{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T18:10:46Z","timestamp":1772302246523,"version":"3.50.1"},"reference-count":42,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,28]],"date-time":"2022-12-28T00:00:00Z","timestamp":1672185600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Basic Research Program of China","award":["12005015"],"award-info":[{"award-number":["12005015"]}]},{"name":"National Basic Research Program of China","award":["11974205"],"award-info":[{"award-number":["11974205"]}]},{"name":"National Basic Research Program of China","award":["11774197"],"award-info":[{"award-number":["11774197"]}]},{"name":"National Basic Research Program of China","award":["2017YFA0303700"],"award-info":[{"award-number":["2017YFA0303700"]}]},{"name":"National Basic Research Program of China","award":["2018B030325002"],"award-info":[{"award-number":["2018B030325002"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12005015"],"award-info":[{"award-number":["12005015"]}],"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":["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\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2018B030325002"],"award-info":[{"award-number":["2018B030325002"]}],"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":["12005015"],"award-info":[{"award-number":["12005015"]}],"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":["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"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2018B030325002"],"award-info":[{"award-number":["2018B030325002"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Key Research and Development Program of Guangdong province","award":["12005015"],"award-info":[{"award-number":["12005015"]}]},{"name":"Key Research and Development Program of Guangdong province","award":["11974205"],"award-info":[{"award-number":["11974205"]}]},{"name":"Key Research and Development Program of Guangdong province","award":["11774197"],"award-info":[{"award-number":["11774197"]}]},{"name":"Key Research and Development Program of Guangdong province","award":["2017YFA0303700"],"award-info":[{"award-number":["2017YFA0303700"]}]},{"name":"Key Research and Development Program of Guangdong province","award":["2018B030325002"],"award-info":[{"award-number":["2018B030325002"]}]},{"name":"Beijing Advanced Innovation Center for Future Chip","award":["12005015"],"award-info":[{"award-number":["12005015"]}]},{"name":"Beijing Advanced Innovation Center for Future Chip","award":["11974205"],"award-info":[{"award-number":["11974205"]}]},{"name":"Beijing Advanced Innovation Center for Future Chip","award":["11774197"],"award-info":[{"award-number":["11774197"]}]},{"name":"Beijing Advanced Innovation Center for Future Chip","award":["2017YFA0303700"],"award-info":[{"award-number":["2017YFA0303700"]}]},{"name":"Beijing Advanced Innovation Center for Future Chip","award":["2018B030325002"],"award-info":[{"award-number":["2018B030325002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Solving the eigenproblems of Hermitian matrices is a significant problem in many fields. The quantum resonant transition (QRT) algorithm has been proposed and demonstrated to solve this problem using quantum devices. To better realize the capabilities of the QRT with recent quantum devices, we improve this algorithm and develop a new procedure to reduce the time complexity. Compared with the original algorithm, it saves one qubit and reduces the complexity with error \u03f5 from O(1\/\u03f52) to O(1\/\u03f5). Thanks to these optimizations, we can obtain the energy spectrum and ground state of the effective Hamiltonian of the water molecule more accurately and in only 20 percent of the time in a four-qubit processor compared to previous work. More generally, for non-Hermitian matrices, a singular-value decomposition has essential applications in more areas, such as recommendation systems and principal component analysis. The QRT has also been used to prepare singular vectors corresponding to the largest singular values, demonstrating its potential for applications in quantum machine learning.<\/jats:p>","DOI":"10.3390\/e25010061","type":"journal-article","created":{"date-parts":[[2022,12,29]],"date-time":"2022-12-29T02:56:50Z","timestamp":1672282610000},"page":"61","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Quantum Multi-Round Resonant Transition Algorithm"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8546-0270","authenticated-orcid":false,"given":"Fan","family":"Yang","sequence":"first","affiliation":[{"name":"State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China"},{"name":"Beijing Academy of Quantum Information Sciences, Beijing 100193, China"}]},{"given":"Xinyu","family":"Chen","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1011-9198","authenticated-orcid":false,"given":"Dafa","family":"Zhao","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China"}]},{"given":"Shijie","family":"Wei","sequence":"additional","affiliation":[{"name":"Beijing Academy of Quantum Information Sciences, Beijing 100193, China"}]},{"given":"Jingwei","family":"Wen","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China"}]},{"given":"Hefeng","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, School of Science, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}]},{"given":"Tao","family":"Xin","sequence":"additional","affiliation":[{"name":"Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China"}]},{"given":"Guilu","family":"Long","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China"},{"name":"Beijing Academy of Quantum Information Sciences, Beijing 100193, China"},{"name":"Tsinghua National Laboratory for Information Science and Technology, Beijing 100084, China"},{"name":"Collaborative Innovation Center of Quantum Matter, Beijing 100084, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.cplett.2004.06.001","article-title":"A coupled cluster and full configuration interaction study of CN and CN-","volume":"393","author":"Olsen","year":"2004","journal-title":"Chem. Phys. Lett."},{"key":"ref_2","unstructured":"Kitaev, A.Y. (1995). Quantum measurements and the Abelian Stabilizer Problem. arXiv."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5162","DOI":"10.1103\/PhysRevLett.83.5162","article-title":"Quantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors","volume":"83","author":"Abrams","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","unstructured":"Farhi, E., Goldstone, J., Gutmann, S., and Sipser, M. (2000). Quantum Computation by Adiabatic Evolution. arXiv."},{"key":"ref_5","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_6","unstructured":"Wang, H., Yu, S., and Xiang, H. (2020). A multi-step quantum algorithm for solving problems with a special structure. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"040305","DOI":"10.1103\/PRXQuantum.3.040305","article-title":"Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary Matrices","volume":"3","author":"Dong","year":"2022","journal-title":"PRX Quantum"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"123025","DOI":"10.1088\/1367-2630\/ab5c60","article-title":"Quantum eigenvalue estimation via time series analysis","volume":"21","author":"Somma","year":"2019","journal-title":"New J. Phys."},{"key":"ref_9","unstructured":"Zeng, P., Sun, J., and Yuan, X. (2021). Universal quantum algorithmic cooling on a quantum computer. arXiv."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/ncomms5213","article-title":"A variational eigenvalue solver on a photonic quantum processor","volume":"5","author":"Peruzzo","year":"2014","journal-title":"Nat. Commun."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1486935","DOI":"10.34133\/2020\/1486935","article-title":"A Full Quantum Eigensolver for Quantum Chemistry Simulations","volume":"2020","author":"Wei","year":"2020","journal-title":"Research"},{"key":"ref_12","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_13","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_14","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_15","doi-asserted-by":"crossref","first-page":"1189","DOI":"10.1007\/s11128-016-1263-6","article-title":"Duality quantum computer and the efficient quantum simulations","volume":"15","author":"Wei","year":"2016","journal-title":"Quantum Inf. Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1303","DOI":"10.1088\/0253-6102\/50\/6\/11","article-title":"Duality computing in quantum computers","volume":"50","author":"Yang","year":"2008","journal-title":"Commun. Theor. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11433-021-1844-7","article-title":"Quantum gradient descent algorithms for nonequilibrium steady states and linear algebraic systems","volume":"65","author":"Liang","year":"2022","journal-title":"Sci. China Phys. Mech."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Lv, P., Wei, S.J., Xie, H.N., and Long, G.L. (2022). QCSH: A Full Quantum Computer Nuclear Shell-Model Package. arXiv.","DOI":"10.1007\/s11433-022-2044-5"},{"key":"ref_19","unstructured":"Wen, J., Xiao, J., Li, H., Wei, S., and Long, G. (2021). A full circuit-based quantum algorithm for excited-states in quantum chemistry. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"090504","DOI":"10.1103\/PhysRevLett.122.090504","article-title":"Quantum Simulation of Resonant Transitions for Solving the Eigenproblem of an Effective Water Hamiltonian","volume":"122","author":"Li","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"062304","DOI":"10.1103\/PhysRevA.85.062304","article-title":"Quantum algorithm for obtaining the energy spectrum of a physical system","volume":"85","author":"Wang","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"052334","DOI":"10.1103\/PhysRevA.93.052334","article-title":"Quantum algorithm for obtaining the eigenstates of a physical system","volume":"93","author":"Wang","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"020502","DOI":"10.1103\/PhysRevB.87.020502","article-title":"Tunneling spectroscopy using a probe qubit","volume":"87","author":"Berkley","year":"2013","journal-title":"Phys. Rev. B"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6603","DOI":"10.1038\/srep06603","article-title":"Adiabatic Quantum Simulation of Quantum Chemistry","volume":"4","author":"Babbush","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/S0167-2789(98)00046-3","article-title":"Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing","volume":"120","author":"Cory","year":"1998","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6857","DOI":"10.1038\/srep06857","article-title":"Experimental Optimal Single Qubit Purification in an NMR Quantum Information Processor","volume":"4","author":"Hou","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1016\/j.scib.2017.03.006","article-title":"Experimental study of Forrelation in nuclear spins","volume":"62","author":"Li","year":"2017","journal-title":"Sci. Bull."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1016\/S0375-9601(02)01479-2","article-title":"The quantum state tomography on an NMR system","volume":"305","author":"Lee","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"190501","DOI":"10.1103\/PhysRevLett.110.190501","article-title":"Experimental Realization of Nonadiabatic Holonomic Quantum Computation","volume":"110","author":"Feng","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"052302","DOI":"10.1103\/PhysRevA.69.052302","article-title":"State interrogation in nuclear magnetic resonance quantum-information processing","volume":"69","author":"Leskowitz","year":"2004","journal-title":"Phys. Rev. A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"022305","DOI":"10.1103\/PhysRevA.88.022305","article-title":"Experimental simulation of anyonic fractional statistics with an NMR quantum-information processor","volume":"88","author":"Feng","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"032307","DOI":"10.1103\/PhysRevA.96.032307","article-title":"Optimal design of measurement settings for quantum-state-tomography experiments","volume":"96","author":"Li","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_33","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_34","doi-asserted-by":"crossref","first-page":"012328","DOI":"10.1103\/PhysRevA.78.012328","article-title":"Liquid-state nuclear magnetic resonance as a testbed for developing quantum control methods","volume":"78","author":"Ryan","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"7555","DOI":"10.1073\/pnas.1619152114","article-title":"Elucidating reaction mechanisms on quantum computers","volume":"114","author":"Reiher","year":"2017","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"190502","DOI":"10.1103\/PhysRevLett.108.190502","article-title":"Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions","volume":"108","author":"Casanova","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s00220-006-0150-x","article-title":"Efficient quantum algorithms for simulating sparse Hamiltonians","volume":"270","author":"Berry","year":"2007","journal-title":"Commun. Math. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"010501","DOI":"10.1103\/PhysRevLett.118.010501","article-title":"Optimal Hamiltonian Simulation by Quantum Signal Processing","volume":"118","author":"Low","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"110501","DOI":"10.1103\/PhysRevLett.120.110501","article-title":"Quantum Simulation of Electronic Structure with Linear Depth and Connectivity","volume":"120","author":"Kivlichan","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_40","unstructured":"Levine, I. (2000). Quantum Chemistry, Prentice Hall Inc."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"5388","DOI":"10.1039\/b804804e","article-title":"Quantum algorithm for obtaining the energy spectrum of molecular systems","volume":"10","author":"Wang","year":"2008","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_42","unstructured":"Szabo, A., and Ostlund, N.S. (2012). Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Courier Corporation."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/61\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:54:20Z","timestamp":1760147660000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/61"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,28]]},"references-count":42,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["e25010061"],"URL":"https:\/\/doi.org\/10.3390\/e25010061","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,28]]}}}