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For more general homogeneous linear differential equations d<\/mml:mi><\/mml:mrow>d<\/mml:mi><\/mml:mrow>t<\/mml:mi><\/mml:mrow><\/mml:mfrac>|<\/mml:mo><\/mml:mrow>&#x03C8;<\/mml:mi>(<\/mml:mo>t<\/mml:mi>)<\/mml:mo>&#x27E9;<\/mml:mo>=<\/mml:mo>A<\/mml:mi>(<\/mml:mo>t<\/mml:mi>)<\/mml:mo>|<\/mml:mo><\/mml:mrow>&#x03C8;<\/mml:mi>(<\/mml:mo>t<\/mml:mi>)<\/mml:mo>&#x27E9;<\/mml:mo>,<\/mml:mo>|<\/mml:mo><\/mml:mrow>&#x03C8;<\/mml:mi>(<\/mml:mo>0<\/mml:mn>)<\/mml:mo>&#x27E9;<\/mml:mo>=<\/mml:mo>|<\/mml:mo><\/mml:mrow>&#x03C8;<\/mml:mi>0<\/mml:mn><\/mml:msub>&#x27E9;<\/mml:mo><\/mml:math>, a time-marching based quantum solver can suffer from exponentially vanishing success probability with respect to the number of time steps and is thus considered impractical. We solve this problem by repeatedly invoking a technique called the uniform singular value amplification, and the overall success probability can be lower bounded by a quantity that is independent of the number of time steps. The success probability can be further improved using a compression gadget lemma. This provides a path of designing quantum differential equation solvers that is alternative to those based on quantum linear systems algorithms (QLSA). We demonstrate the performance of the time-marching strategy with a high-order integrator based on the truncated Dyson series. The complexity of the algorithm depends linearly on the amplification ratio, which quantifies the deviation from a unitary dynamics. We prove that the linear dependence on the amplification ratio attains the query complexity lower bound and thus cannot be improved in the worst case. This algorithm also surpasses existing QLSA based solvers in three aspects: (1) A<\/mml:mi>(<\/mml:mo>t<\/mml:mi>)<\/mml:mo><\/mml:math> does not need to be diagonalizable. (2) A<\/mml:mi>(<\/mml:mo>t<\/mml:mi>)<\/mml:mo><\/mml:math> can be non-smooth, and is only of bounded variation. (3) It can use fewer queries to the initial state |<\/mml:mo><\/mml:mrow>&#x03C8;<\/mml:mi>0<\/mml:mn><\/mml:msub>&#x27E9;<\/mml:mo><\/mml:math>. Finally, we demonstrate the time-marching strategy with a first-order truncated Magnus series, which simplifies the implementation compared to high-order truncated Dyson series approach, while retaining the aforementioned benefits. Our analysis also raises some open questions concerning the differences between time-marching and QLSA based methods for solving differential equations.<\/jats:p>","DOI":"10.22331\/q-2023-03-20-955","type":"journal-article","created":{"date-parts":[[2023,3,20]],"date-time":"2023-03-20T13:55:56Z","timestamp":1679320556000},"page":"955","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":64,"title":["Time-marching based quantum solvers for time-dependent linear differential equations"],"prefix":"10.22331","volume":"7","author":[{"given":"Di","family":"Fang","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of California, Berkeley, CA 94720, USA"},{"name":"Simons Institute for the Theory of Computing, University of California, Berkeley, CA 94720, USA"},{"name":"Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lin","family":"Lin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of California, Berkeley, CA 94720, USA"},{"name":"Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA"},{"name":"Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yu","family":"Tong","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA"},{"name":"Department of Mathematics, University of California, Berkeley, CA 94720, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2023,3,20]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"A. 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