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Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description.
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
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Jianping Li, General explicit difference formulas for numerical differentiation , Journal of Computational and Applied Mathematics 183
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Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, Quantum algorithm for linear systems of equations , Physical Review Letters 103
2009
Jie Shen and Haijun Yu, Efficient spectral sparse grid methods and applications to high-dimensional elliptic equations II. Unbounded domains , SIAM Journal on Scientific Computing 34
2012
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Yudong Cao, Anargyros Papageorgiou, Iasonas Petras, Joseph Traub, and Sabre Kais, Quantum algorithm and circuit design solving the Poisson equation , New Journal of Physics 15
2013
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B. David Clader, Bryan C. Jacobs, and Chad R. Sprouse, Preconditioned quantum linear system algorithm , Physical Review Letters 110
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Jonathan Welch, Daniel Greenbaum, Sarah Mostame, and Alán Aspuru-Guzik, Efficient quantum circuits for diagonal unitaries without ancillas , New Journal of Physics 16
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Cited alongside, same era.
Lawrence C. Evans, Partial differential equations (2nd ed.) , American Mathematical Society, 2010, doi:10.1090/gsm/019
2010
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Benjamin P. Lanyon, James D. Whitfield, Geoff G. Gillett, Michael E. Goggin, Marcelo P. Almeida, Ivan Kassal, Jacob D. Biamonte, Masoud Mohseni, Ben J. Powell, Marco Barbieri, Alán Aspuru-Guzik, and Andrew G. White, Towards quantum chemistry on a quantum computer , Nature Chemistry 2
2010
Cited alongside, same era.
Jie Shen and Haijun Yu, Efficient spectral sparse grid methods and applications to high-dimensional elliptic problems , SIAM Journal on Scientific Computing 32
2010
Cited alongside, same era.
David Poulin, Angie Qarry, Rolando D. Somma, and Frank Verstraete, Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space , Physical Review Letters 106
2011
Cited alongside, same era.
Jie Shen, Tao Tang, and Li-Lian Wang, Spectral methods: algorithms, analysis and applications , vol. 41, Springer Science & Business Media, 2011, doi:10.1007/978-3-540-71041-7
2011
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Andris Ambainis, Variable time amplitude amplification and quantum algorithms for linear algebra problems , 29th Symposium on Theoretical Aspects of Computer Science, vol. 14, pp. 636–647, LIPIcs, 2012, doi:10.4230/LIPIcs.STACS.2012.636
2012
Cited alongside, same era.
Roger A. Horn and Charles R. Johnson, Matrix analysis , Cambridge University Press, 2012, doi:10.1017/CBO9780511810817
2012
Cited alongside, same era.
2014
Later among the works it cites.
Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma, Simulating Hamiltonian dynamics with a truncated Taylor series , Physical Review Letters 114
2015
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Ashley Montanaro and Sam Pallister, Quantum algorithms and the finite element method , Physical Review A 93
2016
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Ian D. Kivlichan, Nathan Wiebe, Ryan Babbush, and Alán Aspuru-Guzik, Bounding the costs of quantum simulation of many-body physics in real space , Journal of Physics A: Mathematical and Theoretical 50
2017
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Markus Reiher, Nathan Wiebe, Krysta M. Svore, Dave Wecker, and Matthias Troyer, Elucidating reaction mechanisms on quantum computers , Proceedings of the National Academy of Sciences 114
2017
Later among the works it cites.
Pedro C. S. Costa, Stephen Jordan, and Aaron Ostrander, Quantum algorithm for simulating the wave equation , Physical Review A 99
2019
Later among the works it cites.
François Fillion-Gourdeau and Emmanuel Lorin, Simple digital quantum algorithm for symmetric first-order linear hyperbolic systems , Numerical Algorithms 82
2019
Later among the works it cites.
Shengbin Wang, Zhimin Wang, Wendong Li, Lixin Fan, Zhiqiang Wei, and Yongjian Gu, Quantum fast Poisson solver: the algorithm and modular circuit design , Quantum Information Processing 19
2020
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