{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T02:12:31Z","timestamp":1774318351294,"version":"3.50.1"},"reference-count":18,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2024,6,13]],"date-time":"2024-06-13T00:00:00Z","timestamp":1718236800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Research Council Discovery Projects","award":["DP190102633, DP210101367, DP220101602."],"award-info":[{"award-number":["DP190102633, DP210101367, DP220101602."]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>Time-dependent linear differential equations are a common type of problem that needs to be solved in classical physics. Here we provide a quantum algorithm for solving time-dependent linear differential equations with logarithmic dependence of the complexity on the error and derivative. As usual, there is an exponential improvement over classical approaches in the scaling of the complexity with the dimension, with the caveat that the solution is encoded in the amplitudes of a quantum state. Our method is to encode the Dyson series in a system of linear equations, then solve via the optimal quantum linear equation solver. Our method also provides a simplified approach in the case of time-independent differential equations.<\/jats:p>","DOI":"10.22331\/q-2024-06-13-1369","type":"journal-article","created":{"date-parts":[[2024,6,13]],"date-time":"2024-06-13T15:24:46Z","timestamp":1718292286000},"page":"1369","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":23,"title":["Quantum algorithm for time-dependent differential equations using Dyson series"],"prefix":"10.22331","volume":"8","author":[{"given":"Dominic W.","family":"Berry","sequence":"first","affiliation":[{"name":"School of Mathematical and Physical Sciences, Macquarie University, Sydney, New South Wales 2109, Australia"}]},{"given":"Pedro","family":"C. S. Costa","sequence":"additional","affiliation":[{"name":"School of Mathematical and Physical Sciences, Macquarie University, Sydney, New South Wales 2109, Australia"}]}],"member":"9598","published-online":{"date-parts":[[2024,6,13]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Dominic W. Berry. High-order quantum algorithm for solving linear differential equations. Journal of Physics A: Mathematical and Theoretical, 47 (10): 105301, 2014a. 10.1088\/1751-8113\/47\/10\/105301.","DOI":"10.1088\/1751-8113\/47\/10\/105301"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Dominic W Berry. High-order quantum algorithm for solving linear differential equations. Journal of Physics A: Mathematical and Theoretical, 47 (10): 105301, 2014b. 10.1088\/1751-8113\/47\/10\/105301.","DOI":"10.1088\/1751-8113\/47\/10\/105301"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Dominic W. Berry, Andrew M. Childs, Aaron Ostrander, and Guoming Wang. Quantum algorithm for linear differential equations with exponentially improved dependence on precision. Communications in Mathematical Physics, 356 (3): 1057\u20131081, Dec 2017. ISSN 1432-0916. 10.1007\/s00220-017-3002-y.","DOI":"10.1007\/s00220-017-3002-y"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Rajendra Bhatia, Tanvi Jain, and Yongdo Lim. On the bures\u2013wasserstein distance between positive definite matrices. Expositiones Mathematicae, 37 (2): 165\u2013191, 2019. ISSN 0723-0869. https:\/\/doi.org\/10.1016\/j.exmath.2018.01.002.","DOI":"10.1016\/j.exmath.2018.01.002"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Andrew M. Childs and Jin-Peng Liu. Quantum spectral methods for differential equations. Communications in Mathematical Physics, 375 (2): 1427\u20131457, February 2020. 10.1007\/s00220-020-03699-z.","DOI":"10.1007\/s00220-020-03699-z"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Andrew M. Childs, Robin Kothari, and Rolando D. Somma. Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing, 46 (6): 1920\u20131950, 2017. 10.1137\/16M1087072.","DOI":"10.1137\/16M1087072"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Andrew M. Childs, Jin-Peng Liu, and Aaron Ostrander. High-precision quantum algorithms for partial differential equations. Quantum, 5: 574, November 2021. 10.22331\/q-2021-11-10-574.","DOI":"10.22331\/q-2021-11-10-574"},{"key":"7","doi-asserted-by":"publisher","unstructured":"B. D. Clader, B. C. Jacobs, and C. R. Sprouse. Preconditioned quantum linear system algorithm. Physical Review Letters, 110: 250504, Jun 2013. 10.1103\/PhysRevLett.110.250504.","DOI":"10.1103\/PhysRevLett.110.250504"},{"key":"8","doi-asserted-by":"publisher","unstructured":"Pedro C.S. Costa, Dong An, Yuval R. Sanders, Yuan Su, Ryan Babbush, and Dominic W. Berry. Optimal scaling quantum linear-systems solver via discrete adiabatic theorem. PRX Quantum, 3: 040303, Oct 2022. 10.1103\/PRXQuantum.3.040303.","DOI":"10.1103\/PRXQuantum.3.040303"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Di Fang, Lin Lin, and Yu Tong. Time-marching based quantum solvers for time-dependent linear differential equations. Quantum, 7: 955, March 2023. ISSN 2521-327X. 10.22331\/q-2023-03-20-955.","DOI":"10.22331\/q-2023-03-20-955"},{"key":"10","doi-asserted-by":"publisher","unstructured":"Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. Physical Review Letters, 103 (15): 150502, Oct 2009. ISSN 1079-7114. 10.1103\/physrevlett.103.150502.","DOI":"10.1103\/physrevlett.103.150502"},{"key":"11","doi-asserted-by":"publisher","unstructured":"M\u00e1ria Kieferov\u00e1, Artur Scherer, and Dominic W. Berry. Simulating the dynamics of time-dependent Hamiltonians with a truncated Dyson series. Physical Review A, 99 (4): 042314, 4 2019. ISSN 2469-9926. 10.1103\/PhysRevA.99.042314.","DOI":"10.1103\/PhysRevA.99.042314"},{"key":"12","doi-asserted-by":"publisher","unstructured":"Emrah K\u0131l\u0131\u00e7 and Pantelimon Stanica. The inverse of banded matrices. Journal of Computational and Applied Mathematics, 237 (1): 126\u2013135, 2013. 10.1016\/j.cam.2012.07.018.","DOI":"10.1016\/j.cam.2012.07.018"},{"key":"13","doi-asserted-by":"publisher","unstructured":"Hari Krovi. Improved quantum algorithms for linear and nonlinear differential equations. Quantum, 7: 913, February 2023. ISSN 2521-327X. 10.22331\/q-2023-02-02-913.","DOI":"10.22331\/q-2023-02-02-913"},{"key":"14","doi-asserted-by":"publisher","unstructured":"Guang Hao Low and Nathan Wiebe. Hamiltonian Simulation in the Interaction Picture. arXiv:1805.00675, 2018. 10.48550\/arXiv.1805.00675.","DOI":"10.48550\/arXiv.1805.00675"},{"key":"15","doi-asserted-by":"publisher","unstructured":"Ashley Montanaro and Sam Pallister. Quantum algorithms and the finite element method. Physical Review A, 93: 032324, Mar 2016. 10.1103\/PhysRevA.93.032324.","DOI":"10.1103\/PhysRevA.93.032324"},{"key":"16","doi-asserted-by":"publisher","unstructured":"Yuval R. Sanders, Dominic W. Berry, Pedro C. S. Costa, Louis W. Tessler, Nathan Wiebe, Craig Gidney, Hartmut Neven, and Ryan Babbush. Compilation of fault-tolerant quantum heuristics for combinatorial optimization. PRX Quantum, 1 (2): 020312\u2013020382, 7 2020. 10.1103\/PRXQuantum.1.020312.","DOI":"10.1103\/PRXQuantum.1.020312"},{"key":"17","doi-asserted-by":"publisher","unstructured":"Yuan Su, Dominic W. Berry, Nathan Wiebe, Nicholas Rubin, and Ryan Babbush. Fault-tolerant quantum simulations of chemistry in first quantization. PRX Quantum, 2: 040332, Nov 2021. 10.1103\/PRXQuantum.2.040332.","DOI":"10.1103\/PRXQuantum.2.040332"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2024-06-13-1369\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2024,6,13]],"date-time":"2024-06-13T15:24:55Z","timestamp":1718292295000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2024-06-13-1369\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,13]]},"references-count":18,"URL":"https:\/\/doi.org\/10.22331\/q-2024-06-13-1369","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,13]]},"article-number":"1369"}}