quantum systems","volume":"6","author":"Wei","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1920","DOI":"10.1137\/16M1087072","article-title":"Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision","volume":"46","author":"Childs","year":"2017","journal-title":"SIAM J. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3457","DOI":"10.1103\/PhysRevA.52.3457","article-title":"Elementary gates for quantum computation","volume":"52","author":"Barenco","year":"1995","journal-title":"Phys. Rev. A At. Mol. Opt. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"032324","DOI":"10.1103\/PhysRevA.93.032324","article-title":"Quantum algorithms and the finite element method","volume":"93","author":"Montanaro","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"210501","DOI":"10.1103\/PhysRevLett.113.210501","article-title":"Fixed-point quantum search with an optimal number of queries","volume":"113","author":"Yoder","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1897","DOI":"10.1007\/s11128-012-0498-0","article-title":"Quantum search with certainty based on modified Grover algorithms: Optimum choice of parameters","volume":"12","author":"Toyama","year":"2013","journal-title":"Quantum Inf. Process."},{"key":"ref_28","unstructured":"Kondrashuk, I., Notte-Cuello, E.A., Poblete-Cantellano, M., and Rojas-Medar, M.A. (2015). On the time-dependent grade-two model for the magnetohydrodynamic flow: 2D case. arXiv."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/62\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:55:01Z","timestamp":1760147701000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/62"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,29]]},"references-count":28,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["e25010062"],"URL":"https:\/\/doi.org\/10.3390\/e25010062","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,29]]}}}