Quantum information and the pcp theorem

Research output: Contribution to journalArticle

4 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)
Pages (from-to)462-489
Number of pages28
JournalAlgorithmica (New York)
Volume55
Issue number3
DOIs
StatePublished - Nov 1 2009
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Computer Science(all)
  • Computer Science Applications
  • Applied Mathematics

Keywords

  • Low degree test
  • Probabilistically checkable proofs
  • Quantum advice
  • Quantum computation
  • Quantum information
  • Quantum interactive proofs

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