Zhengfeng Ji
Abstract
Note from the Research Highlights Co-Chairs: A Research Highlights paper appearing in Communications is usually peer-reviewed prior to publication. The following paper is unusual in that it is still under review. However, the result has generated enormous excitement in the research community, and came strongly nominated by SIGACT, a nomination seconded by external reviewers.
The complexity class NP characterizes the collection of computational problems that have efficiently verifiable solutions. With the goal of classifying computational problems that seem to lie beyond NP, starting in the 1980s complexity theorists have considered extensions of the notion of efficient verification that allow for the use of randomness (the class MA), interaction (the class IP), and the possibility to interact with multiple proofs, or provers (the class MIP). The study of these extensions led to the celebrated PCP theorem and its applications to hardness of approximation and the design of cryptographic protocols.
In this work, we study a fourth modification to the notion of efficient verification that originates in the study of quantum entanglement. We prove the surprising result that every problem that is recursively enumerable, including the Halting problem, can be efficiently verified by a classical probabilistic polynomial-time verifier interacting with two all-powerful but noncommunicating provers sharing entanglement. The result resolves long-standing open problems in the foundations of quantum mechanics (Tsirelson's problem) and operator algebras (Connes' embedding problem).
- Babai, L., Fortnow, L., Lund, C. Non-deterministic exponential time has two-prover interactive protocols. Comput. Complex. 1, 1 (1991), 3--40.Google Scholar
Cross Ref
- Bavarian, M., Vidick, T., Yuen, H. Hardness amplification for entangled games via anchoring. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, ACM, New York, NY, USA, 2017, 303--316.Google ScholarDigital Library
- Ben-Sasson, E., Goldreich, O., Harsha, P., Sudan, M., Vadhan, S. Robust PCPs of proximity, shorter PCPs, and applications to coding. SIAM J. Comput. 36, 4 (2006), 889--974.Google ScholarDigital Library
- Ben-Sasson, E., Sudan, M. Simple PCPs with poly-log rate and query complexity. In Proceedings of the Thirty-seventh Annual ACM Symposium on Theory of Computing, ACM, New York, NY, USA, 2005, 266--275.Google ScholarDigital Library
- Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 15 (1969), 880.Google ScholarCross Ref
- Cleve, R., Hoyer, P., Toner, B., Watrous, J. Consequences and limits of nonlocal strategies. In Proceedings. 19th IEEE Annual Conference on Computational Complexity 2004, IEEE, Washington, DC, USA, 2004, 236--249.Google ScholarCross Ref
- Colbeck, R. Quantum and relativistic protocols for secure multi-party computation. Ph. D. Thesis, 2009.Google Scholar
- Connes, A. Classification of injective factors cases II1, II∞, IIIλ, λ ≠ 1. Ann. Math. 104 (1976), 73--115.Google ScholarCross Ref
- Ekert, A.K. Quantum cryptography based on bell's theorem. Phys. Rev. Lett. 67, 6 (1991), 661.Google ScholarCross Ref
- Fritz, T., Netzer, T., Thom, A. Can you compute the operator norm? Proc. Am. Math. Soc. 142, 12 (2014), 4265--4276.Google ScholarCross Ref
- Gemmel, P., Lipton, R., Rubinfeld, R., Sudan, M., Wigderson, A. Self testing/correcting for polynomials and for approximate functions. In Proceedings of the 23rd STOC. ACM, New York, NY, USA, 1991, 32--42.Google Scholar
- Goldwasser, S., Kalai, Y.T., Rothblum, G.N. Delegating computation: interactive proofs for muggles. J. ACM 62, 4 (2015), 1--64.Google ScholarDigital Library
- Ito, T., Vidick, T. A multi-prover interactive proof for NEXP sound against entangled provers. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, IEEE, Washington, DC, USA, 2012, 243--252.Google ScholarDigital Library
- Ji, Z., Natarajan, A., Vidick, T., Wright, J., Yuen, H. Quantum soundness of the classical low individual degree test. arXiv preprint arXiv:2009.12982 (2020).Google Scholar
- Kleene, S.C. Introduction to Metamathematics. J. Symb. Log. 19, 3 (1954), 215--216.Google Scholar
- Mermin, D. Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett. 65, 27 (1990), 3373.Google ScholarCross Ref
- Natarajan, A., Vidick, T. Two-player entangled games are NP-hard. In Proceedings of the 33rd Computational Complexity Conference, Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2018, 20.Google Scholar
- Natarajan, A., Wright, J. NEEXP ⊆ MIP*. arXiv preprint arXiv:1904.05870v3 (2019).Google Scholar
- Ozawa, N. About the Connes embedding conjecture. Jpn. J. Math. 8, 1 (2013), 147--183.Google ScholarCross Ref
- Peres, A. Incompatible results of quantum measurements. Phys. Lett. A 151, 3-4 (1990), 107--108.Google ScholarCross Ref
- Rogers Jr., H. Theory of Recursive Functions and Effective Computability. MIT Press, Cambridge, MA, USA, 1987.Google ScholarDigital Library
- Tsirelson, B.S. Some results and problems on quantum bell-type inequalities. Hadronic J. Suppl. 8, 4 (1993), 329--345.Google Scholar
- Tsirelson, B.S. Bell inequalities and operator algebras, 2006. Problem statement from website of open problems at TU Braunschweig, 2006. Available at http://web.archive.org/web/20090414083019/ http://www.imaph.tu-bs.de/qi/problems/33.html.Google Scholar
Index Terms
- MIP* = RE
Recommendations
Infeasibility of instance compression and succinct PCPs for NP
STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computingThe OR-SAT problem asks, given Boolean formulae Φ1,...,Φm each of size at most n, whether at least one of the Φi's is satisfiable. We show that there is no reduction from OR-SAT to any set A where the length of the output is bounded by a polynomial in n,...
Collapse of the Hierarchy of Constant-Depth Exact Quantum Circuits
We study the quantum complexity class $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0 of quantum operations implementable exactly by constant-depth polynomial-size quantum circuits with unbounded fan-out gates. Our main result is that the quantum OR ...
Impossibility results on weakly black-box hardness amplification
FCT'07: Proceedings of the 16th international conference on Fundamentals of Computation TheoryWe study the task of hardness amplification which transforms a hard function into a harder one. It is known that in a high complexity class such as exponential time, one can convert worst-case hardness into average-case hardness. However, in a lower ...
Comments