Abstract

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed to be able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems that are generally thought to be hard on classical computers and that have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, for example, the number of digits of the integer to be factored.

MSC codes

  1. 81P10
  2. 11Y05
  3. 68Q10
  4. 03D10

Keywords

  1. algorithmic number theory
  2. prime factorization
  3. discrete logarithms
  4. Church's thesis
  5. quantum computers
  6. foundations of quantum mechanics
  7. spin systems
  8. Fourier transforms

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References

1.
D. S. Abrams and S. Lloyd (1997), Simulation of many‐body Fermi systems on a universal quantum computer, Phys. Rev. Lett., 79, pp. 2586–2589.
2.
Leonard Adleman, Algorithmic number theory—the complexity contribution, IEEE Comput. Soc. Press, Los Alamitos, CA, 1994, 88–113
3.
L. M. Adleman and K. S. McCurley (1994), Open problems in number‐theoretic complexity II, in Algorithmic Number Theory, Proc. 1994 Algorithmic Number Theory Symposium, Ithaca, NY, Lecture Notes in Computer Science 877, L. M. Adleman and M.‐D. Huang, eds., Springer‐Verlag, Berlin, pp. 291–322.
4.
Aharonov and Ben‐Or (1997) Fault tolerant quantum computation with constant error in Proc. 29th Annual ACM Symposium on Theory of Computing, ACM, New York, pp. 176–188.
5.
A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter (1995a), Elementary gates for quantum computation, Phys. Rev. A, 52, pp. 3457–3467.
6.
A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa (1995b), Conditional quantum dynamics and logic gates, Phys. Rev. Lett., 74, pp. 4083–4086.
7.
David Beckman, Amalavoyal Chari, Srikrishna Devabhaktuni, John Preskill, Efficient networks for quantum factoring, Phys. Rev. A (3), 54 (1996), 1034–1063
8.
P. Benioff (1980), The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines, J. Statist. Phys., 22, pp. 563–591.
9.
Paul Benioff, Quantum mechanical Hamiltonian models of Turing machines, J. Statist. Phys., 29 (1982), 515–546
10.
P. Benioff (1982b), Quantum mechanical Hamiltonian models of Turing machines that dissipate no energy, Phys. Rev. Lett., 48, pp. 1581–1585.
11.
C. H. Bennett (1973), Logical reversibility of computation, IBM J. Res. Develop., 17, pp. 525–532.
12.
Charles Bennett, Time/space trade‐offs for reversible computation, SIAM J. Comput., 18 (1989), 766–776
13.
Charles Bennett, Ethan Bernstein, Gilles Brassard, Umesh Vazirani, Strengths and weaknesses of quantum computing, SIAM J. Comput., 26 (1997), 1510–1523
14.
C. H. Bennett and G. Brassard (1984), Quantum cryptography: Public key distribution and coin tossing, in Proc. IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, pp. 175–179.
15.
C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters (1993), Teleporting an unknown quantum state via dual classical and Einstein‐Podolsky‐Rosen channelsPhys. Rev. Lett., 70, pp. 1895–1898.
16.
C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wooters (1996), Purification of noisy entanglement and faithful teleportation via noisy channels, Phys. Rev. Lett., 76, pp. 722–725.
17.
Charles Bennett, Peter Shor, Quantum information theory, IEEE Trans. Inform. Theory, 44 (1998), 2724–2742, Information theory: 1948–1998
18.
Ethan Bernstein, Umesh Vazirani, Quantum complexity theory, SIAM J. Comput., 26 (1997), 1411–1473
19.
André Berthiaume, Gilles Brassard, The quantum challenge to structural complexity theory, IEEE Comput. Soc. Press, Los Alamitos, CA, 1992, 132–137
20.
André Berthiaume, Gilles Brassard, Oracle quantum computing, J. Modern Opt., 41 (1994), 2521–2535
21.
A. Berthiaume, D. Deutsch, and R. Jozsa (1994), The stabilisation of quantum computations, in Proc. Workshop on Physics of Computation: PhysComp ’94, IEEE Computer Society Press, Los Alamitos, CA, pp. 60–62.
22.
M. Biafore (1994), Can quantum computers have simple Hamiltonians, in Proc. Workshop on Physics of Computation: PhysComp ’94, IEEE Computer Society Press, Los Alamitos, CA, pp. 63–68.
23.
Dan Boneh, Richard Lipton, Quantum cryptanalysis of hidden linear functions (extended abstract), Lecture Notes in Comput. Sci., Vol. 963, Springer, Berlin, 1995, 424–437
24.
G. Brassard (1993), Quantum cryptography: A bibliography, SIGACT News, 24:3, pp. 16–20.
A more recent version is available online at: http://www.iro.umontreal.ca/∼crepeau/Biblio‐QC.html.
25.
H. Buhrman, R. Cleve and A. Wigderson (1998), Quantum vs. classical communication and computation, in Proc. of the 30th Annual ACM Symposium on Theory of Computing, ACM Press, New York, pp. 63–69.
26.
A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane (1997), Quantum error correction via orthogonal geometry, Physical Review Letters, 78, pp. 405–408.
27.
A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane (1998), Quantum error correction via codes over GF(4), IEEE Transactions on Information Theory, 44, pp. 1369–1387.
28.
A. R. Calderbank and P. W. Shor (1996), Good quantum error‐correcting codes exist, Phys. Rev. A, 54, pp. 1098–1106.
29.
J. F. Canny and J. Reif (1987), New lower bound techniques for robot motion planning problems, in Proc. 28th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, LosAlamitos, CA, pp. 49–60.
30.
J. Choi, J. Sellen, and C.‐K. Yap (1995), Precision‐sensitive Euclidean shortest path in 3‐space, in Proc. 11th Annual Symposium on Computational Geometry, Association for Computing Machinery, New York, pp. 350–359.
31.
I. Chuang, R. Laflamme, P. Shor, W. Zurek, Quantum computers, factoring, and decoherence, Science, 270 (1995), 1633–1635
32.
I. L. Chuang and Y. Yamamoto (1995), A simple quantum computer, Phys. Rev. A, 52, pp. 3489–3496.
33.
A. Church (1936), An unsolvable problem of elementary number theory, Amer. J. Math., 58, pp. 345–363.
34.
J. I. Cirac and P. Zoller (1995), Quantum computations with cold trapped ions, Phys. Rev. Lett., 74, pp. 4091–4094.
35.
R. Cleve (1994), A note on computing Fourier transforms by quantum programs, preprint.
36.
R. Cleve and H. Buhrman (1997), Substituting quantum entanglement for communication, Phys. Rev. A, 56, pp. 1201–1204.
37.
D. Coppersmith (1994), An Approximate Fourier Transform Useful in Quantum Factoring, IBM Research Report RC 19642.
38.
D. G. Cory, A. F. Fahmy, and T. F. Havel (1997), Ensemble quantum computing by nuclear magnetic resonance spectroscopy, Proc. Nat. Acad. Sci., 94, pp. 1634–1639.
39.
D. Deutsch (1985), Quantum theory, the Church–Turing principle and the universal quantum computer, Proc. Roy. Soc. London Ser. A, 400, pp. 96–117.
40.
D. Deutsch, Quantum computational networks, Proc. Roy. Soc. London Ser. A, 425 (1989), 73–90
41.
D. Deutsch, A. Barenco, and A. Ekert (1995), Universality of quantum computation, Proc. Roy. Soc. London Ser. A, 449, pp. 669–677.
42.
David Deutsch, Richard Jozsa, Rapid solution of problems by quantum computation, Proc. Roy. Soc. London Ser. A, 439 (1992), 553–558
43.
D. P. DiVincenzo (1995), Two‐bit gates are universal for quantum computation, Phys. Rev. A, 51, pp. 1015–1022.
44.
A. Ekert and R. Jozsa (1996), Shor’s quantum algorithm for factorising numbers, Rev. Mod. Phys., 68, pp. 733–753.
45.
A. Ekert and C. Macchiavello (1996), Error correction in quantum communication, Phys. Rev. Lett., 77, pp. 2585–2588.
46.
Richard Feynman, Simulating physics with computers, Internat. J. Theoret. Phys., 21 (1981/82), 467–488, Physics of computation, Part II (Dedham, Mass., 1981)
47.
R. Feynman (1986), Quantum mechanical computers, Found. Phys., 16, pp. 507–531;
originally published in Optics News (February 1985), pp. 11–20.
48.
Edward Fredkin, Tommaso Toffoli, Conservative logic, Internat. J. Theoret. Phys., 21 (1981/82), 219–253, Physics of computation, Part I (Dedham, Mass., 1981)
49.
N. A. Gershenfeld and I. L. Chuang (1997), Bulk spin resonance quantum computation, Science, 275, pp. 350–356.
50.
D. M. Gordon (1993), Discrete logarithms in GF(p) using the number field sieve, SIAM J. Discrete Math., 6, pp. 124–139.
51.
D. Gottesman (1996), A class of quantum error‐correcting codes saturating the quantum Hamming bound, Phys. Rev. A, 54, pp. 1862–1868.
52.
R. B. Griffiths and C.‐S. Niu (1996), Semiclassical Fourier tranform for quantum computation, Phys. Rev. Lett., 76, pp. 3228–3231.
53.
L. K. Grover (1997), Quantum mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett., 78, pp. 325–328.
54.
Lov Grover, A framework for fast quantum mechanical algorithms, ACM, New York, 1999, 53–62
55.
Ivan Niven, Herbert Zuckerman, Hugh Montgomery, An introduction to the theory of numbers, John Wiley & Sons Inc., 1991xiv+529
56.
J. Hartmanis and J. Simon (1974), On the power of multiplication in random access machines, in Proc. 15th Annual Symposium on Switching and Automata Theory, IEEE Computer Society, Long Beach, CA, pp. 13–23.
57.
B. E. Kane (1998), A silicon‐based nuclear spin quantum computer, Nature, 393, pp. 133–137.
58.
A. Karatsuba and Yu. Ofman (1962), Multiplication of multidigit numbers on automata, Dokl. Akad. Nauk SSSR, 145, pp. 293–294 (in Russian);
Sov. Phys. Dokl., 7 (1963), pp. 595–596 (English translation).
59.
A. Kitaev (1997), Quantum error correction with imperfect gates, manuscript.
60.
E. Knill (1995), personal communication.
61.
E. Knill and R. Laflamme (1997), A theory of quantum error‐correcting codes, Phys. Rev. A, 55, pp. 900–911.
62.
E. Knill, R. Laflamme, and W. H. Zurek (1998), Resiliant quantum computation, Science, 279, pp. 342–345.
63.
D. E. Knuth (1981), The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd ed., Addison‐Wesley, Reading, MA.
64.
R. Landauer (1995), Is quantum mechanics useful?Philos. Trans. Roy. Soc. London Ser. A, 353, pp. 367–376.
65.
R. Landauer (1997), Is quantum mechanically coherent computation useful?, in Proc. Drexel‐4 Symposium on Quantum Nonintegrability—Quantum Classical Correspondence, D. H. Feng and B‐L. Hu, eds., International Press, Cambridge, MA, to appear.
66.
Y. Lecerf (1963), Machines de Turing réversibles. Récursive insolubilité en nN de l’équation u=θu, où θ est un isomorphisme de codes, C. R. Acad. Française Sci., 257, pp. 2597–2600.
67.
A. Lenstra, H. Lenstra, The development of the number field sieve, Lecture Notes in Mathematics, Vol. 1554, Springer‐Verlag, 1993viii+131
68.
A. Lenstra, H. Lenstra, Jr., M. Manasse, J. Pollard, The number field sieve, Lecture Notes in Math., Vol. 1554, Springer, Berlin, 1993, 11–42
69.
Robert Levine, Alan Sherman, A note on Bennett’s time‐space tradeoff for reversible computation, SIAM J. Comput., 19 (1990), 673–677
70.
S. Lloyd (1993), A potentially realizable quantum computer, Science, 261, pp. 1569–1571.
71.
S. Lloyd (1994), Envisioning a quantum supercomputer, Science, 263, p. 695.
72.
S. Lloyd (1995), Almost any quantum logic gate is universal, Phys. Rev. Lett., 75, pp. 346–349.
73.
Seth Lloyd, Universal quantum simulators, Science, 273 (1996), 1073–1078
74.
N. Margolus (1986), Quantum computation, Ann. New York Acad. Sci., 480, pp. 487–497.
75.
N. Margolus (1990), Parallel quantum computation, in Complexity, Entropy and the Physics of Information, Santa Fe Institute Studies in the Sciences of Complexity, Vol. VIII, W. H. Zurek, ed., Addison‐Wesley, Reading, MA, pp. 273–287.
76.
D. Mayers (1998), Unconditional security in quantum cryptography, manuscript, LANL e‐print quant‐ph/9802025.
77.
D. Mayers and A. C.‐C. Yao (1998), Quantum cryptography with imperfect apparatus, in Proc. 39th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, to appear.
78.
G. L. Miller (1976), Riemann’s hypothesis and tests for primality, J. Comput. System Sci., 13, pp. 300–317.
79.
A. M. Odlyzko (1995), personal communication.
80.
G. Palma, Kalle‐Antti Suominen, Artur Ekert, Quantum computers and dissipation, Proc. Roy. Soc. London Ser. A, 452 (1996), 567–584
81.
A. Peres (1993), Quantum Theory: Concepts and Methods, Kluwer Academic Publishers, Dordrecht, The Netherlands.
82.
Carl Pomerance, Fast, rigorous factorization and discrete logarithm algorithms, Perspect. Comput., Vol. 15, Academic Press, Boston, MA, 1987, 119–143
83.
E. Post (1936), Finite combinatory processes. Formulation I, J. Symbolic Logic, 1, pp. 103–105.
84.
J. Preskill (1998), Lecture notes for Physics 229, California Institute of Technology, available online at http://www.theory.caltech.edu/people/preskill/ph229/.
85.
Daniel Gottesman, Fault‐tolerant quantum computation with higher‐dimensional systems, Chaos Solitons Fractals, 10 (1999), 1749–1758
86.
R. L. Rivest, A. Shamir, and L. Adleman (1978), A method of obtaining digital signatures and public‐key cryptosystems, Comm. Assoc. Comput. Mach., 21, pp. 120–126.
87.
Lee Rubel, Digital simulation of analog computation and Church’s thesis, J. Symbolic Logic, 54 (1989), 1011–1017
88.
Arnold Schönhage, Asymptotically fast algorithms for the numerical multiplication and division of polynomials with complex coefficients, Lecture Notes in Comput. Sci., Vol. 144, Springer, Berlin, 1982, 3–15
89.
A. Schönhage, A. F. W. Grotefeld, and E. Vetter (1994), Fast Algorithms: A Multitape Turing Machine Implementation, B. I. Wissenschaftsverlag, Mannheim, Germany.
90.
A. Schönhage and V. Strassen (1971), Schnelle Multiplikation grosser Zahlen, Computing, 7, pp. 281–292.
91.
P. W. Shor (1994), Algorithms for quantum computation: Discrete logarithms and factoring, in Proc. 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, pp. 124–134.
92.
P. W. Shor (1995), Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A, 52, pp. 2493–2496.
93.
Peter Shor, Fault‐tolerant quantum computation, IEEE Comput. Soc. Press, Los Alamitos, CA, 1996, 56–65
94.
P. W. Shor (1998) Quantum computing Documenta Mathematica, Extra Volume Proceedings ICM, Vol. I, pp. 305–324.
95.
Daniel Simon, On the power of quantum computation, SIAM J. Comput., 26 (1997), 1474–1483
96.
Tycho Sleator, Harald Weinfurter, Realizable universal quantum logic gates, Phys. Rev. Lett., 74 (1995), 4087–4090
97.
R. Solovay (1995), personal communication.
98.
A. Steane, Error correcting codes in quantum theory, Phys. Rev. Lett., 77 (1996), 793–797
99.
Andrew Steane, Multiple‐particle interference and quantum error correction, Proc. Roy. Soc. London Ser. A, 452 (1996), 2551–2577
100.
J. Steinbach, J. Twamley, Motional quantum error correction, J. Modern Opt., 47 (2000), 453–485, Physics of quantum information
101.
K. Steiglitz (1988), Two non‐standard paradigms for computation: Analog machines and cellular automata, in Performance Limits in Communication Theory and Practice, Proc. NATO Advanced Study Institute, Il Ciocco, Castelvecchio Pascoli, Tuscany, Italy, 1986, J. K. Skwirzynski, ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 173–192.
102.
W. G. Teich, K. Obermayer, and G. Mahler (1988), Structural basis of multistationary quantum systems II: Effective few‐particle dynamics, Phys. Rev. B, 37, pp. 8111–8121.
103.
Tommaso Toffoli, Reversible computing, Lecture Notes in Comput. Sci., Vol. 85, Springer, Berlin, 1980, 632–644
104.
A. M. Turing (1936), On computable numbers, with an application to the Entscheidungsproblem, in Proc. London Math. Soc. (2), 42, pp. 230–265;
corrections in Proc. London Math. Soc. (2), 43 (1937), pp. 544–546.
105.
W. G. Unruh (1995), Maintaining coherence in quantum computers, Phys. Rev. A, 51, pp. 992–997.
106.
Peter van Emde Boas, Machine models and simulations, Elsevier, Amsterdam, 1990, 1–66
107.
A. Vergis, K. Steiglitz, and B. Dickinson (1986), The complexity of analog computation, Math. Comput. Simulation, 28, pp. 91–113.
108.
Andrew Yao, Quantum circuit complexity, IEEE Comput. Soc. Press, Los Alamitos, CA, 1993, 352–361
109.
C. Zalka (1998), Efficient simulation of quantum systems by quantum computers, Proc. Roy. Soc. London Ser. A, 454, pp. 313–322.

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Published In

cover image SIAM Review
SIAM Review
Pages: 303 - 332
ISSN (online): 1095-7200

History

Published online: 4 August 2006

MSC codes

  1. 81P10
  2. 11Y05
  3. 68Q10
  4. 03D10

Keywords

  1. algorithmic number theory
  2. prime factorization
  3. discrete logarithms
  4. Church's thesis
  5. quantum computers
  6. foundations of quantum mechanics
  7. spin systems
  8. Fourier transforms

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