NP-complete Problems and Physical Reality

Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and “anthropic computing.” The section on soap bubbles even includes some “experimental” results. While I do not believe that any of the proposals will let us solve NP-complete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics.

[1]  D. Bohm.  A SUGGESTED INTERPRETATION OF THE QUANTUM THEORY IN TERMS OF "HIDDEN" VARIABLES. II , 1952 .

[2]  Jozef Gruska,et al.  Quantum Computing , 2008, Wiley Encyclopedia of Computer Science and Engineering.

[3]  John Gill,et al.  Relativizations of the P =? NP Question , 1975, SIAM J. Comput..

[4]  室 章治郎.  Michael R.Garey/David S.Johnson 著, "COMPUTERS AND INTRACTABILITY A guide to the Theory of NP-Completeness", FREEMAN, A5判変形判, 338+xii, \5,217, 1979 , 1980 .

[5]  E. Baum.  What Is Thought? (Bradford Books) , 2006 .

[6]  Tien D. Kieu,et al.  Quantum Algorithm for Hilbert's Tenth Problem , 2001, ArXiv.

[7]  Daniel R. Simon,et al.  On the power of quantum computation , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[8]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[9]  J. Bekenstein.  Universal upper bound on the entropy-to-energy ratio for bounded systems , 1981, Jacob Bekenstein.

[10]  Eric Allender,et al.  What can be efficiently reduced to the Kolmogorov-random strings? , 2006, Ann. Pure Appl. Log..

[11]  Eric B. Baum,et al.  What is thought? , 2003 .

[12]  Scott Aaronson,et al.  Quantum lower bound for the collision problem , 2001, STOC '02.

[13]  Scott Aaronson,et al.  Quantum lower bounds for the collision and the element distinctness problems , 2004, JACM.

[14]  B. Jack Copeland,et al.  Hypercomputation , 2004, Minds and Machines.

[15]  Volker Strassen,et al.  Algebraic Complexity Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[16]  R. Bousso.  The Holographic principle , 2002, hep-th/0203101.

[17]  Dominic J. A. Welsh,et al.  The Computational Complexity of the Tutte Plane: the Bipartite Case , 1992, Combinatorics, Probability and Computing.

[18]  D. Abrams,et al.  NONLINEAR QUANTUM MECHANICS IMPLIES POLYNOMIAL-TIME SOLUTION FOR NP-COMPLETE AND P PROBLEMS , 1998, quant-ph/9801041.

[19]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[20]  Gus Gutoski,et al.  Quantum Interactive Proofs with Competing Provers , 2004, STACS.

[21]  John Preskill,et al.  Comment on "The black hole final state" , 2004 .

[22]  E. Farhi,et al.  A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem , 2001, Science.

[23]  Andris Ambainis,et al.  Quantum lower bounds by quantum arguments , 2000, STOC '00.

[24]  T. P. Singh.  Gravitational collapse, black holes and naked singularities , 1998 .

[25]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[26]  D. Bacon.  Quantum computational complexity in the presence of closed timelike curves , 2003, quant-ph/0309189.

[27]  B. Dickinson,et al.  The complexity of analog computation , 1986 .

[28]  Michael Sipser,et al.  The history and status of the P versus NP question , 1992, STOC '92.

[29]  Stathis Zachos,et al.  Does co-NP Have Short Interactive Proofs? , 1987, Inf. Process. Lett..

[30]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[31]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[32]  Ronald de Wolf,et al.  Quantum lower bounds by polynomials , 2001, JACM.

[33]  S. Aaronson.  Quantum computing and hidden variables , 2004, quant-ph/0408035.

[34]  Gilles Brassard,et al.  Strengths and Weaknesses of Quantum Computing , 1997, SIAM J. Comput..

[35]  Lane A. Hemaspaandra,et al.  Threshold Computation and Cryptographic Security , 1993, ISAAC.

[36]  J. Polchinski,et al.  Weinberg's nonlinear quantum mechanics and the Einstein-Podolsky-Rosen paradox. , 1991, Physical review letters.

[37]  Umesh V. Vazirani,et al.  How powerful is adiabatic quantum computation? , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[38]  Daniel A. Spielman,et al.  PP is closed under intersection , 1991, STOC '91.

[39]  Craig Alan Feinstein.  Evidence that P is not equal to NP , 2003, ArXiv.

[40]  John Terning,et al.  INTRODUCTION TO THE ADS/CFT CORRESPONDENCE , 2005 .

[41]  R. Impagliazzo,et al.  P=BPP unless E has sub-exponential circuits: Derandomizing the XOR Lemma , 2002 .

[42]  Scott Aaronson,et al.  Limits on Efficient Computation in the Physical World , 2004, ArXiv.

[43]  Weinberg,et al.  Precision tests of quantum mechanics. , 1989, Physical review letters.

[44]  Antony Valentini.  Subquantum information and computation , 2002 .

[45]  Alexander A. Razborov,et al.  Natural Proofs , 2007 .

[46]  R. Solovay,et al.  Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question , 1975 .

[47]  Ronald L. Graham,et al.  Some NP-complete geometric problems , 1976, STOC '76.

[48]  A. Valentini.  On the pilot-wave theory of classical, quantum and subquantum physics , 1992 .

[49]  E. Farhi,et al.  Quantum Adiabatic Evolution Algorithms versus Simulated Annealing , 2002, quant-ph/0201031.

[50]  M. Born.  Zur Quantenmechanik der Stoßvorgänge , 1926 .

[51]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[52]  H Löe,et al.  The end of science? , 1997, European journal of oral sciences.

[53]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[54]  Nicolas Gisin,et al.  Weinberg's non-linear quantum mechanics and supraluminal communications , 1990 .

[55]  R. Penrose.  Angular Momentum: an Approach to Combinatorial Space-Time , 1971 .

[56]  Deutsch,et al.  Quantum mechanics near closed timelike lines. , 1991, Physical review. D, Particles and fields.

[57]  Gatis Midrijanis.  A Polynomial Quantum Query Lower Bound for the Set Equality Problem , 2004, ICALP.

[58]  J. Maldacena,et al.  The black hole final state , 2003, hep-th/0310281.

[59]  Leonid A. Levin,et al.  Average Case Complete Problems , 1986, SIAM J. Comput..

[60]  Lee Smolin.  The present moment in quantum cosmology: challenges to the arguments for the elimination of time , 2000 .

[61]  Arnold Schönhage,et al.  On the Power of Random Access Machines , 1979, ICALP.

[62]  Jin-Yi Cai,et al.  Circuit minimization problem , 2000, STOC '00.

[63]  Todd A. Brun.  Computers with Closed Timelike Curves Can Solve Hard Problems Efficiently , 2002, ArXiv.

[64]  Michael Larsen,et al.  A Modular Functor Which is Universal¶for Quantum Computation , 2000, quant-ph/0001108.

[65]  Emil T. Akhmedov,et al.  Introduction to the Ads/cft Correspondence , 1999 .

[66]  Greg Egan,et al.  An efficient algorithm for the Riemannian 10j symbols , 2001 .

[67]  Albert R. Meyer,et al.  Cosmological lower bound on the circuit complexity of a small problem in logic , 2002, JACM.

[68]  M. Hogarth.  Non-Turing Computers and Non-Turing Computability , 1994, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association.

[69]  Ben Reichardt,et al.  The quantum adiabatic optimization algorithm and local minima , 2004, STOC '04.

[70]  Lance Fortnow.  One complexity theorist's view of quantum computing , 2000, Electron. Notes Theor. Comput. Sci..

[71]  Scott Aaronson,et al.  Limitations of quantum advice and one-way communication , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[72]  M. Freedman,et al.  Simulation of Topological Field Theories¶by Quantum Computers , 2000, quant-ph/0001071.

[73]  Scott Aaronson,et al.  Quantum computing, postselection, and probabilistic polynomial-time , 2004, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[74]  Scott Aaronson.  Quantum Computing and Hidden Variables II: The Complexity of Sampling Histories , 2004 .