Quantum multiprover interactive proof systems with entanglement MIP* are much more powerful than their 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 EPR states, where each EPR state is affected by an arbitrarily small constant amount of noise, the resulting complexity class is contained in 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 Fr\'echet derivatives or which are Lipschitz continous.

翻译：量子多方交互式证明系统（MIP*）因其共享纠缠特性而远强于经典对应系统MIP（Babai等 '91, Ji等 '20）：MIP = NEXP，而量子类MIP*等于RE——包含停机问题的类。这一优势源于MIP*中的证明者可共享无界量子纠缠。然而，Qin与Yao近期的研究（'21, '23）表明，若证明者共享态存在噪声，该优势将显著减弱。本文致力于精确刻画噪声对量子多方交互式证明系统计算能力的影响。我们研究量子双证明者单轮交互系统MIP*[poly, O(1)]（验证者发送多项式比特给证明者，证明者返回常数比特），并证明在此模型中噪声会完全消除共享纠缠带来的计算优势。具体而言，我们证明：若允许证明者共享任意数量的EPR态，且每个EPR态受任意小常数噪声影响，所得复杂度类被包含于NEXP = MIP。这一结果显著改进了Qin与Yao（'21）此前最优的NEEEXP（非确定性三指数时间）界限。进一步通过对比实验确证该能力降级源于噪声而非O(1)答案规模：允许无噪声EPR态可使该类获得RE = MIP*[poly, poly]的全部能力。研究过程中，我们独立发展了两种技术工具：其一，针对具有低阶Pauli矩阵傅里叶分解特征的指数规模矩阵，提出新型确定性正定性检测器；其二，建立关于具有有界三阶Fr\'echet导数或Lipschitz连续性的光滑矩阵函数的不变性原理。