Guest Column: NP-complete problems and physical reality (original) (raw)
Published: 01 March 2005 Publication History
Abstract
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.
References
[1]
S. Aaronson. Quantum lower bound for the collision problem. In Proc. ACM STOC, pages 635--642, 2002. quant-ph/0111102.
[2]
S. Aaronson. Limitations of quantum advice and one-way communication. Theory of Computing, 2004. To appear. Conference version in Proc. IEEE Complexity 2004, pp. 320--332. quant-ph/0402095.
[3]
S. Aaronson. Limits on Efficient Computation in the Physical World. PhD thesis, University of California, Berkeley, 2004.
[4]
S. Aaronson. Quantum computing and hidden variables. Accepted to Phys. Rev. A. quant-ph/0408035 and quant-ph/0408119, 2004.
[5]
S. Aaronson. Quantum computing, postselection, and probabilistic polynomial-time. Submitted. quant-ph/0412187, 2004.
[6]
D. S. Abrams and S. Lloyd. Nonlinear quantum mechanics implies polynomial-time solution for NP-complete and #P problems. Phys. Rev. Lett., 81:3992--3995, 1998. quant-ph/9801041.
[7]
D. Aharonov, V. Jones, and Z. Landau. On the quantum algorithm for approximating the Jones polynomial. Unpublished, 2005.
[8]
E. Allender, H. Buhrman, and M. Koucký. What can be efficiently reduced to the Kolmogorov-random strings? In Proc. Intl. Symp. on Theoretical Aspects of Computer Science (STACS), pages 584--595, 2004. To appear in Annals of Pure and Applied Logic.
[9]
A. Ambainis. Quantum lower bounds by quantum arguments. J. Comput. Sys. Sci., 64:750--767, 2002. Earlier version in ACM STOC 2000. quant-ph/0002066.
[10]
D. Bacon. Quantum computational complexity in the presence of closed timelike curves. quant-ph/0309189, 2003.
[11]
T. Baker, J. Gill, and R. Solovay. Relativizations of the P=?NP question. SIAM J. Comput., 4:431--442, 1975.
[12]
E. B. Baum. What Is Thought? Bradford Books, 2004.
[13]
R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. de Wolf. Quantum lower bounds by polynomials. J. ACM, 48(4):778--797, 2001. Earlier version in IEEE FOCS 1998, pp. 352--361. quant-ph/9802049.
[14]
J. E. Beasley. OR-Library (test data sets for operations research problems), 1990. At www.brunel.ac.uk/depts/ma/research/jeb/info.html.
[15]
R. Beigel, N. Reingold, and D. Spielman. PP is closed under intersection. J. Comput. Sys. Sci., 50(2):191--202, 1995.
[16]
J. D. Bekenstein. A universal upper bound on the entropy to energy ratio for bounded systems. Phys. Rev. D, 23(2):287--298, 1981.
[17]
C. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computing. SIAM J. Comput., 26(5):1510--1523, 1997. quant-ph/9701001.
[18]
L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation. Springer-Verlag, 1997.
[19]
J. de Boer. Introduction to the AdS/CFT correspondence. University of Amsterdam Institute for Theoretical Physics (ITFA) Technical Report 03-02, 2003.
[20]
D. Bohm. A suggested interpretation of the quantum theory in terms of "hidden" variables. Phys. Rev., 85:166--193, 1952.
[21]
R. B. Boppana, J. Håstad, and S. Zachos. Does co-NP have short interactive proofs? Inform. Proc. Lett., 25:127--132, 1987.
[22]
M. Born. Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik, 37:863--867, 1926. English translation in Quantum Theory and Measurement (J. A. Wheeler and W. H. Zurek, eds.), Princeton, 1983, pp. 52--55.
[23]
R. Bousso. The holographic principle. Reviews of Modern Physics, 74(3), 2002. hep-th/0203101.
[24]
S. Bringsjord and J. Taylor. P=NP. 2004. cs.CC/0406056.
[25]
T. Brun. Computers with closed timelike curves can solve hard problems. Foundations of Physics Letters, 16:245--253, 2003. gr-qc/0209061.
[26]
P. Bürgisser, M. Clausen, and M. A. Shokrollahi. Algebraic Complexity Theory. Springer-Verlag, 1997.
[27]
J. D. Christensen and C. Egan. An efficient algorithm for the Riemannian 10j symbols. Classical and Quantum Gravity, 19:1184--1193, 2002. gr-qc/0110045.
[28]
J. Copeland. Hypercomputation. Minds and Machines, 12:461--502, 2002.
[29]
D. Deutsch. Quantum mechanics near closed timelike lines. Phys. Rev. D, 44:3197--3217, 1991.
[30]
G. Egan. Quarantine: A Novel of Quantum Catastrophe. Eos, 1995. First printing 1992.
[31]
E. Farhi, J. Goldstone, and S. Gutmann. Quantum adiabatic evolution algorithms versus simulated annealing. quant-ph/0201031, 2002.
[32]
E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda. A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science, 292:472--476, 2001. quant-ph/0104129.
[33]
C. Feinstein. Post to comp. theory newsgroup on July 8, 2004.
[34]
C. Feinstein. Evidence that P is not equal to NP. cs.CC/.0310060, 2003.
[35]
L. Fortnow. One complexity theorist's view of quantum computing. Theoretical Comput. Sci., 292(3):597--610, 2003.
[36]
M. Freedman, A. Kitaev, and Z. Wang. Simulation of topological quantum field theories by quantum computers. Commun. Math. Phys., 227:587--603, 2002. quant-ph/0001071.
[37]
M. Freedman, M. Larsen, and Z. Wang. A modular functor which is universal for quantum computation. Commun. Math. Phys., 227:605--622, 2002. quant-ph/0001108.
[38]
M. R. Garey, R. L. Graham, and D. S. Johnson. Some NP-complete geometric problems. In Proc. ACM STOC, pages 10--22, 1976.
[39]
M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979.
[40]
W. Gasarch. The P=?NP poll. SIGACT News, 33(2):34--47, June 2002.
[41]
N. Gisin. Weinberg's non-linear quantum mechanics and superluminal communications. Phys. Lett. A, 143:1--2, 1990.
[42]
D. Gottesman and J. Preskill. Comment on "The black hole final state". J. High Energy Phys., (0403:026), 2004. hep-th/0311269.
[43]
L. K. Grover. A fast quantum mechanical algorithm for database search. In Proc. ACM STOC, pages 212--219, 1996. quant-ph/9605043.
[44]
G. Gutoski and J. Watrous. Quantum interactive proofs with competing provers. To appear in STACS 2005. cs.CC/0412102, 2004.
[45]
Y. Han, L. Hemaspaandra, and T. Thierauf. Threshold computation and cryptographic security. SIAM J. Comput., 26(1):59--78, 1997.
[46]
M. Hogarth. Non-Turing computers and non-Turing computability. Biennial Meeting of the Philosophy of Science Association, 1:126--138, 1994.
[47]
J. Horgan. The End of Science. Helix Books, 1997.
[48]
G. Horowitz and J. Maldacena. The black hole final state. J. High Energy Phys., (0402:008), 2004. hep-th/0310281.
[49]
R. Impagliazzo and A. Wigderson. P=BPP unless E has subexponential circuits: derandomizing the XOR Lemma. In Proc. ACM STOC, pages 220--229, 1997.
[50]
Clay Math Institute. Millennium prize problems, 2000. www.claymath.org/millennium/.
[51]
V. Kabanets and J.-Y. Cai. Circuit minimization problem. In Proc. ACM STOC, pages 73--79, 2000. TR99-045.
[52]
R. M. Karp and R. J. Lipton. Turing machines that take advice. Enseign. Math., 28:191--201, 1982.
[53]
T. D. Kieu. Quantum algorithm for hilbert's tenth problem. Intl. Journal of Theoretical Physics, 42:1461--1478, 2003. quant-ph/0110136.
[54]
L. A. Levin. Average case complete problems. SIAM J. Comput., 15(1):285--286, 1986.
[55]
L. A. Levin. Polynomial time and extravagant models, in The tale of one-way functions. Problems of Information Transmission, 39(1):92--103, 2003. cs.CR/0012023.
[56]
G. Midrijanis. A polynomial quantum query lower bound for the set equality problem. In Proc. Intl. Colloquium on Automata, Languages, and Programming (ICALP), pages 996--1005, 2004. quant-ph/0401073.
[57]
M. Nielsen and I. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000.
[58]
R. Penrose. Angular momentum: an approach to combinatorial spacetime. In T. Bastin, editor, Quantum Theory and Beyond. Cambridge, 1971.
[59]
J. Polchinski. Weinberg's nonlinear quantum mechanics and the Einstein-Podolsky-Rosen paradox. Phys. Rev. Lett., 66:397--400, 1991.
[60]
J. Preskill. Quantum computation and the future of physics. Talk at UC Berkeley, May 10, 2002.
[61]
A. A. Razborov and S. Rudich. Natural proofs. J. Comput. Sys Sci., 55(1):24--35, 1997.
[62]
B. Reichardt. The quantum adiabatic optimization algorithm and local minima. In Proc. ACM STOC, pages 502--510, 2004.
[63]
A. Schönhage. On the power of random access machines. In Proc. Intl. Colloquium on Automata, Languages, and Programming (ICALP), pages 520--529, 1979.
[64]
Y. Shi. Quantum lower bounds for the collision and the element distinctness problems. In Proc. IEEE FOCS, pages 513--519, 2002. quant-ph/0112086.
[65]
P. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput., 26(5):1484--1509, 1997. Earlier version in IEEE FOCS 1994. quant-ph/9508027.
[66]
D. Simon. On the power of quantum computation. In Proc. IEEE FOCS, pages 116--123, 1994.
[67]
T. P. Singh. Gravitational collapse, black holes, and naked singularities. In Proceedings of Discussion Workshop on Black Holes, Bangalore, India, Dec 1997. gr-qc/9805066.
[68]
M. Sipser. The history and status of the P versus NP question. In Proc. ACM STOC, pages 603--618, 1992.
[69]
L. Smolin. The present moment in quantum cosmology: challenges to arguments for the elimination of time. In R. Durie, editor, Time and the Instant. Clinamen Press, 2000. gr-qc/0104097.
[70]
L. J. Stockmeyer and A. R. Meyer. Cosmological lower bound on the circuit complexity of a small problem in logic. J. ACM, 49(6):753--784, 2002.
[71]
A. Valentini. On the Pilot-Wave Theory of Classical, Quantum, and Subquantum Physics. PhD thesis, International School for Advanced Studies, 1992.
[72]
A. Valentini. Subquantum information and computation. Pramana J. Physics, 59(2):269--277, 2002. quant-ph/0203049.
[73]
W. van Dam, M. Mosca, and U. Vazirani. How powerful is adiabatic quantum computation? In Proc. IEEE FOCS, pages 279--287, 2001. quant-ph/0206003.
[74]
A. Vergis, K. Steiglitz, and B. Dickinson. The complexity of analog computation. Mathematics and Computers in Simulation, 28(91--113), 1986.
[75]
D. L. Vertigan and D. J. A. Welsh. The computational complexity of the Tutte plane: the bipartite case. Combinatorics, Probability, and Computing, 1(2), 1992.
[76]
S. Weinberg. Precision tests of quantum mechanics. Phys. Rev. Lett., 62:485--488, 1989.
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Published In
ACM SIGACT News Volume 36, Issue 1
March 2005
101 pages
Copyright © 2005 Author.
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Association for Computing Machinery
New York, NY, United States
Publication History
Published: 01 March 2005
Published in SIGACT Volume 36, Issue 1
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Scott Aaronson
University of Rochester, Rochester, NY