The Computational Advantage of MIP* Vanishes in the Presence of Noise

TL;DR

The paper investigates how noise in shared quantum entanglement affects the computational power of MIP* protocols. It proves that with arbitrarily many noisy MES copies and constant-size answers, MIP* collapses to NEXP, implying the quantum advantage evaporates under depolarizing noise. Conversely, in the noiseless setting, MIP* retains the full power of RE, and the authors develop an invariance principle for matrix functions and a positivity tester, enabling efficient, dimension-reduced verifications. Two main technical advances—a Hadamard-code–based answer reduction and a deterministic PSD tester—enable NP-hardness results for noisy games and an RE-in-MIP* construction with O(1) answers, highlighting a sharp boundary between robustness to noise and quantum advantage. The work thus links noise tolerance, self-testing limitations, and the hardness landscape of multi-prover quantum interactive proofs, guiding future studies on fault tolerance and robust self-testing in quantum complexity.

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.

Figures (5)

• Figure 1: Positivity testing algorithm
• Figure 2: Nondeterministic algorithm solving the noisy nonlocal game value problem
• Figure 3: The noisy $\mathrm{MIP}^*$ verifier $V^*$ from an $\mathrm{MIP}$ verifier $V = (\mathsf{Alg}_Q,\mathsf{Alg}_A)$
• Figure 4: Subset tester for the Hadamard code
• Figure 5: The answer reduced verifier $V^{AR}$.

Theorems & Definitions (103)

• Theorem 1.1: Informal
• Corollary 1.2
• Theorem 1.3: \ref{['thm:re']}
• Theorem : informal
• Definition 2.1: Quantum maximal correlation
• Definition 2.3
• Remark 2.5
• Definition 2.8
• Definition 2.11
• Definition 2.13