arXiv 0810.51092008-10quant-ph

NP与QMA_log（2）

NP vs QMA_log(2)

Salman Beigi

备注：: 10 pages. Thanks to referees, the main result is now stated in terms of 3-SAT instead of NP. Clearer proofs. To appear in Quantum Information and Computation
发表信息：: Quant. Inf. Comp. 10, 0141-0151 (2010)

摘要：

虽然人们认为$\NP$ -困难问题不太可能采用有效的量子算法，但已经表明，量子验证器可以解决给定“短”量子证明的$\NP$ -完全问题;更准确地说，$\NP\subseteq \QMA_{\log}(2)$，其中$\QMA_{\log}(2)$表示量子梅林-阿瑟博弈类，其中有两个未纠缠的证明器向验证器发送两个对数大小的量子见证。包含$\NP\subseteq \QMA_{\log}(2)$已经由Blier和Tapp通过陈述具有完美完备性和间隙$\frac{1}{24n^6}$的3-着色的量子Merlin-Arthur协议来证明。此外，Aaronson { \it et al.}已经通过考虑对数大小的$\widetilde{O}(\sqrt{n})$证据，显示了具有恒定间隙的上述包含。然而，我们仍然不知道具有恒定间隙的$\QMA_{\log}(2)$是否包含$\NP$。本文证明了3-SAT允许一个$\QMA_{\log}(2)$协议，对于每个常数$ε>0$，该协议具有间隙$\frac{1}{n^{3+ε}}$.

英文摘要：

Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve $\NP$-complete problems given a "short" quantum proof; more precisely, $\NP\subseteq \QMA_{\log}(2)$ where $\QMA_{\log}(2)$ denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion $\NP\subseteq \QMA_{\log}(2)$ has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap $\frac{1}{24n^6}$. Moreover, Aaronson {\it et al.} have shown the above inclusion with a constant gap by considering $\widetilde{O}(\sqrt{n})$ witnesses of logarithmic size. However, we still do not know if $\QMA_{\log}(2)$ with a constant gap contains $\NP$. In this paper, we show that 3-SAT admits a $\QMA_{\log}(2)$ protocol with the gap $\frac{1}{n^{3+ε}}$ for every constant $ε>0$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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