Quantum information and the PCP theorem

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

7 Scopus citations

Abstract

Our main result is that the membership x ∈ SAT (for x of length n) can be proved by a logarithmic-size quantum state |ψ〉, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |ψ〉 the verifier only needs to read one block of the classical proof. This shows that if a short quantum witness is available then a (classical) PCP with only one query is possible. Our second result is that the class QIP / qpoly contains all languages. That is, for any language L (even non-recursive), the membership x ∈ L (for x of length n) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state |ψ L,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. The advice |ψ L, n> given to the verifier can also be replaced by a classical probabilistic advice, as long as this advice is kept as a secret from the prover. Our result can hence be interpreted as: the class IP/rpoly contains all languages. For the proof of the second result, we introduce the quantum low-degree-extension of a string of bits. The main result requires an additional machinery of quantum low-degree-test.

Original languageEnglish (US)
Title of host publicationProceedings - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005
Pages459-468
Number of pages10
DOIs
StatePublished - Dec 1 2005
Externally publishedYes
Event46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005 - Pittsburgh, PA, United States
Duration: Oct 23 2005Oct 25 2005

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2005
ISSN (Print)0272-5428

Other

Other46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005
CountryUnited States
CityPittsburgh, PA
Period10/23/0510/25/05

All Science Journal Classification (ASJC) codes

  • Engineering(all)

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  • Cite this

    Raz, R. (2005). Quantum information and the PCP theorem. In Proceedings - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005 (pp. 459-468). [1530738] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS; Vol. 2005). https://doi.org/10.1109/SFCS.2005.62