Abstract

The quantum model of computation is a model, analogous to the probabilistic Turing machine (PTM), in which the normal laws of chance are replaced by those obeyed by particles on a quantum mechanical scale, rather than the rules familiar to us from the macroscopic world. We present here a problem of distinguishing between two fairly natural classes of functions, which can provably be solved exponentially faster in the quantum model than in the classical probabilistic one, when the function is given as an oracle drawn equiprobably from the uniform distribution on either class. We thus offer compelling evidence that the quantum model may have significantly more complexity theoretic power than the PTM. In fact, drawing on this work, Shor has recently developed remarkable new quantum polynomial-time algorithms for the discrete logarithm and integer factoring problems.

MSC codes

  1. 03D15
  2. 68Q10
  3. 81P10

Keywords

  1. quantum computation
  2. complexity theory
  3. oracles

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References

1.
Paul Benioff, Quantum mechanical Hamiltonian models of Turing machines, J. Statist. Phys., 29 (1982), 515–546
2.
C. H. Bennett, Logical reversibility of computation, IBM J. Res. Develop., 17 (1973), pp. 525–532.
3.
Charles Bennett, Ethan Bernstein, Gilles Brassard, Umesh Vazirani, Strengths and weaknesses of quantum computing, SIAM J. Comput., 26 (1997), 1510–1523
4.
André Berthiaume, Gilles Brassard, The quantum challenge to structural complexity theory, IEEE Comput. Soc. Press, Los Alamitos, CA, 1992, 132–137
5.
André Berthiaume, Gilles Brassard, Oracle quantum computing, J. Modern Opt., 41 (1994), 2521–2535
6.
A. Barenco, C. Bennett, R. Cleve, D. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, Elementary gates for quantum computation, Phys. Rev. A, 52 (1995), pp. 3457–3467.
7.
Ethan Bernstein, Umesh Vazirani, Quantum complexity theory, SIAM J. Comput., 26 (1997), 1411–1473
8.
Benny Chor, Oded Goldreich, Unbiased bits from sources of weak randomness and probabilistic communication complexity, SIAM J. Comput., 17 (1988), 230–261, Special issue on cryptography
9.
D. Deutsch, Quantum theory, the Church‐Turing principle and the universal quantum computer, Proc. Roy. Soc. London Ser. A, 400 (1985), 97–117
10.
D. Deutsch, Quantum computational networks, Proc. Roy. Soc. London Ser. A, 425 (1989), 73–90
11.
D. DiVincenzo, Two‐bit gates are universal for quantum computation, Phys. Rev. A, 51 (1995), pp. 1015–1022.
12.
David Deutsch, Richard Jozsa, Rapid solution of problems by quantum computation, Proc. Roy. Soc. London Ser. A, 439 (1992), 553–558
13.
Christoph Dürr, Huong Lê Thanh, Miklos Santha, A decision procedure for well‐formed linear quantum cellular automata, Lecture Notes in Comput. Sci., Vol. 1046, Springer, Berlin, 1996, 281–292
14.
Richard Feynman, Simulating physics with computers, Internat. J. Theoret. Phys., 21 (1981/82), 467–488, Physics of computation, Part II (Dedham, Mass., 1981)
15.
Richard Feynman, Quantum mechanical computers, Found. Phys., 16 (1986), 507–531
16.
Lov Grover, A fast quantum mechanical algorithm for database search, ACM, New York, 1996, 212–219
17.
Yves Lecerf, Machines de Turing réversibles. Récursive insolubilité en nN de l’équation u=θnu, où θ est un “isomorphisme de codes”, C. R. Acad. Sci. Paris, 257 (1963), 2597–2600
18.
Peter Shor, Algorithms for quantum computation: discrete logarithms and factoring, IEEE Comput. Soc. Press, Los Alamitos, CA, 1994, 124–134
19.
Peter Shor, Fault‐tolerant quantum computation, IEEE Comput. Soc. Press, Los Alamitos, CA, 1996, 56–65
20.
Daniel Simon, On the power of quantum computation, IEEE Comput. Soc. Press, Los Alamitos, CA, 1994, 116–123
21.
U. V. Vazirani and V. V. Vazirani, Random polynomial time is equal to slightly‐random polynomial time, in Proc. 26th IEEE Symp. on Foundations of Computer Science, Portland, OR, 1985, pp. 417–428.
22.
Andrew Yao, Quantum circuit complexity, IEEE Comput. Soc. Press, Los Alamitos, CA, 1993, 352–361

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Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 1474 - 1483
ISSN (online): 1095-7111

History

Published online: 28 July 2006

MSC codes

  1. 03D15
  2. 68Q10
  3. 81P10

Keywords

  1. quantum computation
  2. complexity theory
  3. oracles

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