[论文解读] The Computational Advantage of MIP* Vanishes in the Presence of Noise

Yangjing Dong, Honghao Fu+4 more|arXiv (Cornell University)|Dec 7, 2023

Quantum Computing Algorithms and Architecture被引用 3

一句话总结

该论文表明，当共享的EPR态中引入噪声时，具有量子纠缠的多证明者量子交互式证明系统（MIP*）会失去其计算优势。即使噪声极小，复杂度类也会从RE（不可判定问题）坍缩至NEXP，表明噪声完全破坏了该模型中纠缠的计算能力。该结果确立了一个清晰的临界点：噪声消除了MIP*的优势，而无噪声的纠缠则保留了该优势。

ABSTRACT

Quantum multiprover interactive proof systems with entanglement MIP* are much more powerful than its classical counterpart MIP (Babai et al. '91, Ji et al. '20): while MIP = NEXP, the quantum class MIP* is equal to RE, a class including the halting problem. This is because the provers in MIP* can share unbounded quantum entanglement. However, recent works of Qin and Yao '21 and '23 have shown that this advantage is significantly reduced if the provers' shared state contains noise. This paper attempts to exactly characterize the effect of noise on the computational power of quantum multiprover interactive proof systems. We investigate the quantum two-prover one-round interactive system MIP*[poly, O(1)], where the verifier sends polynomially many bits to the provers and the provers send back constantly many bits. We show noise completely destroys the computational advantage given by shared entanglement in this model. Specifically, we show that if the provers are allowed to share arbitrarily many noisy EPR states, where each EPR state is affected by an arbitrarily small constant amount of noise, the resulting complexity class is equivalent to NEXP = MIP. This improves significantly on the previous best-known bound of NEEEXP (nondeterministic triply exponential time) by Qin and Yao '21. We also show that this collapse in power is due to the noise, rather than the O(1) answer size, by showing that allowing for noiseless EPR states gives the class the full power of RE = MIP*[poly, poly]. Along the way, we develop two technical tools of independent interest. First, we give a new, deterministic tester for the positivity of an exponentially large matrix, provided it has a low-degree Fourier decomposition in terms of Pauli matrices. Secondly, we develop a new invariance principle for smooth matrix functions having bounded third-order Fréchet derivatives or which are Lipschitz continous.

研究动机与目标

• 精确刻画噪声对具有纠缠的多证明者量子交互式证明系统计算能力的影响。
• 确定MIP*的量子优势在MIP*[poly, O(1)]模型中是否对真实噪声条件具有鲁棒性。
• 解决MIP*坍缩至NEXP的主要原因是否为O(1)答案大小限制，或噪声本身。
• 开发新的分析工具——特别是针对低度矩阵的正定性检测器，以及矩阵函数的不变性原理——以应用于噪声非局域游戏。

提出的方法

• 设计了一种确定性、低复杂度的检测器，用于检测在泡利基中具有低度傅里叶展开的指数级大矩阵的正定性。
• 提出了一种针对具有有界三阶弗雷歇导数或利普希茨连续性的光滑矩阵函数的新不变性原理。
• 应用去随机化的不变性原理，利用伪随机性模拟噪声下的量子关联。
• 在非局域游戏中使用哈达玛码实现答案压缩，以在最小化答案大小的同时保持可靠性。
• 结合预言化、并行重复与迭代答案压缩，构建出具有O(1)答案大小与多项式时间复杂度的协议。
• 证明了具有恒定噪声率的噪声EPR态会使MIP*[poly, O(1)]坍缩至NEXP，而无噪声EPR态则保持完整的MIP*能力（即RE），表明噪声是导致坍缩的关键因素。

实验结果

研究问题

• RQ1 MIP*的量子优势是否会在共享EPR态中存在任意小的噪声时消失？
• RQ2 MIP*坍缩至NEXP是由于噪声，还是由于MIP*[poly, O(1)]模型中的O(1)答案大小约束？
• RQ3 能否为具有低度泡利分解的指数级大矩阵构造出确定性的正定性检测器？
• RQ4 在噪声与伪随机采样下，光滑矩阵函数的行为受何种不变性原理支配？
• RQ5 能否优化答案压缩技术，使其在噪声MIP*协议中保持多项式时间复杂度的同时实现O(1)答案大小？

主要发现

• EPR态中的噪声——即使为任意小的恒定噪声——也会完全摧毁MIP*的量子优势，使复杂度类坍缩至NEXP。
• 在具有噪声EPR态的MIP*[poly, O(1)]模型中，其复杂度等价于MIP = NEXP，优于先前的上界NEEEXP。
• 无噪声EPR态保留了MIP*的全部能力，实现RE = MIP*[poly, poly]，确认噪声而非答案大小是导致坍缩的关键因素。
• 开发了一种针对在泡利基中具有低度傅里叶展开的矩阵的新型确定性正定性检测器，其时间复杂度在输入大小上为多项式。
• 建立了一种针对具有有界三阶弗雷歇导数或利普希茨连续性的矩阵函数的新不变性原理，使噪声环境下的鲁棒分析成为可能。
• 通过哈达玛码与预言化实现的迭代答案压缩，可在噪声环境下实现具有O(1)答案大小、多项式问题大小与多项式时间复杂度的协议。