A lattice problem in quantum NP

D. Aharonov, O. Regev

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We consider coGapSVP√n, a gap version of the shortest vector in a lattice problem. This problem is known to be in AM ∩ coNP but is not known to be in NP or in MA. We prove that it lies inside QMA, the quantum analogue of NP. This is the first non-trivial upper bound on the quantum complexity of a lattice problem. The proof relies on two novel ideas. First, we give a new characterization of QMA, called QMA+ formulation allows us to circumvent a problem which arises commonly in the context of QMA: the prover might use entanglement between different copies of the same state in order to cheat. The second idea involves using estimations of autocorrelation functions for verification. We make the important observation that autocorrelation functions are positive definite functions and using properties of such functions we severely restrict the prover's possibility to cheat. We hope that these ideas will lead to further developments in the field.

Original languageEnglish (US)
Title of host publicationProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
PublisherIEEE Computer Society
Pages210-219
Number of pages10
Volume2003-January
ISBN (Print)0769520405
DOIs
StatePublished - 2003
Event44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003 - Cambridge, United States
Duration: Oct 11 2003Oct 14 2003

Other

Other44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003
CountryUnited States
CityCambridge
Period10/11/0310/14/03

Fingerprint

Autocorrelation

Keywords

  • Algorithm design and analysis
  • Autocorrelation
  • Computer science
  • Cryptography
  • Lattices
  • Polynomials
  • Quantum computing
  • Quantum mechanics
  • Upper bound

ASJC Scopus subject areas

  • Computer Science(all)

Cite this

Aharonov, D., & Regev, O. (2003). A lattice problem in quantum NP. In Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS (Vol. 2003-January, pp. 210-219). [1238195] IEEE Computer Society. https://doi.org/10.1109/SFCS.2003.1238195

A lattice problem in quantum NP. / Aharonov, D.; Regev, O.

Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS. Vol. 2003-January IEEE Computer Society, 2003. p. 210-219 1238195.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Aharonov, D & Regev, O 2003, A lattice problem in quantum NP. in Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS. vol. 2003-January, 1238195, IEEE Computer Society, pp. 210-219, 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003, Cambridge, United States, 10/11/03. https://doi.org/10.1109/SFCS.2003.1238195
Aharonov D, Regev O. A lattice problem in quantum NP. In Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS. Vol. 2003-January. IEEE Computer Society. 2003. p. 210-219. 1238195 https://doi.org/10.1109/SFCS.2003.1238195
Aharonov, D. ; Regev, O. / A lattice problem in quantum NP. Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS. Vol. 2003-January IEEE Computer Society, 2003. pp. 210-219
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