Title | High-precision quantum algorithms for partial differential equations |

Publication Type | Journal Article |

Year of Publication | 2021 |

Authors | Childs, AM, Liu, J-P, Ostrander, A |

Journal | Quantum 5, 574 |

Volume | 5 |

Issue | 574 |

Date Published | 11/4/2021 |

Abstract | 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. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity poly(1/ε), where ε is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be poly(d,log(1/ε)), where d is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations. |

URL | https://arxiv.org/abs/2002.07868 |

DOI | 10.22331/q-2021-11-10-574 |