### Abstract

We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time. The complexity of the algorithm is polynomial in the logarithm of the inverse error, an exponential improvement over previous quantum algorithms for this problem. Our result builds upon recent advances in quantum linear systems algorithms by encoding the simulation into a sparse, well-conditioned linear system that approximates evolution according to the propagator using a Taylor series. Unlike with finite difference methods, our approach does not require additional hypotheses to ensure numerical stability.

Language | English |
---|---|

Pages | 1057-1081 |

Number of pages | 25 |

Journal | Communications in Mathematical Physics |

Volume | 356 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Dec 2017 |

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### Cite this

*Communications in Mathematical Physics*,

*356*(3), 1057-1081. https://doi.org/10.1007/s00220-017-3002-y

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*Communications in Mathematical Physics*, vol. 356, no. 3, pp. 1057-1081. https://doi.org/10.1007/s00220-017-3002-y

**Quantum algorithm for linear differential equations with exponentially improved dependence on precision.** / Berry, Dominic W.; Childs, Andrew M.; Ostrander, Aaron; Wang, Guoming.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Quantum algorithm for linear differential equations with exponentially improved dependence on precision

AU - Berry, Dominic W.

AU - Childs, Andrew M.

AU - Ostrander, Aaron

AU - Wang, Guoming

PY - 2017/12/1

Y1 - 2017/12/1

N2 - We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time. The complexity of the algorithm is polynomial in the logarithm of the inverse error, an exponential improvement over previous quantum algorithms for this problem. Our result builds upon recent advances in quantum linear systems algorithms by encoding the simulation into a sparse, well-conditioned linear system that approximates evolution according to the propagator using a Taylor series. Unlike with finite difference methods, our approach does not require additional hypotheses to ensure numerical stability.

AB - We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time. The complexity of the algorithm is polynomial in the logarithm of the inverse error, an exponential improvement over previous quantum algorithms for this problem. Our result builds upon recent advances in quantum linear systems algorithms by encoding the simulation into a sparse, well-conditioned linear system that approximates evolution according to the propagator using a Taylor series. Unlike with finite difference methods, our approach does not require additional hypotheses to ensure numerical stability.

UR - http://www.scopus.com/inward/record.url?scp=85030661144&partnerID=8YFLogxK

U2 - 10.1007/s00220-017-3002-y

DO - 10.1007/s00220-017-3002-y

M3 - Article

VL - 356

SP - 1057

EP - 1081

JO - Communications in Mathematical Physics

T2 - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -