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Andrew M. Childs, Robin Kothari, and Rolando D. Somma, Quantum linear systems algorithm with exponentially improved dependence on precision , SIAM Journal on Computing (2017), 46, 1920–1950, arXiv:1511.02306
Original
1950
Earlier work this paper cites.
Tom M. Apostol, Calculus, Volume II , Wiley (1969)
1969
Earlier work this paper cites.
David Kahaner, Cleve Moler, and Stephen Nash, Numerical methods and software , Prentice Hall (1989)
1989
Earlier work this paper cites.
Raimer Kress, Numerical analysis , Springer (1998)
1998
Earlier work this paper cites.
Esmail Babolian and Mohammad Mahdi Hosseini, A modified spectral method for numerical solution of ordinary differential equations with non-analytic solution , Applied Mathematics and Computation (2002), 132(2), 341-351
2002
Earlier work this paper cites.
R. Byron Bird, Transport phenomena , Applied Mechanics Reviews (2002), 55, no. 1, R1–R4
2002
Earlier work this paper cites.
Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp, Quantum amplitude amplification and estimation , Contemporary Mathematics (2002), 305, 53-74, arXiv:quant-ph/0005055
2002
Earlier work this paper cites.
Stewart Harris, An introduction to the theory of the Boltzmann equation , Dover (2004)
2004
Earlier work this paper cites.
Bo Thidé, Electromagnetic field theory , Upsilon Books (2004)
2004
Earlier work this paper cites.
Vladimir Antonovich Zorich, Mathematical analysis II , Springer (2004)
Original
2004
Earlier work this paper cites.
Mohammad Mahdi Hosseini, A modified pseudospectral method for numerical solution of ordinary differential equations systems , Applied Mathematics and Computation (2006), 176 (2), 470-475
2006
Earlier work this paper cites.

Similar

Vivek V. Shende, Stephen S. Bullock, and Igor L. Markov, Synthesis of quantum-logic circuits , IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems (2006), 25(6) 1000-1010, arXiv:quant-ph/0406176
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2009
Cited alongside, same era.

Then

B. David Clader, Bryan C. Jacobs, and Chad R. Sprouse, Preconditioned quantum linear system algorithm , Physical Review Letters (2013), 110.25, 250504, arXiv:1301.2340
Original
2013
Later among the works it cites.
Sangtae Kim and Seppo J. Karrila, Microhydrodynamics: principles and selected applications , Courier Corporation (2013)
2013
Later among the works it cites.
Dominic W. Berry, High-order quantum algorithm for solving linear differential equations , Journal of Physics A (2014), 47, no. 10, 105301, arXiv:1010.2745
Original
2014
Later among the works it cites.
Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma, Exponential improvement in precision for simulating sparse Hamiltonians , Proceedings of the 46th Annual ACM Symposium on Theory of Computing (2014), pp. 283-292, arXiv:1312.1414
Original
2014
Later among the works it cites.

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2019

Quantum spectral methods for differential equations

Childs, Andrew M., Liu, Jin-Peng

Understand

Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space.

Such algorithms can produce a quantum state proportional to the solution of a $d$-dimensional system of linear equations or linear differential equations with complexity $\mathrm{poly}(\log d)$.
While several of these algorithms approximate the solution to within $\epsilon$ with complexity $\mathrm{poly}(\log(1/\epsilon))$, no such algorithm was previously known for differential equations with time-dependent coefficients.
Here we develop a quantum algorithm for linear ordinary differential equations based on so-called spectral methods, an alternative to finite difference methods that approximates the solution globally.

Built on

Andrew M. Childs, Robin Kothari, and Rolando D. Somma, Quantum linear systems algorithm with exponentially improved dependence on precision , SIAM Journal on Computing (2017), 46, 1920–1950, arXiv:1511.02306
Original
1950
Earlier work this paper cites.
Tom M. Apostol, Calculus, Volume II , Wiley (1969)
1969
Earlier work this paper cites.
David Kahaner, Cleve Moler, and Stephen Nash, Numerical methods and software , Prentice Hall (1989)
1989
Earlier work this paper cites.
Raimer Kress, Numerical analysis , Springer (1998)
1998
Earlier work this paper cites.
Esmail Babolian and Mohammad Mahdi Hosseini, A modified spectral method for numerical solution of ordinary differential equations with non-analytic solution , Applied Mathematics and Computation (2002), 132(2), 341-351
2002
Earlier work this paper cites.
R. Byron Bird, Transport phenomena , Applied Mechanics Reviews (2002), 55, no. 1, R1–R4
2002
Earlier work this paper cites.
Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp, Quantum amplitude amplification and estimation , Contemporary Mathematics (2002), 305, 53-74, arXiv:quant-ph/0005055
2002
Earlier work this paper cites.
Stewart Harris, An introduction to the theory of the Boltzmann equation , Dover (2004)
2004
Earlier work this paper cites.
Bo Thidé, Electromagnetic field theory , Upsilon Books (2004)
2004
Earlier work this paper cites.
Vladimir Antonovich Zorich, Mathematical analysis II , Springer (2004)
Original
2004
Earlier work this paper cites.
Mohammad Mahdi Hosseini, A modified pseudospectral method for numerical solution of ordinary differential equations systems , Applied Mathematics and Computation (2006), 176 (2), 470-475
2006
Earlier work this paper cites.

Similar

Vivek V. Shende, Stephen S. Bullock, and Igor L. Markov, Synthesis of quantum-logic circuits , IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems (2006), 25(6) 1000-1010, arXiv:quant-ph/0406176
2006
Cited alongside, same era.
Dominic W. Berry, Graeme Ahokas, Richard Cleve, and Barry C. Sanders, Efficient quantum algorithms for simulating sparse Hamiltonians , Communications in Mathematical Physics (2007), 270, no. 2, 359-371, arXiv:quant-ph/0508139
2007
Cited alongside, same era.
Calin-Ioan Gheorghiu, Spectral methods for differential problems , Casa Cartii de Stiinta Publishing House (2007)
2007
Cited alongside, same era.
Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, Quantum algorithm for linear systems of equations , Physical Review Letters (2009), 103, no. 15, 150502, arXiv:0811.3171
Original
2009
Cited alongside, same era.

Then

B. David Clader, Bryan C. Jacobs, and Chad R. Sprouse, Preconditioned quantum linear system algorithm , Physical Review Letters (2013), 110.25, 250504, arXiv:1301.2340
Original
2013
Later among the works it cites.
Sangtae Kim and Seppo J. Karrila, Microhydrodynamics: principles and selected applications , Courier Corporation (2013)
2013
Later among the works it cites.
Dominic W. Berry, High-order quantum algorithm for solving linear differential equations , Journal of Physics A (2014), 47, no. 10, 105301, arXiv:1010.2745
Original
2014
Later among the works it cites.
Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma, Exponential improvement in precision for simulating sparse Hamiltonians , Proceedings of the 46th Annual ACM Symposium on Theory of Computing (2014), pp. 283-292, arXiv:1312.1414
Original
2014
Later among the works it cites.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Andrew M. Childs, On the relationship between continuous- and discrete-time quantum walk , Communications in Mathematical Physics (2010), 294, no. 2, 581–603, arXiv:0810.0312

Original

2010

Cited alongside, same era.

Lawrence C. Evans, Partial differential equations (2nd ed.) , American Mathematical Society (2010)

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 (2011), 106, 170501, arXiv:1102.1360

Original

2011

Cited alongside, same era.

Jie Shen, Tao Tang, and Li-Lian Wang, Spectral methods: algorithms, analysis and applications , Springer Science & Business Media (2011)

2011

Cited alongside, same era.

R. K. Michael Thambynayagam, The diffusion handbook: Applied solutions for engineers , McGraw-Hill Professional (2011)

2011

Cited alongside, same era.

Andris Ambainis, Variable time amplitude amplification and quantum algorithms for linear algebra problems , Proceedings of the 29th Symposium on Theoretical Aspects of Computer Science (2012), 636-647, arXiv:1010.4458

Original

2012

Cited alongside, same era.

John David Jackson, Classical electrodynamics , John Wiley & \& Sons (2012)

2012

Cited alongside, same era.

Juan Miguel Arrazola, Timjan Kalajdzievski, Christian Weedbrook, and Seth Lloyd, Quantum algorithm for non-homogeneous linear partial differential equations , arXiv:1809.02622

Original

Cited in the paper.

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 (2015), 114, no. 9, 090502, arXiv:1412.4687

Original

2015

Later among the works it cites.

Dominic W. Berry, Andrew M. Childs, and Robin Kothari, Hamiltonian simulation with nearly optimal dependence on all parameters , Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (2015), pp. 792-809, arXiv:1501.01715

Original

2015

Later among the works it cites.

Dominic W. Berry and Leonardo Novo, Corrected quantum walk for optimal Hamiltonian simulation , Quantum Information and Computation (2016), 16, no. 15-16, 1295, arXiv:1606.03443

Original

2016

Later among the works it cites.

Dominic W. Berry, Andrew M. Childs, Aaron Ostrander, Guoming Wang, Quantum algorithm for linear differential equations with exponentially improved dependence on precision , Communications in Mathematical Physics (2017), 356, no. 3, 1057-1081, arXiv:1701.03684