On preparing ground states of gapped hamiltonians: An efficient quantum lovász local lemma

Andras Pal Gilyen, Or Sattath

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

8 Scopus citations

Abstract

A frustration-free local Hamiltonian has the property that its ground state minimises the energy of all local terms simultaneously. In general, even deciding whether a Hamiltonian is frustration-free is a hard task, as it is closely related to the QMA1-complete quantum satisfiability problem (QSAT) - the quantum analogue of SAT, which is the archetypal NP-complete problem in classical computer science. This connection shows that the frustration-free property is not only relevant to physics but also to computer science.The Quantum Lovsz Local Lemma (QLLL) provides a sufficient condition for frustration-freeness. Is there an efficient way to prepare a frustration-free state under the conditions of the QLLL? Previous results showed that the answer is positive if all local terms commute. These works were based on Mosers compression argument which was the original analysis technique of the celebrated resampling algorithm. We generalise and simplify the compression argument, so that it provides a simplified version of the previous quantum results, and improves on some classical results as well.More importantly, we improve on the previous constructive results by designing an algorithm that works efficiently for non-commuting terms as well, assuming that the system is uniformly gapped, by which we mean that the system and all its subsystems have an inverse polynomial energy gap. Similarly to the previous results, our algorithm has the charming feature that it uses only local measurement operations corresponding to the local Hamiltonian terms.

Original languageEnglish
Title of host publicationProceedings - 58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017
PublisherInstitute of Electrical and Electronics Engineers
Pages439-450
Number of pages12
ISBN (Electronic)9781538634646
DOIs
StatePublished - 10 Nov 2017
Externally publishedYes
Event58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017 - Berkeley, United States
Duration: 15 Oct 201717 Oct 2017

Publication series

NameAnnual Symposium on Foundations of Computer Science - Proceedings
Volume2017-October
ISSN (Print)0272-5428

Conference

Conference58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017
Country/TerritoryUnited States
CityBerkeley
Period15/10/1717/10/17

Keywords

  • Local Hamiltonians
  • Moser-Tardos Algorithm
  • QMA
  • QSAT
  • Quantum Algorithm
  • Quantum Lovász Local Lemma

ASJC Scopus subject areas

  • General Computer Science

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