메뉴 건너뛰기




Volumn 26, Issue 5, 1997, Pages 1484-1509

Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer

Author keywords

Algorithmic number theory; Church's thesis; Discrete logarithms; Foundations of quantum mechanics; Fourier transforms; Prime factorization; Quantum computers; Spin systems

Indexed keywords


EID: 0142051871     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539795293172     Document Type: Article
Times cited : (6361)

References (83)
  • 1
    • 33749896857 scopus 로고
    • Algorithmic number theory - The complexity contribution
    • IEEE Computer Society Press, Los Alamitos, CA
    • L. M. ADLEMAN (1994), Algorithmic number theory - The complexity contribution, in Proc. 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, pp. 88-113.
    • (1994) Proc. 35th Annual Symposium on Foundations of Computer Science , pp. 88-113
    • Adleman, L.M.1
  • 2
    • 84874976434 scopus 로고
    • Open problems in number-theoretic complexity II
    • Algorithmic Number Theory, Proc. 1994 Algorithmic Number Theory Symposium, Ithaca, NY, L. M. Adleman and M.-D. Huang, eds., Springer-Verlag, Berlin
    • 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.
    • (1994) Lecture Notes in Computer Science , vol.877 , pp. 291-322
    • Adleman, L.M.1    Mccurley, K.S.2
  • 4
    • 5744237957 scopus 로고
    • Conditional quantum dynamics and logic gates
    • A. BARENCO, D. DEUTSCH, A. EKERT, AND R. JOZSA (1995b), Conditional quantum dynamics and logic gates, Phys. Rev. Lett., 74, pp. 4083-4086.
    • (1995) Phys. Rev. Lett. , vol.74 , pp. 4083-4086
    • Barenco, A.1    Deutsch, D.2    Ekert, A.3    Jozsa, R.4
  • 6
    • 34250256884 scopus 로고
    • The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines
    • 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.
    • (1980) J. Statist. Phys. , vol.22 , pp. 563-591
    • Benioff, P.1
  • 7
    • 0347572604 scopus 로고
    • Quantum mechanical Hamiltonian models of Turing machines
    • P. BENIOFF (1982a), Quantum mechanical Hamiltonian models of Turing machines, J. Statist. Phys., 29, pp. 515-546.
    • (1982) J. Statist. Phys. , vol.29 , pp. 515-546
    • Benioff, P.1
  • 8
    • 4243610080 scopus 로고
    • Quantum mechanical Hamiltonian models of Turing machines that dissipate no energy
    • P. BENIOFF (1982b), Quantum mechanical Hamiltonian models of Turing machines that dissipate no energy, Phys. Rev. Lett., 48, pp. 1581-1585.
    • (1982) Phys. Rev. Lett. , vol.48 , pp. 1581-1585
    • Benioff, P.1
  • 9
    • 0015680909 scopus 로고
    • Logical reversibility of computation
    • C. H. BENNETT (1973), Logical reversibility of computation, IBM J. Res. Develop., 17, pp. 525-532.
    • (1973) IBM J. Res. Develop. , vol.17 , pp. 525-532
    • Bennett, C.H.1
  • 10
    • 0024714327 scopus 로고
    • Time/space trade-offs for reversible computation
    • C. H. BENNETT (1989), Time/space trade-offs for reversible computation, SIAM J. Comput., 18, pp. 766-776.
    • (1989) SIAM J. Comput. , vol.18 , pp. 766-776
    • Bennett, C.H.1
  • 13
    • 0027245270 scopus 로고
    • Quantum complexity theory
    • Association for Computing Machinery, New York
    • E. BERNSTEIN AND U. VAZIRANI (1993), Quantum complexity theory, in Proc. 25th Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, New York, pp. 11-20; SIAM J. Comput., 26 (1997), pp. 1411-1473.
    • (1993) Proc. 25th Annual ACM Symposium on Theory of Computing , pp. 11-20
    • Bernstein, E.1    Vazirani, U.2
  • 14
    • 5544255325 scopus 로고    scopus 로고
    • E. BERNSTEIN AND U. VAZIRANI (1993), Quantum complexity theory, in Proc. 25th Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, New York, pp. 11-20; SIAM J. Comput., 26 (1997), pp. 1411-1473.
    • (1997) SIAM J. Comput. , vol.26 , pp. 1411-1473
  • 15
    • 0027040009 scopus 로고
    • The quantum challenge to structural complexity theory
    • IEEE Computer Society Press, Los Alamitos, CA
    • A. BERTHIAUME AND G. BRASSARD (1992), The quantum challenge to structural complexity theory, in Proc. 7th Annual Structure in Complexity Theory Conference, IEEE Computer Society Press, Los Alamitos, CA, pp. 132-137.
    • (1992) Proc. 7th Annual Structure in Complexity Theory Conference , pp. 132-137
    • Berthiaume, A.1    Brassard, G.2
  • 16
    • 0001644966 scopus 로고
    • Oracle quantum computing
    • A. BERTHIAUME AND G. BRASSARD (1994), Oracle quantum computing, J. Modern Opt., 41, pp. 2521-2535.
    • (1994) J. Modern Opt. , vol.41 , pp. 2521-2535
    • Berthiaume, A.1    Brassard, G.2
  • 18
    • 0040580869 scopus 로고
    • Can quantum computers have simple Hamiltonians
    • IEEE Computer Society Press, Los Alamitos, CA
    • 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.
    • (1994) Proc. Workshop on Physics of Computation: PhysComp '94 , pp. 63-68
    • Biafore, M.1
  • 19
    • 84957668600 scopus 로고
    • Quantum cryptanalysis of hidden linear functions
    • Proc. 15th Annual International Cryptology Conference, Santa Barbara, CA, D. Coppersmith, ed. Springer-Verlag, Berlin
    • D. BONEH AND R. J. LIPTON (1995), Quantum cryptanalysis of hidden linear functions, Advances in Cryptology - CRYPTO '95, Proc. 15th Annual International Cryptology Conference, Santa Barbara, CA, D. Coppersmith, ed. Springer-Verlag, Berlin, pp. 424-437.
    • (1995) Advances in Cryptology - CRYPTO '95 , pp. 424-437
    • Boneh, D.1    Lipton, R.J.2
  • 20
    • 0023601193 scopus 로고
    • New lower bound techniques for robot motion planning problems
    • IEEE Computer Society Press, Los Alamitos, CA
    • 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, Los Alamitos, CA, pp. 49-60.
    • (1987) Proc. 28th Annual Symposium on Foundations of Computer Science , pp. 49-60
    • Canny, J.F.1    Reif, J.2
  • 21
    • 85010084001 scopus 로고
    • Precision-sensitive Euclidean shortest path in 3-space
    • Association for Computing Machinery, New York
    • 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.
    • (1995) Proc. 11th Annual Symposium on Computational Geometry , pp. 350-359
    • Choi, J.1    Sellen, J.2    Yap, C.-K.3
  • 22
    • 0029639815 scopus 로고
    • Quantum computers, factoring and decoherence
    • I. L. CHUANG, R. LAFLAMME, P. W. SHOR, AND W. H. ZUREK (1995), Quantum computers, factoring and decoherence, Science, 270, pp. 1633-1635.
    • (1995) Science , vol.270 , pp. 1633-1635
    • Chuang, I.L.1    Laflamme, R.2    Shor, P.W.3    Zurek, W.H.4
  • 23
    • 0000300961 scopus 로고
    • A simple quantum computer
    • I. L. CHUANG AND Y. YAMAMOTO (1995), A simple quantum computer, Phys. Rev. A, 52, pp. 3489-3496.
    • (1995) Phys. Rev. A , vol.52 , pp. 3489-3496
    • Chuang, I.L.1    Yamamoto, Y.2
  • 24
    • 0000954072 scopus 로고
    • An unsolvable problem of elementary number theory
    • A. CHURCH (1936), An unsolvable problem of elementary number theory, Amer. J. Math., 58, pp. 345-363.
    • (1936) Amer. J. Math. , vol.58 , pp. 345-363
    • Church, A.1
  • 25
    • 4244043498 scopus 로고
    • Quantum computations with cold trapped ions
    • J. I. CIRAC AND P. ZOLLER (1995), Quantum computations with cold trapped ions, Phys. Rev. Lett., 74, pp. 4091-4094.
    • (1995) Phys. Rev. Lett. , vol.74 , pp. 4091-4094
    • Cirac, J.I.1    Zoller, P.2
  • 28
    • 0022421379 scopus 로고
    • Quantum theory, the Church-Turing principle and the universal quantum computer
    • D. DEUTSCH (1985), Quantum theory, the Church-Turing principle and the universal quantum computer, Proc. Roy. Soc. London Ser. A, 400, pp. 96-117.
    • (1985) Proc. Roy. Soc. London Ser. A , vol.400 , pp. 96-117
    • Deutsch, D.1
  • 29
    • 0002496370 scopus 로고
    • Quantum computational networks
    • D. DEUTSCH (1989), Quantum computational networks, Proc. Roy. Soc. London Ser. A, 425, pp. 73-90.
    • (1989) Proc. Roy. Soc. London Ser. A , vol.425 , pp. 73-90
    • Deutsch, D.1
  • 31
    • 0000119236 scopus 로고
    • Rapid solution of problems by quantum computation
    • D. DEUTSCH AND R. JOZSA (1992), Rapid solution of problems by quantum computation, Proc. Roy. Soc. London Ser. A, 439, pp. 553-558.
    • (1992) Proc. Roy. Soc. London Ser. A , vol.439 , pp. 553-558
    • Deutsch, D.1    Jozsa, R.2
  • 32
    • 33751021446 scopus 로고
    • Two-bit gates are universal for quantum computation
    • D. P. DIVINCENZO (1995), Two-bit gates are universal for quantum computation, Phys. Rev. A, 51, pp. 1015-1022.
    • (1995) Phys. Rev. A , vol.51 , pp. 1015-1022
    • Divincenzo, D.P.1
  • 33
    • 0030520263 scopus 로고    scopus 로고
    • Shor's quantum algorithm for factorising numbers
    • A. EKERT AND R. JOZSA (1996), Shor's quantum algorithm for factorising numbers, Rev. Mod. Phys., 68, pp. 733-753.
    • (1996) Rev. Mod. Phys. , vol.68 , pp. 733-753
    • Ekert, A.1    Jozsa, R.2
  • 34
    • 51249180339 scopus 로고
    • Simulating physics with computers
    • R. FEYNMAN (1982), Simulating physics with computers, Internat. J. Theoret. Phys., 21, pp. 467-488.
    • (1982) Internat. J. Theoret. Phys. , vol.21 , pp. 467-488
    • Feynman, R.1
  • 35
    • 5344230456 scopus 로고
    • Quantum mechanical computers
    • R. FEYNMAN (1986), Quantum mechanical computers, Found. Phys., 16, pp. 507-531; originally published in Optics News (February 1985), pp. 11-20.
    • (1986) Found. Phys. , vol.16 , pp. 507-531
    • Feynman, R.1
  • 36
    • 5344230456 scopus 로고
    • February
    • R. FEYNMAN (1986), Quantum mechanical computers, Found. Phys., 16, pp. 507-531; originally published in Optics News (February 1985), pp. 11-20.
    • (1985) Optics News , pp. 11-20
  • 38
    • 0000303380 scopus 로고
    • Discrete logarithms in GF(p) using the number field sieve
    • D. M. GORDON (1993), Discrete logarithms in GF(p) using the number field sieve, SIAM J. Discrete Math., 6, pp. 124-139.
    • (1993) SIAM J. Discrete Math. , vol.6 , pp. 124-139
    • Gordon, D.M.1
  • 39
    • 0001260174 scopus 로고    scopus 로고
    • Semiclassical Fourier tranform for quantum computation
    • R. B. GRIFFITHS AND C.-S. NIU (1996), Semiclassical Fourier tranform for quantum computation, Phys. Rev. Lett., 76, pp. 3228-3231.
    • (1996) Phys. Rev. Lett. , vol.76 , pp. 3228-3231
    • Griffiths, R.B.1    Niu, C.-S.2
  • 41
    • 85115201693 scopus 로고
    • On the power of multiplication in random access machines
    • IEEE Computer Society, Long Beach, CA
    • 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.
    • (1974) Proc. 15th Annual Symposium on Switching and Automata Theory , pp. 13-23
    • Hartmanis, J.1    Simon, J.2
  • 42
    • 0005395785 scopus 로고
    • Multiplication of multidigit numbers on automata
    • in Russian
    • 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).
    • (1962) Dokl. Akad. Nauk SSSR , vol.145 , pp. 293-294
    • Karatsuba, A.1    Ofman, Yu.2
  • 43
    • 0001464763 scopus 로고
    • English translation
    • 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).
    • (1963) Sov. Phys. Dokl. , vol.7 , pp. 595-596
  • 44
    • 5544279124 scopus 로고
    • personal communication
    • E. KNILL (1995), personal communication.
    • (1995)
    • Knill, E.1
  • 48
    • 0000192783 scopus 로고
    • nu, où θ est un isomorphisme de codes
    • nu, où θ est un isomorphisme de codes, C. R. Acad. Française Sci., 257, pp. 2597-2600.
    • (1963) C. R. Acad. Française Sci. , vol.257 , pp. 2597-2600
    • Lecerf, Y.1
  • 49
    • 0003291192 scopus 로고
    • The Development of the Number Field Sieve
    • Springer-Verlag, Berlin
    • A. K. LENSTRA AND H. W. LENSTRA, JR., EDS. (1993), The Development of the Number Field Sieve, Lecture Notes in Mathematics 1554, Springer-Verlag, Berlin.
    • (1993) Lecture Notes in Mathematics , vol.1554
    • Lenstra, A.K.1    Lenstra Jr., H.W.2
  • 52
    • 0025476530 scopus 로고
    • A note on Bennett's time-space tradeoff for reversible computation
    • R. Y. LEVINE AND A. T. SHERMAN (1990), A note on Bennett's time-space tradeoff for reversible computation, SIAM J. Comput., 19, pp. 673-677.
    • (1990) SIAM J. Comput. , vol.19 , pp. 673-677
    • Levine, R.Y.1    Sherman, A.T.2
  • 53
    • 0027660531 scopus 로고
    • A potentially realizable quantum computer
    • S. LLOYD (1993), A potentially realizable quantum computer, Science, 261, pp. 1569-1571.
    • (1993) Science , vol.261 , pp. 1569-1571
    • Lloyd, S.1
  • 54
    • 0028269694 scopus 로고
    • Envisioning a quantum supercomputer
    • S. LLOYD (1994), Envisioning a quantum supercomputer, Science, 263, p. 695.
    • (1994) Science , vol.263 , pp. 695
    • Lloyd, S.1
  • 55
    • 4043102370 scopus 로고
    • Almost any quantum logic gate is universal
    • S. LLOYD (1995), Almost any quantum logic gate is universal, Phys. Rev. Lett., 75, pp. 346-349.
    • (1995) Phys. Rev. Lett. , vol.75 , pp. 346-349
    • Lloyd, S.1
  • 56
  • 58
    • 0017217375 scopus 로고
    • Riemann's hypothesis and tests for primality
    • G. L. MILLER (1976), Riemann's hypothesis and tests for primality, J. Comput. System Sci., 13, pp. 300-317.
    • (1976) J. Comput. System Sci. , vol.13 , pp. 300-317
    • Miller, G.L.1
  • 59
    • 5544269276 scopus 로고
    • personal communication
    • A. M. ODLYZKO (1995), personal communication.
    • (1995)
    • Odlyzko, A.M.1
  • 61
  • 62
    • 0040235097 scopus 로고
    • Fast, rigorous factorization and discrete logarithm algorithms
    • Discrete Algorithms and Complexity, 1986, Kyoto, D. S. Johnson, T. Nishizeki, A. Nozaki, and H. S. Wilf, eds., Academic Press, New York
    • C. POMERANCE (1987), Fast, rigorous factorization and discrete logarithm algorithms, in Discrete Algorithms and Complexity, Proc. Japan-US Joint Seminar, 1986, Kyoto, D. S. Johnson, T. Nishizeki, A. Nozaki, and H. S. Wilf, eds., Academic Press, New York, pp. 119-143.
    • (1987) Proc. Japan-US Joint Seminar , pp. 119-143
    • Pomerance, C.1
  • 63
    • 0000350658 scopus 로고
    • Finite combinatory processes. Formulation I
    • E. POST (1936), Finite combinatory processes. Formulation I, J. Symbolic Logic, 1, pp. 103-105.
    • (1936) J. Symbolic Logic , vol.1 , pp. 103-105
    • Post, E.1
  • 64
    • 0017930809 scopus 로고
    • A method of obtaining digital signatures and public-key cryptosystems
    • 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.
    • (1978) Comm. Assoc. Comput. Mach. , vol.21 , pp. 120-126
    • Rivest, R.L.1    Shamir, A.2    Adleman, L.3
  • 65
    • 0001638540 scopus 로고
    • Digital simulation of analog computation and Church's thesis
    • L. A. RUBEL (1989), Digital simulation of analog computation and Church's thesis, J. Symbolic Logic, 54, pp. 1011-1017.
    • (1989) J. Symbolic Logic , vol.54 , pp. 1011-1017
    • Rubel, L.A.1
  • 66
    • 0020226762 scopus 로고
    • Asymptotically fast algorithms for the numerical multiplication and division of polynomials with complex coefficients
    • Computer Algebra EUROCAM '82, J. Calmet, ed., Springer-Verlag, Berlin
    • A. SCHÖNHAGE (1982), Asymptotically fast algorithms for the numerical multiplication and division of polynomials with complex coefficients, in Computer Algebra EUROCAM '82, Lecture Notes in Computer Science 144, J. Calmet, ed., Springer-Verlag, Berlin, pp. 3-15.
    • (1982) Lecture Notes in Computer Science , vol.144 , pp. 3-15
    • Schönhage, A.1
  • 68
    • 34250459760 scopus 로고
    • Schnelle Multiplikation grosser Zahlen
    • A. SCHÖNHAGE AND V. STRASSEN (1971), Schnelle Multiplikation grosser Zahlen, Computing, 7, pp. 281-292.
    • (1971) Computing , vol.7 , pp. 281-292
    • Schönhage, A.1    Strassen, V.2
  • 69
    • 85115374351 scopus 로고
    • Algorithms for quantum computation: Discrete logarithms and factoring
    • IEEE Computer Society Press, Los Alamitos, CA
    • 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.
    • (1994) Proc. 35th Annual Symposium on Foundations of Computer Science , pp. 124-134
    • Shor, P.W.1
  • 70
    • 33746738958 scopus 로고
    • Scheme for reducing decoherence in quantum computer memory
    • P. W. SHOR (1995), Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A, 52, pp. 2493-2496.
    • (1995) Phys. Rev. A , vol.52 , pp. 2493-2496
    • Shor, P.W.1
  • 71
    • 0002007763 scopus 로고
    • On the power of quantum computation
    • IEEE Computer Society Press, Los Alamitos, CA
    • D. SIMON (1994), On the power of quantum computation, in Proc. 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, pp. 116-123; SIAM J. Comput., 26 (1997), pp. 1474-1483.
    • (1994) Proc. 35th Annual Symposium on Foundations of Computer Science , pp. 116-123
    • Simon, D.1
  • 72
    • 0009438294 scopus 로고    scopus 로고
    • D. SIMON (1994), On the power of quantum computation, in Proc. 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, pp. 116-123; SIAM J. Comput., 26 (1997), pp. 1474-1483.
    • (1997) SIAM J. Comput. , vol.26 , pp. 1474-1483
  • 73
    • 0000709068 scopus 로고
    • Realizable universal quantum logic gates
    • T. SLEATOR AND H. WEINFURTER (1995), Realizable universal quantum logic gates, Phys. Rev. Lett., 74, pp. 4087-4090.
    • (1995) Phys. Rev. Lett. , vol.74 , pp. 4087-4090
    • Sleator, T.1    Weinfurter, H.2
  • 74
    • 5544320583 scopus 로고
    • personal communication
    • R. SOLOVAY (1995), personal communication.
    • (1995)
    • Solovay, R.1
  • 75
    • 4243181943 scopus 로고
    • Two non-standard paradigms for computation: Analog machines and cellular automata
    • Performance Limits in Communication Theory and Practice, Il Ciocco, Castelvecchio Pascoli, Tuscany, Italy, 1986, J. K. Skwirzynski, ed., Kluwer Academic Publishers, Dordrecht, The Netherlands
    • 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.
    • (1988) Proc. NATO Advanced Study Institute , pp. 173-192
    • Steiglitz, K.1
  • 76
    • 3743085708 scopus 로고
    • Structural basis of multistationary quantum systems II: Effective few-particle dynamics
    • 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.
    • (1988) Phys. Rev. B , vol.37 , pp. 8111-8121
    • Teich, W.G.1    Obermayer, K.2    Mahler, G.3
  • 77
    • 84978092325 scopus 로고
    • Reversible computing
    • Automata, Languages and Programming, 7th Colloquium, J. W. de Bakker and J. van Leeuwen, eds., Springer-Verlag, Berlin
    • T. TOFFOLI (1980), Reversible computing, in Automata, Languages and Programming, 7th Colloquium, Lecture Notes in Computer Science 84, J. W. de Bakker and J. van Leeuwen, eds., Springer-Verlag, Berlin, pp. 632-644.
    • (1980) Lecture Notes in Computer Science , vol.84 , pp. 632-644
    • Toffoli, T.1
  • 78
    • 84960561455 scopus 로고
    • On computable numbers, with an application to the Entscheidungsproblem
    • 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.
    • (1936) Proc. London Math. Soc. (2) , vol.42 , pp. 230-265
    • Turing, A.M.1
  • 79
    • 84960561455 scopus 로고
    • 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.
    • (1937) Proc. London Math. Soc. (2) , vol.43 , pp. 544-546
  • 80
    • 5544280946 scopus 로고
    • Maintaining coherence in quantum computers
    • W. G. UNRUH (1995), Maintaining coherence in quantum computers, Phys. Rev. A, 51, pp. 992-997.
    • (1995) Phys. Rev. A , vol.51 , pp. 992-997
    • Unruh, W.G.1
  • 81
    • 0001900109 scopus 로고
    • Machine models and simulations
    • J. van Leeuwen, ed., Elsevier, Amsterdam
    • P. VAN EMDE BOAS (1990), Machine models and simulations, in Handbook of Theoretical Computer Science, Vol. A, J. van Leeuwen, ed., Elsevier, Amsterdam, pp. 1-66.
    • (1990) Handbook of Theoretical Computer Science , vol.A , pp. 1-66
    • Van Emde Boas, P.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.