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The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise

Authors Yangjing Dong , Honghao Fu , Anand Natarajan , Minglong Qin , Haochen Xu , Penghui Yao

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Author Details

Yangjing Dong
  • State Key Laboratory for Novel Software Technology, Nanjing University, China
Honghao Fu
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Anand Natarajan
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Minglong Qin
  • State Key Laboratory for Novel Software Technology, Nanjing University, China
Haochen Xu
  • State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China
Penghui Yao
  • State Key Laboratory for Novel Software Technology, Nanjing University, China
  • Hefei National Laboratory, China

Funding

  • Dong, Yangjing: supported by National Natural Science Foundation of China (Grant No. 62332009, 12347104, 61972191) and Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302901).
  • Fu, Honghao: supported by the US National Science Foundation QLCI program (grant OMA-2016245).
  • Qin, Minglong: supported by National Natural Science Foundation of China (Grant No. 62332009, 12347104, 61972191) and Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302901).
  • Xu, Haochen: supported by the Key Research Program of the Chinese Academy of Sciences, Grant NO. ZDRW-XX-2022-1
  • Yao, Penghui: supported by National Natural Science Foundation of China (Grant No. 62332009, 12347104, 61972191) and Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302901).

Acknowledgements

P.Y. would like to thank the discussion with Zhengfeng Ji. Part of the work was done when H.F. and H.X. visited Nanjing University.

Cite AsGet BibTex

Yangjing Dong, Honghao Fu, Anand Natarajan, Minglong Qin, Haochen Xu, and Penghui Yao. The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 30:1-30:71, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CCC.2024.30

Abstract

The class MIP^* of quantum multiprover interactive proof systems with entanglement is much more powerful than its classical counterpart MIP [Babai et al., 1991; Zhengfeng Ji et al., 2020; Zhengfeng Ji et al., 2020]: 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 [Qin and Yao, 2021; Qin and Yao, 2023] 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 equivalent to NEXP = MIP. This improves significantly on the previous best-known bound of NEEEXP (nondeterministic triply exponential time) [Qin and Yao, 2021]. 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 continuous.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • Interactive proofs
  • Quantum complexity theory
  • Quantum entanglement
  • Fourier analysis
  • Matrix analysis
  • Invariance principle
  • Derandomization
  • PCP
  • Locally testable code
  • Positivity testing

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