Exponential improvement in precision for simulating sparse Hamiltonians

Dominic Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, Rolando D. Somma

Research output: Contribution to journalArticleResearchpeer-review

Abstract

We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a d-sparse Hamiltonian H acting 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(τ(log2(τ/ε)/log log(τ/ε))n)additional 2-qubit gates, where τ = d2Hmaxt. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for timevarying 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 continuous- and 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 simplify the analysis of this conversion, avoiding the need for a complex fault-correction procedure. Our simplification 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.
LanguageEnglish
Article numbere8
Pages1-40
Number of pages40
JournalForum of Mathematics, Sigma
Volume5
DOIs
Publication statusPublished - 2 Mar 2017

Fingerprint

Hamiltonians
Amplification
Derivatives

Cite this

Berry, Dominic ; Childs, Andrew M. ; Cleve, Richard ; Kothari, Robin ; Somma, Rolando D. / Exponential improvement in precision for simulating sparse Hamiltonians. In: Forum of Mathematics, Sigma. 2017 ; Vol. 5. pp. 1-40.
@article{040bd2ac3d404e17b56cd21f7dc6a7fa,
title = "Exponential improvement in precision for simulating sparse Hamiltonians",
abstract = "We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a d-sparse Hamiltonian H acting 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(τ(log2(τ/ε)/log log(τ/ε))n)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 timevarying 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 continuous- and 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 simplify the analysis of this conversion, avoiding the need for a complex fault-correction procedure. Our simplification 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.",
author = "Dominic Berry and Childs, {Andrew M.} and Richard Cleve and Robin Kothari and Somma, {Rolando D.}",
year = "2017",
month = "3",
day = "2",
doi = "10.1017/fms.2017.2",
language = "English",
volume = "5",
pages = "1--40",
journal = "Forum of Mathematics, Sigma",
issn = "2050-5094",
publisher = "Cambridge University Press",

}

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

In: Forum of Mathematics, Sigma, Vol. 5, e8, 02.03.2017, p. 1-40.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - Exponential improvement in precision for simulating sparse Hamiltonians

AU - Berry, Dominic

AU - Childs, Andrew M.

AU - Cleve, Richard

AU - Kothari, Robin

AU - Somma, Rolando D.

PY - 2017/3/2

Y1 - 2017/3/2

N2 - We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a d-sparse Hamiltonian H acting 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(τ(log2(τ/ε)/log log(τ/ε))n)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 timevarying 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 continuous- and 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 simplify the analysis of this conversion, avoiding the need for a complex fault-correction procedure. Our simplification 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 dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a d-sparse Hamiltonian H acting 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(τ(log2(τ/ε)/log log(τ/ε))n)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 timevarying 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 continuous- and 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 simplify the analysis of this conversion, avoiding the need for a complex fault-correction procedure. Our simplification 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=85057234056&partnerID=8YFLogxK

U2 - 10.1017/fms.2017.2

DO - 10.1017/fms.2017.2

M3 - Article

VL - 5

SP - 1

EP - 40

JO - Forum of Mathematics, Sigma

T2 - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

SN - 2050-5094

M1 - e8

ER -