Theory of Computing
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Title : Quantum Versus Classical Proofs and Advice
Authors : Scott Aaronson and Greg Kuperberg
Volume : 3
Number : 7
Pages : 129-157
URL : https://theoryofcomputing.org/articles/v003a007
Abstract
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This paper studies whether quantum proofs are more powerful
than classical proofs, or in complexity terms, whether
QMA = QCMA. We prove three results about this question.
First, we give a *quantum oracle separation* between
QMA and QCMA. More concretely, we show that any quantum
algorithm needs Omega(sqrt{2^n/m+1}) queries to find an
n-qubit *marked state* |psi>, even if given an m-bit
classical description of |psi> together with a quantum
black box that recognizes |psi>. Second, we give an
explicit QCMA protocol that nearly achieves this lower
bound. Third, we show that, in the one previously-known
case where quantum proofs seemed to provide an exponential
advantage, *classical* proofs are basically just as powerful.
In particular, Watrous gave a QMA protocol for verifying
non-membership in finite groups. Under plausible group-theoretic
assumptions, we give a QCMA protocol for the same problem.
Even with no assumptions, our protocol makes only polynomially
many queries to the group oracle. We end with some conjectures
about quantum versus classical oracles, and about the
possibility of a *classical* oracle separation between
QMA and QCMA.