TR08-051 Authors: Scott Aaronson, Salman Beigi, Andrew Drucker, Bill Fefferman, Peter Shor

Publication: 2nd May 2008 20:53

Downloads: 2060

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The class QMA(k), introduced by Kobayashi et al., consists

of all languages that can be verified using k unentangled quantum

proofs. Many of the simplest questions about this class have remained

embarrassingly open: for example, can we give any evidence that k

quantum proofs are more powerful than one? Can we show any upper bound

on QMA(k), besides the trivial NEXP? Does QMA(k)=QMA(2) for k>=2? Can

QMA(k) protocols be amplified to exponentially small error? In this

paper, we make progress on all of the above questions. First, we give

a protocol by which a verifier can be convinced that a 3SAT formula of

size n is satisfiable, with constant soundness, given ~O(sqrt(n))

unentangled quantum witnesses with O(log n) qubits each. Our protocol

relies on Dinur's version of the PCP Theorem and is inherently

non-relativizing. Second, we show that assuming the famous Additivity

Conjecture from quantum information theory, any QMA(2) protocol can be

amplified to exponentially small error, and QMA(k)=QMA(2) for all

k>=2. Third, we give evidence that QMA(2) is contained in PSPACE, by

showing that this would follow from "strong amplification" of QMA(2)

protocols. Finally, we prove the nonexistence of "perfect

disentanglers" for simulating multiple Merlins with one.