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 Fr\'echet derivatives or which are Lipschitz continous.

翻译：具有纠缠的量子多方交互证明系统 MIP* 比其经典对应物 MIP 强大得多（Babai 等人 '91，Ji 等人 '20）：虽然 MIP = NEXP，但量子类 MIP* 等于 RE，这是一个包含停机问题的复杂性类。这是因为 MIP* 中的证明者可以共享无界的量子纠缠态。然而，Qin 和 Yao '21 及 '23 的近期研究表明，如果证明者共享的量子态包含噪声，这种优势将显著减弱。本文试图精确刻画噪声对量子多方交互证明系统计算能力的影响。我们研究了量子双证明者单轮交互系统 MIP*[poly, O(1)]，其中验证者向证明者发送多项式数量的比特，而证明者返回常数数量的比特。我们证明，在该模型中，噪声完全破坏了共享纠缠所带来的计算优势。具体而言，我们证明，如果允许证明者共享任意多个有噪声的 EPR 态，且每个 EPR 态受到任意小的常数量级噪声影响，则所得的复杂性类等价于 NEXP = MIP。这显著改进了 Qin 和 Yao '21 之前已知的最佳上界 NEEEXP（非确定性三重指数时间）。我们还通过证明允许使用无噪声 EPR 态可使该类具有 RE = MIP*[poly, poly] 的全部能力，表明这种能力塌缩是由于噪声，而非 O(1) 的答案长度所致。在此过程中，我们开发了两个具有独立价值的技术工具。首先，我们给出了一种新的、确定性的测试方法，用于判断一个指数级大矩阵是否为正定矩阵，前提是该矩阵在泡利矩阵基下具有低阶傅里叶分解。其次，我们针对具有有界三阶 Fréchet 导数或 Lipschitz 连续的光滑矩阵函数，建立了一个新的不变性原理。