Quantum computations: Algorithms and error correction

Russian Mathematical Surveys

Volumn 52, Issue 6, 1997, Pages 1191-1249

  Kitaev, A Yu ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0040052002     PISSN: 00360279     EISSN: 14684829     Source Type: Journal    
DOI: 10.1070/RM1997v052n06ABEH002155     Document Type: Article

References (56)

1. Yu. I. Manin, The computable and incomputable, Sovetskoe Radio Moscow 1980. (Russian)
2. P. Benioff, "Quantum mechanical Hamiltonian models of Turing machines", J. Statist. Phys. 29 (1982), 515-546.
3. A. Peres, "Reversible logic and quantum computers", Phys. Rev. Ser. A 32 (1985), 3266-3276.
4. R. P. Feynman, "Quantum mechanical computers", Optics News 11 (1985), 11.
5. D. Deutsch, "Quantum theory, the Church-Turing principle and the universal quantum computer", Proc. Roy. Soc. London. Ser. A 400 (1985), 97-117.
6. D. Deutsch, "Quantum computational networks", Proc. Roy. Soc. London. Ser. A 425 (1989), 73-90.
7. A. C.-C. Yao, "Quantum circuit complexity", Proc. 34th Annual Symposium on the Foundations of Computer Science (Los Alamitos, CA), IEEE Comput. Soc. Press, 1993, pp. 352-361.
8. M. R. Garey and D. S. Johnson, Computers and intractibility, W. H. Freeman and Co., San Francisco 1979; Russian transl., Mir, Moscow 1982.
9. Q. A. Turchette, C. J. Hood, W. A. Lange, H. Mabushi, and H. J. Kimble, "Measurement of conditional phase shifts for quantum logic", Phys. Rev. Lett. 75 (1995), 4710-4713.
10. C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, "Demonstration of a fundamental quantum logic gate", Phys. Rev. Lett. 75 (1995), 4714-4717.
11. P. W. Shor, "Algorithms for quantum computation: discrete log and factoring", Proc. 35th Annual Symposium on the Foundations of Computer Science (Los Alamitos, CA), IEEE Computer Society Press, 1994, p. 124.
12. W. Diffie and M. E. Hellmann, "New directions in cryptography", IEEE Trans. Information Theory IT-22 (1976), 644-654.
13. D. Simon, "On the power of quantum computation", Proc. 35th Annual Symposium on the Foundations of Computer Science (Los Alamitos, CA), IEEE Computer Society Press, 1994, p. 116.
14. D. Grigoriev, "Testing shift equivalence of polynomials using quantum mechanics", Preprint, 1995.
15. A. Yu. Kitaev, "Quantum measurements and the Abelian stabilizer problem", LANL e-print quant-ph/9511026, http://xxx.lanl.gov.
16. E. Bernstein and U. Vaziriani, "Quantum complexity theory", Proc. 25th Annual ACM Symposium on the Theory of Computing, ACM Press, New York 1993, pp. 11-20.
17. F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes. I, II, North-Holland Publ. Co., Amsterdam-New York-Oxford 1977; Russian transl., Svyaz', Moscow 1979.
18. P. W. Shor, "Scheme for reducing decoherence in quantum computer memory", Phys. Rev. Ser. A 52 (1995), 2493-2496.
19. A. R. Calderbank and P. W. Shor, "Good quantum error-correcting codes exist", LANL e-print quant-ph/9512032, http://xxx.lanl.gov (to appear in Phys. Rev. Ser. A).
20. A. M. Steane, "Multiple-particle interference and quantum error correction", Proc. Roy. Soc. London. Ser. A 452 (1996), 2551-2577; LANL e-print quant-ph/9601029, http://xxx.lanl.gov.
21. R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, "Perfect quantum error correction code", LANL e-print quant-ph/9602019, http://xxx.lanl.gov.
22. B. B. Plenio, V. Vedral, and P. L. Knight, "Optimal realistic quantum error correcting code", LANL e-print quant-ph/9603022, http://xxx.lanl.gov.
23. L. Vaidman, L. Goldenberg, and S. Weisner, "Error prevention scheme with four particles", LANL e-print quant-ph/9603031, http://xxx.lanl.gov.
24. P. W. Shor and J. A. Smolin, "Quantum error-correcting codes need not completely reveal their error syndrome", LANL e-print quant-ph/9604006, http://xxx.lanl.gov.
25. C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, "Mixed state entanglement and quantum error-correcting codes", LANL e-print quant-ph/9604024, http://xxx.lanl.gov.
26. S. L. Braunstein, "Perfect quantum error correction coding in 26 laser pulses", LANL e-print quant-ph/9604036, http://xxx.lanl.gov.
27. D. Gottesman, "A class of quantum error-correcting codes saturating the quantum Hamming bound", LANL e-print quant-ph/9604038, http://xxx.lanl.gov.
28. D. Gottesman, "Stabilizer codes in quantum error correction", LANL e-print quant-ph/9705052, http://xxx.lanl.gov.
29. A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, "Quantum error correction and orthogonal geometry", LANL e-print quant-ph/9605005, http://xxx.lanl.gov.
30. E. M. Rains, R. H. Hardin, P. W. Shor, and N. J. A. Sloane, "A nonadditive quantum code", LANL e-print quant-ph/9703002, http://xxx.lanl.gov.
31. A. S. Kholevo, "Some estimates for the amount of information transmittable by a quantum communication channel", Problemy Peredachi Informatsii 9:3 (1973), 3-11; English transl. in Problems of Information Transmission 9 (1973).
32. A. S. Kholevo, "Some estimates for the amount of information transmittable by a quantum communication channel", Problemy Peredachi Informatsii 9:3 (1973), 3-11; English transl. in Problems of Information Transmission 9 (1973).
33. B. Schumacher, "Sending quantum entanglement through noisy channels", LANL e-print quant-ph/9604023, http://xxx.lanl.gov.
34. B. Schumacher and M. A. Nielsen, "Quantum data processing and error correction", LANL e-print quant-ph/9604022, http://xxx.lanl.gov.
35. S. Lloyd, "The capacity of the noisy quantum channel", LANL e-print quant-ph/9604015, http://xxx.laril.gov.
36. A. Ekert and C. Macchiavello, "Error correction in quantum communication", LANL e-print quant-ph/9602022, http://xxx.lanl.gov.
37. E. Knill and R. Laflamme, "A theory of quantum error-correcting codes", LANL e-print quant-ph/9604034, http://xxx.lanl.gov.
38. P. W. Shor and R. Laflamme, "Quantum analog of the MacWilliams identities in classical coding theory", LANL e-print quant-ph/9610040, http://xxx.lanl.gov.
39. E. M. Rains, "Quantum weight enumerators", LANL e-print quant-ph/9612015, http://xxx.lanl.gov.
40. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels", Phys. Rev. Lett. 70 (1993), 1895-1899.
41. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, "Purification of noisy entanglement and faithful teleportation via noisy channels", Phys. Rev. Lett. 76 (1996), 722-725.
42. C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, "Mixed state entanglement and quantum error correction", Phys. Rev. Ser. A 54 (1996), 3824-3851; LANL e-print quant-ph/9604024, http://xxx.lanl.gov.
43. A. Yu. Kitaev, "Quantum error correction with imperfect gates', in: Quantum communication, computing and measurement, Plenum Press, New York 1997.
44. P. W. Shor, "Fault-tolerant quantum computation", LANL e-print, quant-ph/9605011, http://xxx.lanl.gov.
45. E. Knill and R. Laflamme, "Concatenated quantum codes", LANL e-print quant-ph/9608012, http://xxx.lanl.gov.
46. E. Knill, R. Laflamme, and W. Zurek, "Accuracy threshold for quantum computation", LANL e-print quant-ph/9610011, http://xxx.lanl.gov.
47. D. Aharonov and M. Ben-Or, "Fault tolerant quantum computation with constant error", LANL e-print quant-ph/9611025, http://xxx.lanl.gov.
48. D. Gottesman, "A theory of fault-tolerant quantum computation.", LANL e-print quant-ph/9702029, http://xxx.lanl.gov.
49. A. Yu. Kitaev, "Fault-tolerant quantum computation by anions", LANL e-print quant-ph/9707021, http://xxx.lanl.gov.
50. G. L. Miller, "Riemann's hypothesis and tests for primarity", J. Comput. System Sci. 13 (1976), 300-317.
51. n où θ est un 'isomorphisme de codes'", C.R. Acad. Sci. Paris 257 (1963), 2597-2600.
52. C. H. Bennett, "Logical reversibility of computation", IBM J. Res. Develop. 17 (1973), 525-532.
53. R. Landauer, "Irreversibility and heat generation in the computing process", IBM J. Res. Develop. 5 (1961), 183-191.
54. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margouls, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, "Elementary gates for quantum computation", quant-ph/9503016.
55. R. Solovay, Private communication.
56. C. H. Papadimitriou and K. Steiglitz, Combinatorial optimization: algorithms and complexity, Prentice-Hall, Inc., Englewood Cliffs, NJ 1982; Russian transl., Mir, Moscow 1985.