Exponential improvement in precision for simulating sparse Hamiltonians

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

Research output: Chapter in Book/Report/Conference proceedingConference proceeding contributionResearch

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∥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.

LanguageEnglish
Title of host publicationSTOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing
Place of PublicationNew York
PublisherAssociation for Computing Machinery
Pages283-292
Number of pages10
ISBN (Print)9781450327107
DOIs
Publication statusPublished - 2014
Event4th Annual ACM Symposium on Theory of Computing, STOC 2014 - New York, NY, United States
Duration: 31 May 20143 Jun 2014

Other

Other4th Annual ACM Symposium on Theory of Computing, STOC 2014
CountryUnited States
CityNew York, NY
Period31/05/143/06/14

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Hamiltonians
Amplification
Derivatives

Cite this

Berry, D. W., Childs, A. M., Cleve, R., Kothari, R., & Somma, R. D. (2014). Exponential improvement in precision for simulating sparse Hamiltonians. In 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
Berry, Dominic W. ; Childs, Andrew M. ; Cleve, Richard ; Kothari, Robin ; Somma, Rolando D. / Exponential improvement in precision for simulating sparse Hamiltonians. STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing. New York : Association for Computing Machinery, 2014. pp. 283-292
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Berry, DW, Childs, AM, Cleve, R, Kothari, R & Somma, RD 2014, Exponential improvement in precision for simulating sparse Hamiltonians. in 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.

STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing. New York : Association for Computing Machinery, 2014. p. 283-292.

Research output: Chapter in Book/Report/Conference proceedingConference proceeding contributionResearch

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Berry DW, Childs AM, Cleve R, Kothari R, Somma RD. Exponential improvement in precision for simulating sparse Hamiltonians. In STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing. New York: Association for Computing Machinery. 2014. p. 283-292 https://doi.org/10.1145/2591796.2591854