MIP* = RE

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).

[1]  A. Turing On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[2]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[3]  J. Hartmanis,et al.  On the Computational Complexity of Algorithms , 1965 .

[4]  Richard Edwin Stearns,et al.  Two-Tape Simulation of Multitape Turing Machines , 1966, JACM.

[5]  D. C. Cooper,et al.  Theory of Recursive Functions and Effective Computability , 1969, The Mathematical Gazette.

[6]  A. Shimony,et al.  Proposed Experiment to Test Local Hidden Variable Theories. , 1969 .

[7]  Jr. Hartley Rogers Theory of Recursive Functions and Effective Computability , 1969 .

[8]  A. Connes,et al.  Classification of Injective Factors Cases II 1 , II ∞ , III λ , λ 1 , 1976 .

[9]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[10]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[11]  Charles C. Wang,et al.  An Algorithm to Design Finite Field Multipliers Using a Self-Dual Normal Basis , 1987, IEEE Trans. Computers.

[12]  A. Peres Incompatible results of quantum measurements , 1990 .

[13]  V. Shoup New algorithms for finding irreducible polynomials over finite fields , 1990 .

[14]  Mermin Nd Simple unified form for the major no-hidden-variables theorems. , 1990 .

[15]  Carsten Lund,et al.  Algebraic methods for interactive proof systems , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[16]  Ronitt Rubinfeld,et al.  Self-testing/correcting for polynomials and for approximate functions , 1991, STOC '91.

[17]  H. Lenstra Finding isomorphisms between finite fields , 1991 .

[18]  Ekert,et al.  Quantum cryptography based on Bell's theorem. , 1991, Physical review letters.

[19]  Adi Shamir,et al.  IP = PSPACE , 1992, JACM.

[20]  László Lovász,et al.  Two-prover one-round proof systems: their power and their problems (extended abstract) , 1992, STOC '92.

[21]  B. Tsirelson Some results and problems on quan-tum Bell-type inequalities , 1993 .

[22]  Ronitt Rubinfeld,et al.  Robust Characterizations of Polynomials with Applications to Program Testing , 1996, SIAM J. Comput..

[23]  Ran Raz,et al.  A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP , 1997, STOC '97.

[24]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[25]  C. Caldwell Mathematics of Computation , 1999 .

[26]  P. K. Aravind A simple demonstration of Bell's theorem involving two observers and no probabilities or inequalities , 2002 .

[27]  M. Sudan,et al.  Robust pcps of proximity and shorter pcps , 2004 .

[28]  R. Cleve,et al.  Consequences and limits of nonlocal strategies , 2004 .

[29]  Carsten Lund,et al.  Non-deterministic exponential time has two-prover interactive protocols , 1992, computational complexity.

[30]  Eli Ben-Sasson,et al.  Simple PCPs with poly-log rate and query complexity , 2005, STOC '05.

[31]  Eli Ben-Sasson,et al.  Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding , 2004, SIAM J. Comput..

[32]  Keiji Matsumoto,et al.  Entangled Games are Hard to Approximate , 2007, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[33]  Stephanie Wehner,et al.  The Quantum Moment Problem and Bounds on Entangled Multi-prover Games , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[34]  A. Acín,et al.  A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations , 2008, 0803.4290.

[35]  Eli Ben-Sasson,et al.  Short PCPs with Polylog Query Complexity , 2008, SIAM J. Comput..

[36]  Yael Tauman Kalai,et al.  Delegating computation: interactive proofs for muggles , 2008, STOC.

[37]  Roger Colbeck,et al.  Quantum And Relativistic Protocols For Secure Multi-Party Computation , 2009, 0911.3814.

[38]  Tsuyoshi Ito,et al.  Oracularization and Two-Prover One-Round Interactive Proofs against Nonlocal Strategies , 2008, 2009 24th Annual IEEE Conference on Computational Complexity.

[39]  M. Junge,et al.  Connes' embedding problem and Tsirelson's problem , 2010, 1008.1142.

[40]  Tsuyoshi Ito,et al.  A Multi-prover Interactive Proof for NEXP Sound against Entangled Provers , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[41]  T. Fritz TSIRELSON'S PROBLEM AND KIRCHBERG'S CONJECTURE , 2010, 1008.1168.

[42]  Daniel Panario,et al.  Handbook of Finite Fields , 2013, Discrete mathematics and its applications.

[43]  Narutaka Ozawa,et al.  About the Connes embedding conjecture , 2013 .

[44]  Rajat Mittal,et al.  Characterization of Binary Constraint System Games , 2012, ICALP.

[45]  David Steurer,et al.  A parallel repetition theorem for entangled projection games , 2013, computational complexity.

[46]  Andreas Thom,et al.  Can you compute the operator norm , 2012, 1207.0975.

[47]  Rahul Jain,et al.  A Parallel Repetition Theorem for Entangled Two-Player One-Round Games under Product Distributions , 2013, 2014 IEEE 29th Conference on Computational Complexity (CCC).

[48]  Yael Tauman Kalai,et al.  Delegating computation: interactive proofs for muggles , 2008, STOC.

[49]  Carlos Palazuelos,et al.  Survey on Nonlocal Games and Operator Space Theory , 2015, 1512.00419.

[50]  Valerio Capraro,et al.  Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture , 2013, 1309.2034.

[51]  Zheng-Feng Ji,et al.  Classical verification of quantum proofs , 2015, STOC.

[52]  Thomas Vidick,et al.  Quantum Proofs , 2016, Found. Trends Theor. Comput. Sci..

[53]  Henry Yuen,et al.  A parallel repetition theorem for all entangled games , 2016, Electron. Colloquium Comput. Complex..

[54]  Simone Severini,et al.  Estimating quantum chromatic numbers , 2014, 1407.6918.

[55]  Isaac Goldbring,et al.  COMPUTABILITY AND THE CONNES EMBEDDING PROBLEM , 2016, The Bulletin of Symbolic Logic.

[56]  Thomas Vidick Three-Player Entangled XOR Games are NP-Hard to Approximate , 2016, SIAM J. Comput..

[57]  Jean-Daniel Bancal,et al.  Device-independent parallel self-testing of two singlets , 2015, 1512.02074.

[58]  Zheng-Feng Ji,et al.  Compression of quantum multi-prover interactive proofs , 2016, STOC.

[59]  Thomas Vidick,et al.  Hardness amplification for entangled games via anchoring , 2017, STOC.

[60]  Anand Natarajan,et al.  A quantum linearity test for robustly verifying entanglement , 2017, STOC.

[61]  Vern I. Paulsen,et al.  A synchronous game for binary constraint systems , 2017, 1707.01016.

[62]  Joseph Fitzsimons,et al.  Quantum proof systems for iterated exponential time, and beyond , 2018, Electron. Colloquium Comput. Complex..

[63]  Andrea Coladangelo,et al.  Unconditional separation of finite and infinite-dimensional quantum correlations , 2018, 1804.05116.

[64]  Anand Natarajan,et al.  Two-player entangled games are NP-hard , 2018, Computational Complexity Conference.

[65]  Anand Natarajan,et al.  Low-Degree Testing for Quantum States, and a Quantum Entangled Games PCP for QMA , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[66]  William Slofstra,et al.  THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED , 2017, Forum of Mathematics, Pi.

[67]  Anand Natarajan,et al.  NEEXP in MIP , 2019, FOCS 2019.

[68]  Mikael Rørdam,et al.  Non-closure of Quantum Correlation Matrices and Factorizable Channels that Require Infinite Dimensional Ancilla (With an Appendix by Narutaka Ozawa) , 2018, Communications in Mathematical Physics.

[69]  Vern I. Paulsen,et al.  Non-closure of the Set of Quantum Correlations via Graphs , 2017, Communications in Mathematical Physics.

[70]  William Slofstra,et al.  Tsirelson’s problem and an embedding theorem for groups arising from non-local games , 2016, Journal of the American Mathematical Society.

[71]  Quantum soundness of the classical low individual degree test , 2020, ArXiv.

[72]  Seyed Sajjad Nezhadi,et al.  On the complexity of zero gap MIP , 2020, ICALP.

[73]  Andrea Coladangelo,et al.  A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations , 2019, Quantum.