Document Open Access Logo

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

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

PDF
Thumbnail PDF

File

LIPIcs.CCC.2024.30.pdf
  • Filesize: 1.37 MB
  • 71 pages

Document Identifiers

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Dorit Aharonov and Michael Ben-Or. Fault-tolerant quantum computation with constant error rate. SIAM Journal on Computing, 38(4):1207, 2008. Google Scholar
  2. Dorit Aharonov, Xun Gao, Zeph Landau, Yunchao Liu, and Umesh Vazirani. A polynomial-time classical algorithm for noisy random circuit sampling. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 945-957, 2023. Google Scholar
  3. Rotem Arnon-Friedman and Jean-Daniel Bancal. Device-independent certification of one-shot distillable entanglement. New Journal of Physics, 21(3):033010, 2019. Google Scholar
  4. Rotem Arnon-Friedman, Zvika Brakerski, and Thomas Vidick. Computational entanglement theory. arXiv preprint, 2023. URL: https://arxiv.org/abs/2310.02783.
  5. Rotem Arnon-Friedman and Henry Yuen. Noise-Tolerant Testing of High Entanglement of Formation. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018), volume 107 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:12, Dagstuhl, Germany, 2018. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.11.
  6. Srinivasan Arunachalam and Penghui Yao. Positive spectrahedra: invariance principles and pseudorandom generators. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022, pages 208-221, New York, NY, USA, 2022. Association for Computing Machinery. URL: https://doi.org/10.1145/3519935.3519965.
  7. László Babai, Lance Fortnow, and Carsten Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1(1):3-40, March 1991. URL: https://doi.org/10.1007/BF01200056.
  8. Ainesh Bakshi, Nadiia Chepurko, and Rajesh Jayaram. Testing positive semi-definiteness via random submatrices. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020, pages 1191-1202. IEEE, 2020. Google Scholar
  9. Salman Beigi. A new quantum data processing inequality. Journal of Mathematical Physics, 54(8):082202, 2013. URL: https://doi.org/10.1063/1.4818985.
  10. Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. Journal of Computer and System Sciences, 47(3):549-595, 1993. Google Scholar
  11. Dolev Bluvstein, Simon J Evered, Alexandra A Geim, Sophie H Li, Hengyun Zhou, Tom Manovitz, Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, et al. Logical quantum processor based on reconfigurable atom arrays. Nature, pages 1-3, 2023. Google Scholar
  12. Sergio Boixo, Sergei V Isakov, Vadim N Smelyanskiy, Ryan Babbush, Nan Ding, Zhang Jiang, Michael J Bremner, John M Martinis, and Hartmut Neven. Characterizing quantum supremacy in near-term devices. Nature Physics, 14(6):595-600, 2018. Google Scholar
  13. J. Lawrence Carter and Mark N. Wegman. Universal classes of hash functions (extended abstract). In Proceedings of the Ninth Annual ACM Symposium on Theory of Computing, STOC 1977, pages 106-112, New York, NY, USA, 1977. Association for Computing Machinery. URL: https://doi.org/10.1145/800105.803400.
  14. Sitan Chen, Jordan Cotler, Hsin-Yuan Huang, and Jerry Li. The complexity of NISQ. Nature Communications, 14(1):6001, September 2023. URL: https://doi.org/10.1038/s41467-023-41217-6.
  15. R. Cleve, P. Hoyer, B. Toner, and J. Watrous. Consequences and limits of nonlocal strategies. In Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004., pages 236-249, 2004. URL: https://doi.org/10.1109/CCC.2004.1313847.
  16. Rodney Coleman. Calculus on Normed Vector Spaces. Springer-Verlag, New York, New York, NY, 1997. Google Scholar
  17. Yangjing Dong, Honghao Fu, Anand Natarajan, Minglong Qin, Haochen Xu, and Penghui Yao. The computational advantage of MIP* vanishes in the presence of noise. arXiv preprint, 2023. URL: https://arxiv.org/abs/2312.04360.
  18. Bill Fefferman and Zachary Remscrim. Eliminating intermediate measurements in space-bounded quantum computation. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 1343-1356, 2021. Google Scholar
  19. Honghao Fu. Constant-sized correlations are sufficient to self-test maximally entangled states with unbounded dimension. Quantum, 6:614, January 2022. URL: https://doi.org/10.22331/q-2022-01-03-614.
  20. Badih Ghazi, Pritish Kamath, and Prasad Raghavendra. Dimension reduction for polynomials over gaussian space and applications. In Proceedings of the 33rd Computational Complexity Conference, CCC '18, pages 28:1-28:37, Germany, 2018. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2018.28.
  21. Insu Han, Dmitry Malioutov, Haim Avron, and Jinwoo Shin. Approximating spectral sums of large-scale matrices using stochastic Chebyshev approximations. SIAM Journal on Scientific Computing, 39(4):A1558-A1585, 2017. Google Scholar
  22. Prahladh Harsha, Adam Klivans, and Raghu Meka. An invariance principle for polytopes. J. ACM, 59(6), January 2013. URL: https://doi.org/10.1145/2395116.2395118.
  23. Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, July 2001. URL: https://doi.org/10.1145/502090.502098.
  24. Marcus Isaksson and Elchanan Mossel. Maximally stable Gaussian partitions with discrete applications. Israel Journal of Mathematics, 189(1):347-396, 2012. Google Scholar
  25. Tsuyoshi Ito, Hirotada Kobayashi, and Keiji Matsumoto. Oracularization and two-prover one-round interactive proofs against nonlocal strategies. In Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity, CCC 2009, pages 217-228, Washington, DC, USA, 2009. IEEE Computer Society. URL: https://doi.org/10.1109/CCC.2009.22.
  26. Tsuyoshi Ito and Thomas Vidick. A multi-prover interactive proof for NEXP sound against entangled provers. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, FOCS 2012, pages 243-252. IEEE, 2012. Google Scholar
  27. Zhengfeng Ji. Compression of quantum multi-prover interactive proofs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 289-302, New York, NY, USA, 2017. ACM. URL: https://doi.org/10.1145/3055399.3055441.
  28. Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. MIP^* = RE. arXiv preprint, 2020. URL: https://arxiv.org/abs/2001.04383.
  29. Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. Quantum soundness of the classical low individual degree test. arXiv preprint, 2020. URL: https://arxiv.org/abs/2009.12982.
  30. Daniel M. Kane. A Polylogarithmic PRG for Degree 2 Threshold Functions in the Gaussian Setting. In David Zuckerman, editor, 30th Conference on Computational Complexity (CCC 2015), volume 33 of Leibniz International Proceedings in Informatics (LIPIcs), pages 567-581, Dagstuhl, Germany, 2015. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2015.567.
  31. Zander Kelley and Raghu Meka. Random restrictions and prgs for ptfs in gaussian space. In Proceedings of the 37th Computational Complexity Conference, CCC '22, Dagstuhl, DEU, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.21.
  32. Julia Kempe, Hirotada Kobayashi, Keiji Matsumoto, Ben Toner, and Thomas Vidick. Entangled games are hard to approximate. SIAM Journal on Computing, 40(3):848-877, 2011. URL: https://doi.org/10.1137/090751293.
  33. Julia Kempe, Oded Regev, and Ben Toner. Unique games with entangled provers are easy. SIAM Journal on Computing, 39(7):3207-3229, 2010. URL: https://doi.org/10.1137/090772885.
  34. Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, STOC '02, pages 767-775, New York, NY, USA, 2002. Association for Computing Machinery. URL: https://doi.org/10.1145/509907.510017.
  35. Robert Krauthgamer and Ori Sasson. Property testing of data dimensionality. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2003, pages 18-27, USA, 2003. Society for Industrial and Applied Mathematics. Google Scholar
  36. Raghu Meka and David Zuckerman. Pseudorandom generators for polynomial threshold functions. In Proceedings of the Forty-Second ACM Symposium on Theory of Computing, STOC 2010, pages 427-436, New York, NY, USA, 2010. Association for Computing Machinery. URL: https://doi.org/10.1145/1806689.1806749.
  37. Elchanan Mossel, Ryan O'Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions with low influences: invariance and optimality. In 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005, pages 21-30. IEEE, 2005. Google Scholar
  38. Anand Natarajan and Thomas Vidick. A quantum linearity test for robustly verifying entanglement. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 1003-1015, 2017. Google Scholar
  39. Anand Natarajan and John Wright. NEEXP is Contained in MIP^*. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science, FOCS 2019, pages 510-518. IEEE, 2019. URL: https://doi.org/10.1109/FOCS.2019.00039.
  40. Anand Natarajan and Tina Zhang. Quantum free games. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 1603-1616, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3564246.3585208.
  41. Deanna Needell, William Swartworth, and David P. Woodruff. Testing positive semidefiniteness using linear measurements. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science, FOCS 2022, pages 87-97, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00016.
  42. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, Cambridge, UK, 2013. Google Scholar
  43. Ryan O'Donnell, Rocco A. Servedio, and Li-Yang Tan. Fooling gaussian ptfs via local hyperconcentration. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, pages 1170-1183, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3357713.3384281.
  44. Ryan O’Donnell, Rocco A. Servedio, and Li-Yang Tan. Fooling polytopes. J. ACM, 69(2), January 2022. URL: https://doi.org/10.1145/3460532.
  45. Connor Paddock. Rounding near-optimal quantum strategies for nonlocal games to strategies using maximally entangled states. arXiv preprint, 2022. URL: https://arxiv.org/abs/2203.02525.
  46. Minglong Qin and Penghui Yao. Nonlocal games with noisy maximally entangled states are decidable. SIAM Journal on Computing, 50(6):1800-1891, 2021. Google Scholar
  47. Minglong Qin and Penghui Yao. Decidability of Fully Quantum Nonlocal Games with Noisy Maximally Entangled States. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), volume 261 of Leibniz International Proceedings in Informatics (LIPIcs), pages 97:1-97:20, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.97.
  48. Oded Regev and Liron Schiff. Impossibility of a quantum speed-up with a faulty oracle. In International Colloquium on Automata, Languages, and Programming, pages 773-781. Springer, 2008. Google Scholar
  49. Ben W. Reichardt, Falk Unger, and Umesh Vazirani. Classical command of quantum systems. Nature, 496(7446):456-460, April 2013. URL: https://doi.org/10.1038/nature12035.
  50. Hristo S. Sendov. The higher-order derivatives of spectral functions. Linear Algebra and its Applications, 424(1):240-281, 2007. Special Issue in honor of Roger Horn. URL: https://doi.org/10.1016/j.laa.2006.12.013.
  51. Victor Shoup. New algorithms for finding irreducible polynomials over finite fields. Mathematics of computation, 54(189):435-447, 1990. Google Scholar
  52. Anna Skripka and Anna Tomskova. Multilinear operator integrals. Springer, 2019. Google Scholar
  53. William Slofstra. The set of quantum correlations is not closed. Forum of Mathematics, Pi, 7:e1, 2019. URL: https://doi.org/10.1017/fmp.2018.3.
  54. William Slofstra. Tsirelson’s problem and an embedding theorem for groups arising from non-local games. Journal of the American Mathematical Society, 33:1-56, 2020. URL: https://doi.org//10.1090/jams/929.
  55. Salil P. Vadhan. Pseudorandomness. Foundations and Trends® in Theoretical Computer Science, 7(1–3):1-336, 2012. URL: https://doi.org/10.1561/0400000010.
  56. Thomas Vidick. Almost synchronous quantum correlations. Journal of mathematical physics, 63(2), 2022. Google Scholar
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact