[论文解读] NEEXP in MIP.

Anand Natarajan, John C. Wright|arXiv (Cornell University)|Apr 11, 2019

Cryptography and Data Security参考文献 39被引用 8

一句话总结

该论文证明了NEEXP（非确定性双指数时间）包含于MIP*，表明具有纠缠证明者的量子多证明者交互式证明系统比经典系统强大得多。该协议利用纠缠证明者通过量子测量抽样指数级庞大的问题，借助经典PCP验证答案，通过互补测量利用海森堡不确定性原理确保 soundness。

ABSTRACT

We study multiprover interactive proof systems. The power of classical multiprover interactive proof systems, in which the provers do not share entanglement, was characterized in a famous work by Babai, Fortnow, and Lund (Computational Complexity 1991), whose main result was the equality MIP = NEXP. The power of quantum multiprover interactive proof systems, in which the provers are allowed to share entanglement, has proven to be much more difficult to characterize. The best known lower-bound on MIP* is NEXP, due to Ito and Vidick (FOCS 2012). As for upper bounds, MIP* could be as large as RE, the class of recursively enumerable languages. The main result of this work is the inclusion of NEEXP (nondeterministic doubly exponential time) in MIP*. This is an exponential improvement over the prior lower bound and shows that proof systems with entangled provers are at least exponentially more powerful than classical provers. In our protocol the verifier delegates a classical, exponentially large MIP protocol for NEEXP to two entangled provers: the provers obtain their exponentially large questions by measuring their shared state, and use a classical PCP to certify the correctness of their exponentially-long answers. For the soundness of our protocol, it is crucial that each player should not only sample its own question correctly but also avoid performing measurements that would reveal the other player's sampled question. We ensure this by commanding the players to perform a complementary measurement, relying on the Heisenberg uncertainty principle to prevent the forbidden measurements from being performed.

研究动机与目标

• 建立MIP*的下界，使其超过先前的NEXP界限。
• 证明量子纠缠使证明者能够高效处理指数级庞大的证明。
• 设计一种协议，使纠缠证明者能够模拟NEEXP的经典MIP协议。
• 通过量子互补性防止证明者通过联合测量获知对方的问题，从而确保 soundness。
• 利用量子不确定性在无经典协调的情况下强制执行协议约束。

提出的方法

• 验证者将NEEXP的经典MIP协议委托给两个纠缠的证明者。
• 证明者通过测量其共享的纠缠态来抽样其指数级庞大的问题。
• 使用经典概率可检查证明（PCP）来验证证明者指数级长答案的正确性。
• 对两个证明者强制执行互补测量，以防止其测量会暴露对方问题的基。
• 海森堡不确定性原理确保此类禁止测量无法同时执行。
• 通过量子不可克隆性和不确定性原理维持 soundness，防止通过联合测量作弊。

实验结果

研究问题

• RQ1 MIP*能否实现超越NEXP的下界？
• RQ2 纠缠证明者能否模拟双指数时间问题的经典交互式证明？
• RQ3 如何利用量子互补性在无经典通信的情况下强制执行协议约束？
• RQ4 从计算复杂性角度，MIP*的最大能力是什么？
• RQ5 量子不确定性能否防止多证明者系统中证明者获知彼此的问题？

主要发现

• NEEXP包含于MIP*，建立了相对于MIP*先前NEXP下界的指数级改进。
• 该协议利用纠缠证明者通过量子测量抽样指数级庞大的问题。
• 协议的 soundness 依赖于海森堡不确定性原理，防止通过联合测量作弊。
• 证明者使用经典PCP验证指数级长的答案，保持验证效率。
• 由于互补测量约束，每个证明者无法同时获知对方的问题。
• 该结果表明MIP*严格强于经典MIP，后者受限于NEXP。