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Ultimate physical limits to computation

Computers are physical systems: the laws of physics dictate what they can and cannot do. In particular, the speed with which a physical device can process information is limited by its energy and the amount of information that it can process is limited by the number of degrees of freedom it possesses. Here I explore the physical limits of computation as determined by the speed of light c, the quantum scale and the gravitational constant G. As an example, I put quantitative bounds to the computational power of an ‘ultimate laptop’ with a mass of one kilogram confined to a volume of one litre.

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Figure 1: The ultimate laptop.
Figure 2: Computing at the black-hole limit.

References

  1. 1

    Maxwell, J. C. Theory of Heat (Appleton, London, 1871).

    Google Scholar 

  2. 2

    Smoluchowski, F. Vorträge über die kinetische Theorie der Materie u. Elektrizitat (Leipzig, 1914).

  3. 3

    Szilard, L. Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. Z. Physik 53, 840– 856 (1929).

    Article  ADS  CAS  MATH  Google Scholar 

  4. 4

    Brillouin, L. Science and Information Theory (Academic Press, New York, 1953).

  5. 5

    Landauer, R. Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5, 183–191 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  6. 6

    Keyes, R. W. & Landauer, R. Minimal energy dissipation in logic . IBM J. Res. Dev. 14, 152– 157 (1970).

    Article  Google Scholar 

  7. 7

    Landauer, R. Dissipation and noise-immunity in computation and communication. Nature 335, 779–784 ( 1988).

    Article  ADS  Google Scholar 

  8. 8

    Landauer, R. Information is physical. Phys. Today 44, 23–29 (1991).

    Article  ADS  Google Scholar 

  9. 9

    Landauer, R. The physical nature of information. Phys. Lett. A 217 , 188–193 (1996).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  10. 10

    von Neumann, J. Theory of Self-Reproducing Automata Lect. 3 (Univ. Illinois Press, Urbana, IL, 1966).

  11. 11

    Lebedev, D. S. & Levitin, L. B. Information transmission by electromagnetic field. Inform. Control 9, 1–22 (1966).

    Article  Google Scholar 

  12. 12

    Levitin, L. B. in Proceedings of the 3rd International Symposium on Radio Electronics part 3, 1–15 (Varna, Bulgaria, 1970).

    Google Scholar 

  13. 13

    Levitin, L. B. Physical limitations of rate, depth, and minimum energy in information processing . Int. J. Theor. Phys. 21, 299– 309 (1982).

    Article  Google Scholar 

  14. 14

    Levitin, L. B. Energy cost of information transmission (along the path to understanding) . Physica D 120, 162–167 (1998).

    Article  ADS  Google Scholar 

  15. 15

    Margolus, N. & Levitin, L. B. in Proceedings of the Fourth Workshop on Physics and Computation—PhysComp96 (eds Toffoli, T., Biafore, M. & Leão, J.) (New England Complex Systems Institute, Boston, MA, 1996).

    Google Scholar 

  16. 16

    Margolus, N. & Levitin, L. B. The maximum speed of dynamical evolution. Physica D 120, 188– 195 (1998).

    Article  ADS  Google Scholar 

  17. 17

    Bremermann, H. J. in Self-Organizing Systems (eds Yovits, M. C., Jacobi, G. T. & Goldstein, G. D.) 93–106 (Spartan Books, Washington DC, 1962).

    Google Scholar 

  18. 18

    Bremermann, H. J. in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (eds LeCam, L. M. & Neymen, J.) Vol. 4, 15–20 (Univ. California Press, Berkeley, CA, 1967).

    Google Scholar 

  19. 19

    Bremermann, H. J. Minimum energy requirements of information transfer and computing. Int. J. Theor. Phys. 21, 203–217 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  20. 20

    Bekenstein, J. D. Universal upper bound on the entropy-to-energy ration for bounded systems . Phys. Rev. D 23, 287– 298 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  21. 21

    Bekenstein, J. D. Energy cost of information transfer. Phys. Rev. Lett. 46, 623–626 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  22. 22

    Bekenstein, J. D. Entropy content and information flow in systems with limited energy. Phys. Rev. D 30, 1669–1679 (1984).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  23. 23

    Aharonov, Y. & Bohm, D. Time in the quantum theory and the uncertainty relation for the time and energy domain. Phys. Rev. 122, 1649–1658 ( 1961).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. 24

    Aharonov, Y. & Bohm, D. Answer to Fock concerning the time-energy indeterminancy relation. Phys. Rev. B 134, 1417–1418 (1964).

    Article  ADS  MATH  Google Scholar 

  25. 25

    Anandan, J. & Aharonov, Y. Geometry of quantum evolution. Phys. Rev. Lett. 65, 1697–1700 (1990).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  26. 26

    Peres, A. Quantum Theory: Concepts and Methods (Kluwer, Hingham, 1995).

  27. 27

    Lecerf, Y. Machines de Turing réversibles. C.R. Acad. Sci. 257, 2597–2600 (1963).

    MathSciNet  Google Scholar 

  28. 28

    Bennett, C. H. Logical reversibility of computation. IBM J. Res. Dev. 17, 525–532 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  29. 29

    Bennett, C.H. Thermodynamics of computation—a review. Int. J. Theor. Phys. 21, 905–940 ( 1982).

    Article  CAS  Google Scholar 

  30. 30

    Bennett, C. H. Demons, engines and the second law. Sci. Am. 257, 108 (1987).

    Article  ADS  Google Scholar 

  31. 31

    Fredkin, E. & Toffoli, T. Conservative logic. Int. J. Theor. Phys. 21, 219–253 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  32. 32

    Likharev, K. K. Classical and quantum limitations on energy consumption in computation. Int. J. Theor. Phys. 21, 311–325 (1982).

    Article  Google Scholar 

  33. 33

    Seitz, C. L. et al. in Proceedings of the 1985 Chapel Hill Conference on VLSI (ed. Fuchs, H.) (Computer Science Press, Rockville, MD, 1985).

    Google Scholar 

  34. 34

    Merkle, R. C. Reversible electronic logic using switches. Nanotechnology 34, 21–40 (1993).

    Article  ADS  Google Scholar 

  35. 35

    Younis, S. G. & Knight, T. F. in Proceedings of the 1993 Symposium on Integrated Systems, Seattle, Washington (eds Berrielo, G. & Ebeling, C.) (MIT Press, Cambridge, MA, 1993).

    Google Scholar 

  36. 36

    Lloyd, S. & Pagels, H. Complexity as thermodynamic depth . Ann. Phys. 188, 186–213 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  37. 37

    Lloyd, S. Use of mutual information to decrease entropy—implications for the Second Law of Thermodynamics. Phys. Rev. A 39, 5378–5386 (1989).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  38. 38

    Zurek, W. H. Thermodynamic cost of computation, algorithmic complexity and the information metric. Nature 341, 119– 124 (1989).

    Article  ADS  Google Scholar 

  39. 39

    Leff, H. S. & Rex, A. F. Maxwell's Demon: Entropy, Information, Computing (Princeton Univ. Press, Princeton, 1990).

    Book  Google Scholar 

  40. 40

    Lloyd, S. Quantum mechanical Maxwell's demon. Phys. Rev. A 56 , 3374–3382 (1997).

    Article  ADS  CAS  Google Scholar 

  41. 41

    Benioff, P. The computer as a physical system: a microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines. J. Stat. Phys. 22, 563–591 ( 1980).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. 42

    Benioff, P. Quantum mechanical models of Turing machines that dissipate no energy. Phys. Rev. Lett. 48, 1581–1585 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  43. 43

    Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467 (1982).

    Article  MathSciNet  Google Scholar 

  44. 44

    Feynman, R. P. Quantum mechanical computers. Optics News 11, 11 (1985); reprinted in Found. Phys. 16, 507 (1986).

    Article  Google Scholar 

  45. 45

    Zurek, W. H. Reversibility and stability of information-processing systems. Phys. Rev. Lett. 53, 391–394 (1984).

    Article  ADS  Google Scholar 

  46. 46

    Peres, A. Reversible logic and quantum computers. Phys. Rev. A 32, 3266–3276 (1985).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  47. 47

    Deutsch, D. Quantum-theory, the Church-Turing principle, and the universal quantum computer . Proc. R. Soc. Lond. A 400, 97– 117 (1985).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. 48

    Margolus, N. Quantum computation. Ann. N.Y. Acad. Sci. 480, 487–497 (1986).

    Article  ADS  Google Scholar 

  49. 49

    Deutsch, D. Quantum computational networks. Proc. R. Soc. Lond. A 425, 73–90 (1989).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. 50

    Margolus, N. in Complexity, Entropy, and the Physics of Information, Santa Fe Institute Studies in the Sciences of Complexity Vol. VIII (ed. Zurek, W. H.) 273–288 (Addison Wesley, Redwood City, 1991).

    Google Scholar 

  51. 51

    Lloyd, S. Quantum-mechanical computers and uncomputability. Phys. Rev. Lett. 71, 943–946 ( 1993).

    Article  ADS  CAS  Google Scholar 

  52. 52

    Lloyd, S. A potentially realizable quantum computer. Science 261, 1569–1571 (1993).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  53. 53

    Lloyd, S. Necessary and sufficient conditions for quantum computation. J. Mod. Opt. 41, 2503–2520 (1994).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. 54

    Shor, P. in Proceedings of the 35th Annual Symposium on Foundations of Computer Science (ed. Goldwasser, S.) 124–134 (IEEE Computer Society, Los Alamitos, CA, 1994).

    Book  Google Scholar 

  55. 55

    Lloyd, S. Quantum-mechanical computers. Sci. Am. 273, 140–145 (1995).

    Article  CAS  Google Scholar 

  56. 56

    DiVincenzo, D. Quantum computation. Science 270, 255– 261 (1995).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  57. 57

    DiVincenzo, D. P. 2-Bit gates are universal for quantum computation. Phys. Rev. A 51, 1015–1022 ( 1995).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  58. 58

    Sleator, T. & Weinfurter, H. Realizable universal quantum logic gates. Phys. Rev. Lett. 74, 4087– 4090 (1995).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  59. 59

    Barenco, A. et al. Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 ( 1995).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  60. 60

    Lloyd, S. Almost any quantum logic gate is universal. Phys. Rev. Lett. 75, 346–349 (1995).

    Article  ADS  CAS  Google Scholar 

  61. 61

    Deutsch, D., Barenco, A. & Ekert, A. Universality in quantum computation. Proc. R. Soc. Lond. A 449, 669–677 (1995).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. 62

    Cirac, J. I. & Zoller, P. Quantum computation with cold ion traps. Phys. Rev. Lett. 74, 4091– 4094 (1995).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  63. 63

    Pellizzari, T., Gardiner, S. A., Cirac, J. I. & Zoller, P. Decoherence, continuous observation, and quantum computing—a cavity QED model. Phys. Rev. Lett. 75, 3788– 3791 (1995).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  64. 64

    Turchette, Q. A., Hood, C. J., Lange, W., Mabuchi, H. & Kimble, H. J. Measurement of conditional phase-shifts for quantum logic. Phys. Rev. Lett. 75, 4710– 4713 (1995).

    Article  ADS  MathSciNet  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  65. 65

    Monroe, C., Meekhof, D. M., King, B. E., Itano, W. M. & Wineland, D. J. Demonstration of a fundamental quantum logic gate. Phys. Rev. Lett. 75, 4714–4717 (1995).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  66. 66

    Grover, L. K. in Proceedings of the 28th Annual ACM Symposium on the Theory of Computing 212–218 (ACM Press, New York, 1996 ).

    Google Scholar 

  67. 67

    Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996).

    Article  ADS  MathSciNet  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  68. 68

    Zalka, C. Simulating quantum systems on a quantum computer. Proc. R. Soc. Lond A 454, 313–322 ( 1998).

    Article  ADS  MATH  Google Scholar 

  69. 69

    Shor, P. W. A scheme for reducing decoherence in quantum memory. Phys. Rev. A 52, R2493–R2496 ( 1995).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  70. 70

    Steane, A. M. Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793–797 (1996).

    Article  ADS  MathSciNet  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  71. 71

    Laflamme, R., Miquel, C., Paz, J. P. & Zurek, W. H. Perfect quantum error correcting code. Phys. Rev. Lett. 77, 198–201 (1996).

    Article  ADS  CAS  Google Scholar 

  72. 72

    DiVincenzo, D. P. & Shor, P. W. Fault-tolerant error correction with efficient quantum codes. Phys. Rev. Lett. 77, 3260–3263 ( 1996).

    Article  ADS  CAS  Google Scholar 

  73. 73

    Shor, P. in Proceedings of the 37th Annual Symposium on the Foundations of Computer Science 56–65 (IEEE Computer Society Press, Los Alamitos, CA, 1996).

    Google Scholar 

  74. 74

    Preskill, J. Reliable quantum computers. Proc. R. Soc. Lond. A 454 , 385–410 (1998).

    Article  ADS  MATH  Google Scholar 

  75. 75

    Knill, E., Laflamme, R. & Zurek, W. H. Resilient quantum computation. Science 279, 342–345 ( 1998).

    Article  ADS  CAS  MATH  Google Scholar 

  76. 76

    Cory, D. G., Fahmy, A. F. & Havel, T. F. in Proceedings of the Fourth Workshop on Physics and Computation—PhysComp96 (eds Toffoli, T., Biafore, M. & Leão, J.) 87–91 (New England Complex Systems Institute, Boston, MA, 1996).

    Google Scholar 

  77. 77

    Gershenfeld, N. A. & Chuang, I. L. Bulk spin-resonance quantum computation. Science 275, 350– 356 (1997).

    Article  MathSciNet  CAS  MATH  Google Scholar 

  78. 78

    Chuang, I. L., Vandersypen, L. M. K., Zhou, X., Leung, D. W. & Lloyd, S. Experimental realization of a quantum algorithm. Nature 393, 143– 146 (1998).

    Article  ADS  CAS  Google Scholar 

  79. 79

    Jones, J. A., Mosca, M. & Hansen, R. H. Implementation of a quantum search algorithm on a quantum computer. Nature 393, 344– 346 (1998).

    Article  ADS  Google Scholar 

  80. 80

    Chuang, I. L., Gershenfeld, N. & Kubinec, M. Experimental implementation of fast quantum searching . Phys. Rev. Lett. 80, 3408– 3411 (1998).

    Article  ADS  CAS  Google Scholar 

  81. 81

    Kane, B. A silicon-based nuclear-spin quantum computer. Nature 393, 133 (1998).

    Article  ADS  CAS  Google Scholar 

  82. 82

    Nakamura, Y., Pashkin, Yu. A. & Tsai, J. S. Coherent control of macroscopic quantum states in a single-Cooper-pair box. Nature 398, 786 –788 (1999).

    Article  ADS  CAS  Google Scholar 

  83. 83

    Mooij, J. E. et al. Josephson persistent-current qubit. Science 285, 1036–1039 (1999).

    Article  CAS  Google Scholar 

  84. 84

    Lloyd, S. & Braunstein, S. Quantum computation over continuous variables. Phys. Rev. Lett. 82, 1784– 1787 (1999).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  85. 85

    Abrams, D. & Lloyd, S. Nonlinear quantum mechanics implies polynomial-time solution for NP-complete and P problems. Phys. Rev. Lett. 81, 3992–3995 (1998).

    Article  ADS  CAS  Google Scholar 

  86. 86

    Zel'dovich, Ya. B. & Novikov, I. D. Relativistic Astrophysics (Univ. of Chicago Press, Chicago, 1971 ).

    Google Scholar 

  87. 87

    Novikov, I. D. & Frolov, V. P. Black Holes (Springer, Berlin, 1986).

    Google Scholar 

  88. 88

    Pagels, H. The Cosmic Code: Quantum Physics as the Language of Nature (Simon and Schuster, New York, 1982).

    Google Scholar 

  89. 89

    Coleman, S., Preskill, J. & Wilczek, F. Growing hair on black-holes. Phys. Rev. Lett. 67, 1975–1978 ( 1991).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  90. 90

    Preskill, J. Quantum hair. Phys. Scr. T 36, 258– 264 (1991).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  91. 91

    Fiola, T. M., Preskill, J. & Strominger A. Black-hole thermodynamics and information loss in 2 dimensions. Phys. Rev. D 50, 3987– 4014 (1994).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  92. 92

    Susskind, L. & Uglum, J. Black-hole entropy in canonical quantum-gravity and superstring theory. Phys. Rev. D 50, 2700–2711 (1994).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  93. 93

    Strominger A. & Vafa, C. Microscopic origin of the Bekenstein-Hawking entropy. Phys. Lett. B 37, 99– 104 (1996).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  94. 94

    Das, S. R. & Mathur, S. D. Comparing decay rates for black holes and D-branes. Nucl. Phys. B 478, 561 –576 (1996).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  95. 95

    Page, D. N. Particle emision rates from a black-hole: massless particles form an uncharged non-rotating black-hole. Phys. Rev. D 13, 198 (1976).

    Article  ADS  CAS  Google Scholar 

  96. 96

    Thorne, K. S., Zurek, W. H. & Price R. H. in Black Holes: The Membrane Paradigm Ch. VIII (eds Thorne, K. S., Price, R. H. & Macdonald, D. A.) 280– 340 (Yale Univ. Press, New Haven, CT, 1986).

    Google Scholar 

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Lloyd, S. Ultimate physical limits to computation. Nature 406, 1047–1054 (2000). https://doi.org/10.1038/35023282

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