### Abstract

We describe a method to simulate Hamiltonian evolution on a quantum computer by repeatedly using a superposition of steps of a quantum walk, then applying a correction to the weightings for the numbers of steps of the quantum walk. This correction enables us to obtain complexity which is the same as the lower bound up to double-logarithmic factors for all parameter regimes. The scaling of the query complexity is *O* (*τ* (log log*τ*/log log log*τ)* + log(1/*ε*)) where *τ* := *t*||*H*||_{max}*d*, for *ε* the allowable error, *t* the time, ||*H*||_{max} the max-norm of the Hamiltonian, and *d* the sparseness. This technique should also be useful for improving the scaling of the Taylor series approach to simulation, which is relevant to applications such as quantum chemistry.

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

Pages | 1295–1317 |

Number of pages | 23 |

Journal | Quantum Information and Computation |

Volume | 16 |

Issue number | 15&16 |

Publication status | Published - Nov 2016 |

### Fingerprint

### Keywords

- quantum algorithms
- quantum query complexity
- Hamiltonian simulation
- quantum walk

### Cite this

*Quantum Information and Computation*,

*16*(15&16), 1295–1317.

}

*Quantum Information and Computation*, vol. 16, no. 15&16, pp. 1295–1317.

**Corrected quantum walk for optimal Hamiltonian simulation.** / Berry, Dominic; Novo, Leonardo.

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

TY - JOUR

T1 - Corrected quantum walk for optimal Hamiltonian simulation

AU - Berry, Dominic

AU - Novo, Leonardo

PY - 2016/11

Y1 - 2016/11

N2 - We describe a method to simulate Hamiltonian evolution on a quantum computer by repeatedly using a superposition of steps of a quantum walk, then applying a correction to the weightings for the numbers of steps of the quantum walk. This correction enables us to obtain complexity which is the same as the lower bound up to double-logarithmic factors for all parameter regimes. The scaling of the query complexity is O (τ (log logτ/log log logτ) + log(1/ε)) where τ := t||H||maxd, for ε the allowable error, t the time, ||H||max the max-norm of the Hamiltonian, and d the sparseness. This technique should also be useful for improving the scaling of the Taylor series approach to simulation, which is relevant to applications such as quantum chemistry.

AB - We describe a method to simulate Hamiltonian evolution on a quantum computer by repeatedly using a superposition of steps of a quantum walk, then applying a correction to the weightings for the numbers of steps of the quantum walk. This correction enables us to obtain complexity which is the same as the lower bound up to double-logarithmic factors for all parameter regimes. The scaling of the query complexity is O (τ (log logτ/log log logτ) + log(1/ε)) where τ := t||H||maxd, for ε the allowable error, t the time, ||H||max the max-norm of the Hamiltonian, and d the sparseness. This technique should also be useful for improving the scaling of the Taylor series approach to simulation, which is relevant to applications such as quantum chemistry.

KW - quantum algorithms

KW - quantum query complexity

KW - Hamiltonian simulation

KW - quantum walk

UR - http://www.rintonpress.com/journals/qiconline.html#v16n1516

UR - https://doi.org/10.26421/QIC16.15-16

M3 - Article

VL - 16

SP - 1295

EP - 1317

JO - Quantum Information and Computation

T2 - Quantum Information and Computation

JF - Quantum Information and Computation

SN - 1533-7146

IS - 15&16

ER -