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. • 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 ( √ 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. • 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. • We give evidence that QMA (2) ⊆ PSPACE, by showing that this would follow from "strong amplification" of QMA (2) protocols. • We prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one. 23rd Annual IEEE Conference on Computational Complexity 1093-0159/08 $25.00
By contrast, REF showed that 3-SAT can be solved by a QMA(k) protocol with constant soundness, at the expense of increasing k to O √ n polylog(n) .
The Power of Unentanglement
2008 23rd Annual IEEE Conference on Computational Complexity
2008 23rd Annual IEEE Conference on Computational Complexity
Computer Science Mathematics Physics