Classical verification of quantum proofs

We present a classical interactive protocol that verifies the validity of a quantum witness state for the local Hamiltonian problem. It follows from this protocol that approximating the non-local value of a multi-player one-round game to inverse polynomial precision is QMA-hard. Our work makes an interesting connection between the theory of QMA-completeness and Hamiltonian complexity on one hand and the study of non-local games and Bell inequalities on the other.

[1]  Yaoyun Shi,et al.  Optimal robust quantum self-testing by binary nonlocal XOR games , 2012, 1207.1819.

[2]  E. Specker,et al.  The Problem of Hidden Variables in Quantum Mechanics , 1967 .

[3]  J. Biamonte,et al.  Realizable Hamiltonians for Universal Adiabatic Quantum Computers , 2007, 0704.1287.

[4]  Yaoyun Shi,et al.  Robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices , 2014, STOC.

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

[6]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[7]  Julia Kempe,et al.  The Complexity of the Local Hamiltonian Problem , 2004, FSTTCS.

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

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

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

[11]  Yaoyun Shi,et al.  Optimal Robust Self-Testing by Binary Nonlocal XOR Games , 2013, TQC.

[12]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[13]  Elad Eban,et al.  Interactive Proofs For Quantum Computations , 2017, 1704.04487.

[14]  U. Vazirani,et al.  Certifiable quantum dice , 2012, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[15]  Esther Hänggi,et al.  Device-independent quantum key distribution , 2010, ArXiv.

[16]  Joseph Fitzsimons,et al.  A Multiprover Interactive Proof System for the Local Hamiltonian Problem , 2014, ITCS.

[17]  Ashley Montanaro,et al.  Complexity Classification of Local Hamiltonian Problems , 2013, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[18]  Umesh V. Vazirani,et al.  A classical leash for a quantum system: command of quantum systems via rigidity of CHSH games , 2012, ITCS '13.

[19]  Fang Song,et al.  Zero-Knowledge Proof Systems for QMA , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[20]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[21]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[22]  Umesh V. Vazirani,et al.  Certifiable quantum dice: or, true random number generation secure against quantum adversaries , 2012, STOC '12.

[23]  T. H. Yang,et al.  Robust self-testing of the singlet , 2012, 1203.2976.

[24]  Zheng-Feng Ji,et al.  Binary Constraint System Games and Locally Commutative Reductions , 2013, ArXiv.

[25]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[26]  Stefano Pironio,et al.  Random numbers certified by Bell’s theorem , 2009, Nature.

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

[28]  E. Kashefi,et al.  Unconditionally verifiable blind quantum computation , 2012, 1203.5217.

[29]  J. S. BELLt.  Einstein-Podolsky-Rosen Paradox , 2018 .

[30]  T. Osborne.  Hamiltonian complexity , 2011, Reports on progress in physics. Physical Society.

[31]  Mikhail N. Vyalyi,et al.  Classical and Quantum Computation , 2002, Graduate studies in mathematics.

[32]  Mermin,et al.  Simple unified form for the major no-hidden-variables theorems. , 1990, Physical review letters.

[33]  Peter Høyer,et al.  Consequences and limits of nonlocal strategies , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[34]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

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

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

[37]  Laflamme,et al.  Perfect Quantum Error Correcting Code. , 1996, Physical review letters.

[38]  Irit Dinur,et al.  The PCP theorem by gap amplification , 2006, STOC.

[39]  Andrew Chi-Chih Yao,et al.  Quantum cryptography with imperfect apparatus , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[40]  B. S. Cirel'son.  Quantum generalizations of Bell's inequality , 1980 .

[41]  Numérisation de documents anciens mathématiques.  Bulletin de la Société Mathématique de France , 1873 .

[42]  Silvio Micali,et al.  The Knowledge Complexity of Interactive Proof Systems , 1989, SIAM J. Comput..

[43]  J. Bell.  On the Problem of Hidden Variables in Quantum Mechanics , 1966 .

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

[45]  Matthew McKague,et al.  Self-Testing Graph States , 2010, TQC.

[46]  Dorit Aharonov,et al.  Guest column: the quantum PCP conjecture , 2013, SIGA.

[47]  Umesh V. Vazirani,et al.  Classical command of quantum systems , 2013, Nature.

[48]  Seung Woo Shin,et al.  Quantum Hamiltonian Complexity , 2014, Found. Trends Theor. Comput. Sci..

[49]  Alexei Y. Kitaev,et al.  Parallelization, amplification, and exponential time simulation of quantum interactive proof systems , 2000, STOC '00.

[50]  Dorit Aharonov,et al.  Quantum NP - A Survey , 2002, quant-ph/0210077.

[51]  Frédéric Magniez,et al.  Self-testing of universal and fault-tolerant sets of quantum gates , 2000, STOC '00.

[52]  Yaoyun Shi.  Both Toffoli and controlled-NOT need little help to do universal quantum computing , 2003, Quantum Inf. Comput..

[53]  Dorit Aharonov,et al.  The Quantum PCP Conjecture , 2013, ArXiv.

[54]  Charles H. Bennett,et al.  Mixed-state entanglement and quantum error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[55]  Thomas Vidick,et al.  Three-Player Entangled XOR Games Are NP-Hard to Approximate , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[56]  Keiji Matsumoto,et al.  Quantum multi-prover interactive proof systems with limited prior entanglement , 2001, J. Comput. Syst. Sci..

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

[58]  John Watrous,et al.  PSPACE has constant-round quantum interactive proof systems , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[59]  Avi Wigderson,et al.  Multi-prover interactive proofs: how to remove intractability assumptions , 2019, STOC '88.

[60]  Lance Fortnow,et al.  On the Power of Multi-Prover Interactive Protocols , 1994, Theor. Comput. Sci..

[61]  Elham Kashefi,et al.  Universal Blind Quantum Computation , 2008, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[62]  J. Barrett.  Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality , 2001, quant-ph/0107045.

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

[64]  E. Kashefi,et al.  Unconditionally verifiable blind computation , 2012 .

[65]  Simone Severini,et al.  On the Quantum Chromatic Number of a Graph , 2007, Electron. J. Comb..

[66]  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.

[67]  Barbara M. Terhal,et al.  The complexity of quantum spin systems on a two-dimensional square lattice , 2008, Quantum Inf. Comput..

[68]  H. Briegel,et al.  Measurement-based quantum computation on cluster states , 2003, quant-ph/0301052.

[69]  C. Jordan.  Essai sur la géométrie à $n$ dimensions , 1875 .

[70]  Rahul Jain,et al.  QIP = PSPACE , 2011, JACM.

[71]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[72]  LundCarsten,et al.  Algebraic methods for interactive proof systems , 1992 .

[73]  V. Scarani,et al.  Device-independent security of quantum cryptography against collective attacks. , 2007, Physical review letters.

[74]  Carsten Lund,et al.  Nondeterministic exponential time has two-prover interactive protocols , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.