### Abstract

We provide a quantum algorithm for simulating the dynamicsof sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement overprevious methods. Specifically, we show that a d-sparse Hamiltonian H on n qubits can be simulated for time t with precision ε using O(τ log(τ/ε)/log log(τ/ε)) queries and O(τnlog2(τ/ε)/log log(τ/ε)) additional 2-qubit gates, where τ = d2∥H∥_{max}t. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuousand fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. We also significantly simplify the analysis of this conversion, avoiding the need for a complex fault correction procedure. Oursimplification relies on a new form of "oblivious amplitude amplification" that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error.

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

Title of host publication | STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing |

Place of Publication | New York |

Publisher | Association for Computing Machinery |

Pages | 283-292 |

Number of pages | 10 |

ISBN (Print) | 9781450327107 |

DOIs | |

Publication status | Published - 2014 |

Event | 4th Annual ACM Symposium on Theory of Computing, STOC 2014 - New York, NY, United States Duration: 31 May 2014 → 3 Jun 2014 |

### Other

Other | 4th Annual ACM Symposium on Theory of Computing, STOC 2014 |
---|---|

Country | United States |

City | New York, NY |

Period | 31/05/14 → 3/06/14 |

### Fingerprint

### Cite this

*STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing*(pp. 283-292). New York: Association for Computing Machinery. https://doi.org/10.1145/2591796.2591854

}

*STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing.*Association for Computing Machinery, New York, pp. 283-292, 4th Annual ACM Symposium on Theory of Computing, STOC 2014, New York, NY, United States, 31/05/14. https://doi.org/10.1145/2591796.2591854

**Exponential improvement in precision for simulating sparse Hamiltonians.** / Berry, Dominic W.; Childs, Andrew M.; Cleve, Richard; Kothari, Robin; Somma, Rolando D.

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding contribution › Research

TY - GEN

T1 - Exponential improvement in precision for simulating sparse Hamiltonians

AU - Berry, Dominic W.

AU - Childs, Andrew M.

AU - Cleve, Richard

AU - Kothari, Robin

AU - Somma, Rolando D.

PY - 2014

Y1 - 2014

N2 - We provide a quantum algorithm for simulating the dynamicsof sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement overprevious methods. Specifically, we show that a d-sparse Hamiltonian H on n qubits can be simulated for time t with precision ε using O(τ log(τ/ε)/log log(τ/ε)) queries and O(τnlog2(τ/ε)/log log(τ/ε)) additional 2-qubit gates, where τ = d2∥H∥maxt. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuousand fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. We also significantly simplify the analysis of this conversion, avoiding the need for a complex fault correction procedure. Oursimplification relies on a new form of "oblivious amplitude amplification" that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error.

AB - We provide a quantum algorithm for simulating the dynamicsof sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement overprevious methods. Specifically, we show that a d-sparse Hamiltonian H on n qubits can be simulated for time t with precision ε using O(τ log(τ/ε)/log log(τ/ε)) queries and O(τnlog2(τ/ε)/log log(τ/ε)) additional 2-qubit gates, where τ = d2∥H∥maxt. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuousand fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. We also significantly simplify the analysis of this conversion, avoiding the need for a complex fault correction procedure. Oursimplification relies on a new form of "oblivious amplitude amplification" that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error.

UR - http://www.scopus.com/inward/record.url?scp=84904323967&partnerID=8YFLogxK

U2 - 10.1145/2591796.2591854

DO - 10.1145/2591796.2591854

M3 - Conference proceeding contribution

SN - 9781450327107

SP - 283

EP - 292

BT - STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing

PB - Association for Computing Machinery

CY - New York

ER -