Stoquastic PCP vs. Randomness

Dorit Aharonov, Alex Bredariol Grilo

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

2 Scopus citations

Abstract

The derandomization of MA, the probabilistic version of NP, is a long standing open question. In this work, we connect this problem to a variant of another major problem: The quantum PCP conjecture. Our connection goes through the surprising quantum characterization of MA by Bravyi and Terhal. They proved the MA-completeness of the problem of deciding whether the groundenergy of a uniform stoquastic local Hamiltonian is zero or inverse polynomial. We show that the gapped version of this problem, i.e. deciding if a given uniform stoquastic local Hamiltonian is frustration-free or has energy at least some constant ϵ, is in NP. Thus, if there exists a gap-Amplification procedure for uniform stoquastic Local Hamiltonians (in analogy to the gap amplification procedure for constraint satisfaction problems in the original PCP theorem), then MA = NP (and vice versa). Furthermore, if this gap amplification procedure exhibits some additional (natural) properties, then P = RP. We feel this work opens up a rich set of new directions to explore, which might lead to progress on both quantum PCP and derandomization. We also provide two small side results of potential interest. First, we are able to generalize our result by showing that deciding if a uniform stoquastic Local Hamiltonian has negligible or constant frustration can be also solved in NP. Additionally, our work reveals a new MA-complete problem which we call SetCSP, stated in terms of classical constraints on strings of bits, which we define in the appendix. As far as we know this is the first (arguably) natural MA-complete problem stated in non-quantum CSP language.

Original languageAmerican English
Title of host publicationProceedings - 2019 IEEE 60th Annual Symposium on Foundations of Computer Science, FOCS 2019
PublisherIEEE Computer Society
Pages1000-1023
Number of pages24
ISBN (Electronic)9781728149523
DOIs
StatePublished - Nov 2019
Event60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019 - Baltimore, United States
Duration: 9 Nov 201912 Nov 2019

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2019-November
ISSN (Print)0272-5428

Conference

Conference60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019
Country/TerritoryUnited States
CityBaltimore
Period9/11/1912/11/19

Bibliographical note

Publisher Copyright:
© 2019 IEEE.

Keywords

  • Complexity theory
  • Derandomization
  • Quantum PCP

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