### Abstract

We present an algorithm for sparse Hamiltonian simulation whose complexity is optimal (up to log factors) as a function of all parameters of interest. Previous algorithms had optimal or near-optimal scaling in some parameters at the cost of poor scaling in others. Hamiltonian simulation via a quantum walk has optimal dependence on the sparsity at the expense of poor scaling in the allowed error. In contrast, an approach based on fractional-query simulation provides optimal scaling in the error at the expense of poor scaling in the sparsity. Here we combine the two approaches, achieving the best features of both. By implementing a linear combination of quantum walk steps with coefficients given by Bessel functions, our algorithm's complexity (as measured by the number of queries and 2-qubit gates) is logarithmic in the inverse error, and nearly linear in the product tau of the evolution time, the sparsity, and the magnitude of the largest entry of the Hamiltonian. Our dependence on the error is optimal, and we prove a new lower bound showing that no algorithm can have sub linear dependence on tau.

Original language | English |
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Title of host publication | Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015 |

Place of Publication | Piscataway, NJ |

Publisher | Institute of Electrical and Electronics Engineers (IEEE) |

Pages | 792-809 |

Number of pages | 18 |

ISBN (Electronic) | 9781467381918 |

ISBN (Print) | 9781467381925 |

DOIs | |

Publication status | Published - 11 Dec 2015 |

Event | 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 - Berkeley, United States Duration: 17 Oct 2015 → 20 Oct 2015 |

### Other

Other | 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 |
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Country | United States |

City | Berkeley |

Period | 17/10/15 → 20/10/15 |

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### Cite this

*Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015*(pp. 792-809). [7354428] Piscataway, NJ: Institute of Electrical and Electronics Engineers (IEEE). https://doi.org/10.1109/FOCS.2015.54