Quantum algorithms for algebraic problems

Reviews of Modern Physics

Volumn 82, Issue 1, 2010, Pages 1-52

  Childs, Andrew M ^a   Van Dam, Wim ^b  

^a UNIVERSITY OF WATERLOO   (Canada)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL PROBLEM; FACTORING INTEGERS; QUANTUM ALGORITHMS; QUANTUM COMPUTATION;

ALGEBRA; ALGORITHMS; COMPUTATIONAL EFFICIENCY; COMPUTATIONAL LINGUISTICS; INDIUM COMPOUNDS; QUANTUM COMPUTERS; QUANTUM OPTICS;

QUANTUM THEORY;

EID: 77649127557     PISSN: 00346861     EISSN: 15390756     Source Type: Journal    
DOI: 10.1103/RevModPhys.82.1     Document Type: Article

References (264)

1. Aaronson, S., and A. Ambainis, 2005, Theory Comput. 1, 47, preliminary version in FOCS 2003. 10.4086/toc.2005.v001a004
2. Adleman, L. M., J. DeMarrais, and M.-D. A. Huang, 1997, SIAM J. Comput. 26, 1524. 10.1137/S0097539795293639
3. Adleman, L. M., and M.-D. Huang, 2001, J. Symb. Comput. 32, 171, preliminary version in ANTS-II 1996. 10.1006/jsco.2001.0470
4. Agrawal, M., N. Kayal, and N. Saxena, 2004, Ann. Math. 160, 781. 10.4007/annals.2004.160.781
5. Aharonov, D., 2003, e-print arXiv:quant-ph/0301040.
6. Aharonov, D., and I. Arad, 2006, e-print arXiv:quant-ph/0605181.
7. Aharonov, D., I. Arad, E. Eban, and Z. Landau, 2007, e-print arXiv:quant-ph/0702008.
8. Aharonov, D., and M. Ben-Or, 2008, SIAM J. Comput. 38, 1207, preliminary version in STOC 1997. 10.1137/S0097539799359385
9. Aharonov, D., V. Jones, and Z. Landau, 2009, Algorithmica 55, 395, preliminary version in STOC 2006. 10.1007/s00453-008-9168-0
10. Aharonov, D., W. van Dam, J. Kempe, Z. Landau, S. Lloyd, and O. Regev, 2007, SIAM J. Comput. 37, 166, preliminary version in FOCS 2004. 10.1137/S0097539705447323
11. Ajtai, M., 1996, ACM Symposium on Theory of Computing (ACM, New York), pp. 99-108.
12. Ajtai, M., 1998, ACM Symposium on Theory of Computing (ACM, New York), pp. 10-19.
13. Ajtai, M., and C. Dwork, 1997, ACM Symposium on Theory of Computing (ACM, New York), pp. 284-293.
14. Ajtai, M., R. Kumar, and D. Sivakumar, 2001, ACM Symposium on Theory of Computing (ACM, New York), pp. 601-610.
15. Alagic, G., C. Moore, and A. Russell, 2007, ACM-SIAM Symposium on Discrete Algorithms (SIAM, Philadelphia), pp. 1217-1224;
16. e-print arXiv:quant-ph/0603251.
17. Alexander, J. W., 1923, Proc. Natl. Acad. Sci. U.S.A. 9, 93. 10.1073/pnas.9.3.93
18. Ambainis, A., 2007, SIAM J. Comput. 37, 210, preliminary version in FOCS 2004. 10.1137/S0097539705447311
19. Ambainis, A., A. M. Childs, B. W. Reichardt, R. Špalek, and S. Zhang, 2007, IEEE Symposium on Foundations of Computer Science (IEEE, Los Alamitos, CA), pp. 363-372;
20. e-print arXiv:quant-ph/0703015;
21. e-print arXiv:0704.3628.
22. Ambainis, A., J. Kempe, and A. Rivosh, 2005, ACM-SIAM Symposium on Discrete Algorithms (SIAM, Philadelphia), pp. 1099-1108;
23. e-print arXiv:quant-ph/0402107.
24. Ambainis, A., and R. Špalek, 2006, Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science Vol. 3884 (Springer, Berlin), pp. 172-183;
25. e-print arXiv:quant-ph/0508205.
26. Arad, I., and Z. Landau, 2008, e-print arXiv:0805.0040.
27. Aspuru-Guzik, A., A. D. Dutoi, P. J. Love, and M. Head-Gordon, 2005, Science 309, 1704. 10.1126/science.1113479
28. Babai, L., G. Cooperman, L. Finkelstein, E. Luks, and Á. Seress, 1995, J. Comput. Syst. Sci. 50, 296, preliminary version in STOC 1991. 10.1006/jcss.1995.1024
29. Babai, L., D. Grigoriev, and D. Mount, 1982, ACM Symposium on Theory of Computing (ACM, New York), pp. 310-324.
30. Babai, L., W. M. Kantor, and E. Luks, 1983, IEEE Symposium on Foundations of Computer Science (IEEE, Los Alamitos, CA), pp. 162-171.
31. Babai, L., and E. Szemerédi, 1984, IEEE Symposium on Foundations of Computer Science (IEEE, Los Alamitos, CA), pp. 229-240.
32. Bacon, D., 2008, Quantum Inf. Comput. 8, 438.
33. Bacon, D., A. M. Childs, and W. van Dam, 2005, IEEE Symposium on Foundations of Computer Science (IEEE, Los Alamitos, CA), pp. 469-478;
34. e-print arXiv:quant-ph/0504083.
35. Bacon, D., A. M. Childs, and W. van Dam, 2006, Chicago J. Theor. Comput. Sci. 2006, 2.
36. Barenco, A., A. Ekert, K.-A. Suominen, and P. Törmä, 1996, Phys. Rev. A 54, 139. 10.1103/PhysRevA.54.139
37. Barnum, H., and E. Knill, 2002, J. Math. Phys. 43, 2097. 10.1063/1.1459754
38. Beals, R., 1997, ACM Symposium on Theory of Computing (ACM, New York), pp. 48-53.
39. Beals, R., H. Buhrman, R. Cleve, M. Mosca, and R. de Wolf, 2001, J. ACM 48, 778, preliminary version in FOCS 1998. 10.1145/502090.502097
40. Bennett, C. H., 1973, IBM J. Res. Dev. 17, 525.
41. Bennett, C. H., E. Bernstein, G. Brassard, and U. Vazirani, 1997, SIAM J. Comput. 26, 1510. 10.1137/S0097539796300933
42. Berndt, B. C., R. J. Evans, and K. S. Williams, 1998, Gauss and Jacobi Sums (Wiley, New York).
43. Bernstein, E., and U. Vazirani, 1993, ACM Symposium on Theory of Computing (ACM, New York), pp. 11-20.
44. Bernstein, E., and U. Vazirani, 1997, SIAM J. Comput. 26, 1411, preliminary version in STOC 1993. 10.1137/S0097539796300921
45. Beth, T., 1987, Theor. Comput. Sci. 51, 331. 10.1016/0304-3975(87)90041-7
46. Blum, A., A. Kalai, and H. Wasserman, 2003, J. ACM 50, 506, preliminary version in STOC 1999. 10.1145/792538.792543
47. Boneh, D., and R. Lipton, 1995, Advances in Cryptology, Lecture Notes in Computer Science Vol. 963 (Springer, Berlin), pp. 424-437.
48. Bordewich, M., M. Freedman, L. Lovász, and D. Welsh, 2005, Combin. Probab. Comput. 14, 737. 10.1017/S0963548305007005
49. Born, M., and V. Fock, 1928, Z. Phys. 51, 165. 10.1007/BF01343193
50. Boykin, P. O., T. Mor, M. Pulver, V. Roychowdhury, and F. Vatan, 2000, Inf. Process. Lett. 75, 101. 10.1016/S0020-0190(00)00084-3
51. Brassard, G., P. Høyer, M. Mosca, and A. Tapp, 2002, in Quantum Computation and Information, edited by, S. J. Lomonaco, and H. E. Brandt, AMS Contemporary Mathematics Series Vol. 305 (AMS, Providence, RI), pp. 53-74;
52. e-print arXiv:quant-ph/0005055.
53. Brassard, G., P. Høyer, and A. Tapp, 1997, SIGACT News 28, 14. 10.1145/261342.261346
54. Buchmann, J., 1990, in Séminaire de Théorie des Nombres, Paris 1988-1989, Progress in Mathematics Vol. 91 (Birkhäuser, Boston), pp. 27-41.
55. Buchmann, J., 2004, Introduction to Cryptography, 2nd ed., Undergraduate Texts in Mathematics (Springer-Verlag, New York).
56. Buchmann, J. A., and H. C. Williams, 1989, in Advances in Cryptology, Lecture Notes in Computer Science Vol. 435 (Springer, Berlin), pp. 335-343.
57. Buhler, J. P., H. W. Lenstra, Jr., and C. Pomerance, 1993, The Development of the Number Field Sieve, Lecture Notes in Mathematics Vol. 1554 (Springer, New York), pp. 50-94.
58. Buhrman, H., and R. Špalek, 2006, ACM-SIAM Symposium on Discrete Algorithms (SIAM, Philadelphia), pp. 880-889;
59. e-print arXiv:quant-ph/0409035.
60. Cheung, D., D. Maslov, J. Mathew, and D. Pradhan, 2008, in Proceedings of the Third Workshop on Theory of Quantum Computation, Communication, and Cryptography, Lecture Notes in Computer Science Vol. 5106 (Springer, Berlin), pp. 96-104.
61. Cheung, K. K. H., and M. Mosca, 2001, Quantum Inf. Comput. 1, 26.
62. Childs, A. M., R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D. A. Spielman, 2003, ACM Symposium on Theory of Computing (ACM, New York), pp. 59-68;
63. e-print arXiv:quant-ph/0209131.
64. Childs, A. M., and J. Goldstone, 2004a, Phys. Rev. A 70, 042312. 10.1103/PhysRevA.70.042312
65. Childs, A. M., and J. Goldstone, 2004b, Phys. Rev. A 70, 022314. 10.1103/PhysRevA.70.022314
66. Childs, A. M., L. J. Schulman, and U. V. Vazirani, 2007, IEEE Symposium on Foundations of Computer Science (IEEE, Los Alamitos, CA), pp. 395-404;
67. e-print arXiv:0705.2784.
68. Childs, A. M., and P. Wocjan, 2007, Quantum Inf. Comput. 7, 504.
69. Childs, A. M., and W. van Dam, 2007, ACM-SIAM Symposium on Discrete Algorithms (SIAM, Philadelphia), pp. 1225-1234;
70. e-print arXiv:quant-ph/0507190.
71. Clausen, M., 1989, Theor. Comput. Sci. 67, 55. 10.1016/0304-3975(89) 90021-2
72. Cleve, R., 1994, http://www.cpsc.ucalgary.ca/~cleve/pubs/fourier- transform.ps
73. Cleve, R., 2004, Inf. Comput. 192, 162, preliminary version in CCC 2000. 10.1016/j.ic.2004.04.001
74. Cleve, R., A. Ekert, C. Macchiavello, and M. Mosca, 1998, Proc. R. Soc. London, Ser. A 454, 339. 10.1098/rspa.1998.0164
75. Cleve, R., and J. Watrous, 2000, IEEE Symposium on Foundations of Computer Science (IEEE, Los Alamitos, CA), pp. 526-536;
76. e-print arXiv:quant-ph/0006004.
77. Cohen, H., 1993, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics Vol. 138 (Springer, Berlin).
78. Coppersmith, D., 1994, IBM Research Division Technical Report No. RC 19642 (unpublished);
79. e-print arXiv:quant-ph/0201067.
80. Damgård, I. B., 1990, Advances in Cryptology, Lecture Notes in Computer Science Vol. 403 (Springer-Verlag, New York), pp. 163-172.
81. de Beaudrap, J. N., R. Cleve, and J. Watrous, 2002, Algorithmica 34, 449. 10.1007/s00453-002-0978-1
82. Decker, T., J. Draisma, and P. Wocjan, 2009, Quantum Inf. Comput. 9, 215.
83. den Boer, B., 1990, Advances in Cryptology, Lecture Notes in Computer Science Vol. 403 (Springer, Berlin), pp. 530-539.
84. Deutsch, D., 1985, Proc. R. Soc. London, Ser. A 400, 97. 10.1098/rspa.1985.0070
85. Deutsch, D., 1989, Proc. R. Soc. London, Ser. A 425, 73. 10.1098/rspa.1989.0099
86. Deutsch, D., and R. Jozsa, 1992, Proc. R. Soc. London, Ser. A 439, 553. 10.1098/rspa.1992.0167
87. Diaconis, P., 1988, Group Representations in Probability and Statistics, IMS Lecture Notes-Monograph Series Vol. 11 (Institute of Mathematical Statistics, Hayward, CA).
88. Diaconis, P., and D. Rockmore, 1990, J. Am. Math. Soc. 3, 297. 10.2307/1990955
89. Diffie, W., and M. E. Hellman, 1976, IEEE Trans. Inf. Theory 22, 644. 10.1109/TIT.1976.1055638
90. DiVincenzo, D. P., 1995, Phys. Rev. A 51, 1015. 10.1103/PhysRevA.51.1015
91. Dürr, C., M. Heiligman, P. Høyer, and M. Mhalla, 2006, SIAM J. Comput. 35, 1310, preliminary version in STOC 2000. 10.1137/050644719
92. Ekert, A., and R. Jozsa, 1996, Rev. Mod. Phys. 68, 733. 10.1103/RevModPhys.68.733
93. Ettinger, M., and P. Høyer, 1999, e-print arXiv:quant-ph/9901029.
94. Ettinger, M., and P. Høyer, 2000, Adv. Appl. Math. 25, 239. 10.1006/aama.2000.0699
95. Ettinger, M., P. Høyer, and E. Knill, 1999, e-print arXiv:quant-ph/9901034.
96. Ettinger, M., P. Høyer, and E. Knill, 2004, Inf. Process. Lett. 91, 43. 10.1016/j.ipl.2004.01.024
97. Farhi, E., J. Goldstone, and S. Gutmann, 2008, Theor. Comput. 4, 169. 10.4086/toc.2008.v004a008
98. Farhi, E., J. Goldstone, S. Gutmann, and M. Sipser, 2000, e-print arXiv:quant-ph/0001106.
99. Farhi, E., and S. Gutmann, 1998, Phys. Rev. A 58, 915. 10.1103/PhysRevA.58.915
100. Fenner, S. A., and Y. Zhang, 2008, Proceedings of the Fifth Annual Conference on Theory and Applications of Models of Computation, Lecture Notes in Computer Science Vol. 4978 (Springer, Berlin), pp. 70-81
101. e-print arXiv:quant-ph/0610086.
102. Feynman, R. P., 1982, Int. J. Theor. Phys. 21, 467. 10.1007/BF02650179
103. Filotti, I. S., and J. N. Mayer, 1980, ACM Symposium on Theory of Computing (ACM, New York), pp. 236-243.
104. Fisher, D. S., 1992, Phys. Rev. Lett. 69, 534. 10.1103/PhysRevLett.69.534
105. Flaxman, A. D., and B. Przydatek, 2005, Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science Vol. 3404 (Springer, Berlin), pp. 305-314.
106. Fortnow, L., and J. D. Rogers, 1998, J. Comput. Syst. Sci. 59, 240, preliminary version in CCC 1998. 10.1006/jcss.1999.1651
107. Freedman, M., M. Larsen, and Z. Wang, 2002, Commun. Math. Phys. 227, 605. 10.1007/s002200200645
108. Freedman, M. H., A. Kitaev, M. J. Larsen, and Z. Wang, 2003, Bull., New Ser., Am. Math. Soc. 40, 31. 10.1090/S0273-0979-02-00964-3
109. Freedman, M. H., A. Y. Kitaev, and Z. Wang, 2002, Commun. Math. Phys. 227, 587. 10.1007/s002200200635
110. Friedl, K., G. Ivanyos, F. Magniez, M. Santha, and P. Sen, 2003, ACM Symposium on Theory of Computing (ACM, New York), pp. 1-9
111. e-print arXiv:quant-ph/0211091.
112. Gavinsky, D., 2004, Quantum Inf. Comput. 4, 229.
113. Gordon, D. M., 1993, SIAM J. Discrete Math. 6, 124. 10.1137/0406010
114. Grigni, M., L. J. Schulman, M. Vazirani, and U. Vazirani, 2004, Combinatorica 24, 137, preliminary version in STOC 2001. 10.1007/s00493-004- 0009-8
115. Grover, L. K., 1997, Phys. Rev. Lett. 79, 325, preliminary version in STOC 1996. 10.1103/PhysRevLett.79.325
116. Hales, L., and S. Hallgren, 2000, IEEE Symposium on Foundations of Computer Science (IEEE, Los Alamitos, CA), pp. 515-525.
117. Hales, L. R., 2002, Ph.D. thesis, University of California, Berkeley;
118. e-print arXiv:quant-ph/0212002.
119. Hallgren, S., 2005, ACM Symposium on Theory of Computing (ACM, New York), pp. 468-474.
120. Hallgren, S., 2007, J. ACM 54, 4, preliminary version in STOC 2002.
121. Hallgren, S., C. Moore, M. Rötteler, A. Russell, and P. Sen, 2006, ACM Symposium on Theory of Computing (ACM, New York), pp. 604-617;
122. e-print arXiv:quant-ph/0511148;
123. e-print arXiv:quant-ph/0511149.
124. Hallgren, S., A. Russell, and A. Ta-Shma, 2003, SIAM J. Comput. 32, 916, preliminary version in STOC 2000. 10.1137/S009753970139450X
125. Hamermesh, M., 1989, Group Theory and Its Application to Physical Problems (Dover, New York).
126. Hardy, G. H., and E. M. Wright, 1979, An Introduction to the Theory of Numbers, 5th ed. (Oxford University Press, Oxford).
127. Harrow, A. W., B. Recht, and I. L. Chuang, 2002, J. Math. Phys. 43, 4445. 10.1063/1.1495899
128. Harrow, A. W., and A. Winter, 2006, e-print arXiv:quant-ph/0606131.
129. Hausladen, P., and W. K. Wootters, 1994, J. Mod. Opt. 41, 2385. 10.1080/09500349414552221
130. Hayashi, M., A. Kawachi, and H. Kobayashi, 2008, Quantum Inf. Comput. 8, 345.
131. Hoffmann, C. M., 1982, Group-Theoretic Algorithms and Graph Isomorphism, Lecture Notes in Computer Science Vol. 136 (Springer-Verlag, Berlin).
132. Holevo, A. S., 1973, J. Multivariate Anal. 3, 337. 10.1016/0047-259X(73) 90028-6
133. Høyer, P., 1997, e-print arXiv:quant-ph/9702028.
134. Hulek, K., 2003, Elementary Algebraic Geometry, Student Mathematical Library Vol. 20 (AMS, Providence, RI).
135. Impagliazzo, R., and A. Wigderson, 1997, ACM Symposium on Theory of Computing (ACM, New York), pp. 220-229.
136. Ip, L., 2003, http://lawrenceip.com/papers/hspsdpabstract.html
137. Ireland, K., and M. Rosen, 1990, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics Vol. 84, 2nd ed. (Springer-Verlag, New York).
138. Ivanyos, G., 2008, Quantum Inf. Comput. 8, 579.
139. Ivanyos, G., F. Magniez, and M. Santha, 2003, Int. J. Found. Comput. Sci. 14, 723, preliminary version in SPAA 2001. 10.1142/S0129054103001996
140. Ivanyos, G., L. Sanselme, and M. Santha, 2007, Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science Vol. 4393 (Springer, Berlin), pp. 586-597;
141. e-print arXiv:quant-ph/0701235.
142. Ivanyos, G., L. Sanselme, and M. Santha, 2008, Latin American Symposium on Theoretical Informatics, Lecture Notes in Computer Science Vol. 4957 (Springer, Berlin), pp. 759-771;
143. e-print arXiv:0707.1260.
144. Jaeger, F., D. L. Vertigan, and D. J. A. Welsh, 1990, Math. Proc. Cambridge Philos. Soc. 108, 35. 10.1017/S0305004100068936
145. Jansen, S., M. B. Ruskai, and R. Seiler, 2007, J. Math. Phys. 48, 102111. 10.1063/1.2798382
146. Jones, V. F. R., 1985, Bull., New Ser., Am. Math. Soc. 12, 103. 10.1090/S0273-0979-1985-15304-2
147. Jordan, S. P., and P. Wocjan, 2008, e-print arXiv:0807.4688.
148. Jozsa, R., 2003, Ann. Phys. 306, 241. 10.1016/S0003-4916(03)00067-8
149. Kauffman, L. H., 1987, Topology 26, 395. 10.1016/0040-9383(87)90009-7
150. Kawachi, A., T. Koshiba, H. Nishimura, and T. Yamakami, 2005, Advances in Cryptology, Lecture Notes in Computer Science Vol. 3494 (Springer, Berlin), pp. 268-284;
151. e-print arXiv:quant-ph/0403069.
152. Kaye, P., 2005, Quantum Inf. Comput. 5, 474.
153. Kaye, P., R. Laflamme, and M. Mosca, 2007, An Introduction to Quantum Computing (Oxford University Press, Oxford).
154. Kedlaya, K. S., 2006, Comput. Complexity 15, 1. 10.1007/s00037-006-0204-7
155. Khot, S., 2005, J. ACM 52, 789, preliminary version in FOCS 2004. 10.1145/1089023.1089027
156. Kitaev, A. Y., 1995, e-print arXiv:quant-ph/9511026.
157. Kitaev, A. Y., 1997, Russ. Math. Surveys 52, 1191. 10.1070/ RM1997v052n06ABEH002155
158. Kitaev, A. Y., A. H. Shen, and M. N. Vyalyi, 2002, Classical and Quantum Computation (AMS, Providence, RI).
159. Knill, E., 1995, Los Alamos National Laboratory Technical Report No. LAUR-95-2225 (unpublished);
160. e-print arXiv:quant-ph/9508006.
161. Knill, E., and R. Laflamme, 1998, Phys. Rev. Lett. 81, 5672. 10.1103/PhysRevLett.81.5672
162. Knill, E., R. Laflamme, and W. Zurek, 1996, Los Alamos National Laboratory Technical Report No. LAUR-96-2199 (unpublished);
163. e-print arXiv:quant-ph/9610011.
164. Knill, E., R. Laflamme, and W. Zurek, 1997, Proc. R. Soc. London, Ser. A 454, 365.
165. Köbler, J., U. Schöning, and J. Torán, 1993, The Graph Isomorphism Problem: Its Structural Complexity (Springer, New York).
166. Koblitz, N., 1998, Algebraic Aspects of Cryptography, Algorithms and Computation in Mathematics Vol. 3 (Springer-Verlag, New York).
167. Koiran, P., V. Nesme, and N. Portier, 2007, Theor. Comput. Sci. 380, 115, preliminary version in ICALP 2005. 10.1016/j.tcs.2007.02.057
168. Kuperberg, G., 2005, SIAM J. Comput. 35, 170. 10.1137/S0097539703436345
169. Lauder, A., and D. Wan, 2008, in Algorithmic Number Theory, edited by, J. P. Buhler, and P. Stevenhagen, MSRI Publications Vol. 44 (Cambridge University Press, Cambridge), pp. 579-612.
170. Lenstra, A. K., H. W. Lenstra, Jr., and L. Lovász, 1982, Math. Ann. 261, 515. 10.1007/BF01457454
171. Lenstra, H. W., Jr., 1983, Math. Op. Res. 8, 538. 10.1287/moor.8.4.538
172. Lenstra, H. W., Jr., 2002, Not. Am. Math. Soc. 49, 182.
173. Lidl, R., and H. Niederreiter, 1997, Finite Fields, 2nd ed., Encyclopedia of Mathematics and Its Applications Vol. 20 (Cambridge University Press, Cambridge).
174. Lloyd, S., 1996, Science 273, 1073. 10.1126/science.273.5278.1073
175. Lorenzini, D., 1996, An Invitation to Arithmetic Geometry, Graduate Studies in Mathematics Vol. 9 (AMS, Providence, RI).
176. Luks, E. M., 1982, J. Comput. Syst. Sci. 25, 42. 10.1016/0022-0000(82) 90009-5
177. Magniez, F., and A. Nayak, 2007, Algorithmica 48, 221, preliminary version in ICALP 2005. 10.1007/s00453-007-0057-8
178. Magniez, F., A. Nayak, J. Roland, and M. Santha, 2007, ACM Symposium on Theory of Computing (ACM, New York), pp. 575-584;
179. e-print arXiv:quant-ph/0608026.
180. Magniez, F., M. Santha, and M. Szegedy, 2007, SIAM J. Comput. 37, 413. 10.1137/050643684
181. Manin, Y., 1980, unpublished.
182. Maslen, D. K., and D. N. Rockmore, 1995, ACM-SIAM Symposium on Discrete Algorithms (SIAM, Philadelphia), pp. 253-262.
183. Maurer, U. M., and S. Wolf, 1999, SIAM J. Comput. 28, 1689. 10.1137/S0097539796302749
184. Menezes, A. J., P. C. van Oorschot, and S. A. Vanstone, 1996, Handbook of Applied Cryptography (CRC, Boca Raton, FL).
185. Micciancio, D., 2001, SIAM J. Comput. 30, 2008, preliminary version in FOCS 1998. 10.1137/S0097539700373039
186. Micciancio, D., and S. Goldwasser, 2002, Complexity of Lattice Problems: A Cryptographic Perspective (Kluwer, Boston).
187. Miller, G., 1980, ACM Symposium on Theory of Computing (ACM, New York), pp. 225-235.
188. Miller, G. L., 1976, J. Comput. Syst. Sci. 13, 300, preliminary version in STOC 1975.
189. Moore, C., D. Rockmore, and A. Russell, 2006, ACM Trans. Algorithms 2, 707, preliminary version in SODA 2004. 10.1145/1198513.1198525
190. Moore, C., D. N. Rockmore, A. Russell, and L. J. Schulman, 2007, SIAM J. Comput. 37, 938, preliminary version in SODA 2004. 10.1137/S0097539705447177
191. Moore, C., A. Russell, and L. J. Schulman, 2008, SIAM J. Comput. 37, 1842, preliminary version in FOCS 2005. 10.1137/050644896
192. Moore, C., A. Russell, and P. Sniady, 2007, ACM Symposium on Theory of Computing (ACM, New York), pp. 536-545;
193. e-print arXiv:quant-ph/0612089.
194. Moore, C., A. Russell, and U. Vazirani, 2007, e-print arXiv:quant-ph/0701115.
195. Mosca, M., 1999, Ph.D. thesis, University of Oxford.
196. Mosca, M., and A. Ekert, 1999, Proceedings of the First NASA International Conference on Quantum Computing and Quantum Communication, Lecture Notes in Computer Science Vol. 1509 (Springer-Verlag, Berlin).
197. Mosca, M., and C. Zalka, 2004, Int. J. Quantum Inf. 2, 91. 10.1142/S0219749904000109
198. Nielsen, M. A., and I. L. Chuang, 2000, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge).
199. Okamoto, T., K. Tanaka, and S. Uchiyama, 2000, in Advances in Cryptology, Lecture Notes in Computer Science Vol. 1880 (Springer, Berlin), pp. 147-165.
200. Papadimitriou, C. H., 1994, Computational Complexity (Addison-Wesley, Reading, MA).
201. Pérez-García, D., F. Verstraete, M. M. Wolf, and J. I. Cirac, 2007, Quantum Inf. Comput. 7, 401.
202. Petrank, E., and M. Roth, 1997, IEEE Trans. Inf. Theory 43, 1602. 10.1109/18.623157
203. Pila, J., 1990, Math. Comput. 55, 745. 10.2307/2008445
204. Pólya, G., 1945, How to Solve It: A New Aspect of Mathematical Method (Princeton University Press, Princeton, NJ).
205. Pomerance, C., 1987, in Discrete Algorithms and Complexity, edited by, D. S. Johnson, T. Nishizeki, A. Nozaki, and H. S. Wilf, (Academic, Boston), pp. 119-143.
206. Preskill, J., 1998a, Lecture notes for Ph229: Quantum information and computation, http://www.theory.caltech.edu/people/preskill/ph229.
207. Preskill, J., 1998b, Proc. R. Soc. London, Ser. A 454, 385. 10.1098/rspa.1998.0167
208. Proos, J., and C. Zalka, 2003, Quantum Inf. Comput. 3, 317.
209. Püschel, M., M. Rötteler, and T. Beth, 1999, International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science Vol. 1719 (Springer, Berlin), pp. 148-159;
210. e-print arXiv:quant-ph/9807064.
211. Rabin, M. O., 1980, J. Number Theory 12, 128. 10.1016/0022-314X(80)90084- 0
212. Radhakrishnan, J., M. Rötteler, and P. Sen, 2009, Algorithmica 55, 490. 10.1007/s00453-008-9231-x
213. Regev, O., 2004a, J. ACM 51, 899, preliminary version in STOC 2003. 10.1145/1039488.1039490
214. Regev, O., 2004b, SIAM J. Comput. 33, 738, preliminary version in FOCS 2002. 10.1137/S0097539703440678
215. Regev, O., 2004c, e-print arXiv:quant-ph/0406151.
216. Reichardt, B. W., 2004, ACM Symposium on Theory of Computing (ACM, New York), pp. 502-510.
217. Reichardt, B. W., and R. Špalek, 2008, ACM Symposium on Theory of Computing (ACM, New York), pp. 103-112;
218. e-print arXiv:0710.2630.
219. Rivest, R., A. Shamir, and L. Adleman, 1978, Commun. ACM 21, 120. 10.1145/359340.359342
220. Rockmore, D., 1990, Adv. Appl. Math. 11, 164. 10.1016/0196-8858(90)90008- M
221. Rudolph, T., and L. Grover, 2002, e-print arXiv:quant-ph/0210187.
222. Russell, A., and I. E. Shparlinski, 2004, J. Complex. 20, 404. 10.1016/j.jco.2003.08.019
223. Schmidt, A., and U. Vollmer, 2005, ACM Symposium on Theory of Computing (ACM, New York), pp. 475-480.
224. Schnorr, C. P., 1987, Theor. Comput. Sci. 53, 201. 10.1016/0304-3975(87) 90064-8
225. Schoof, R., 1985, Math. Comput. 44, 483. 10.2307/2007968
226. Sen, P., 2006, IEEE Conference on Computational Complexity (IEEE, Los Alamitos, CA), pp. 274-287;
227. e-print arXiv:quant-ph/0512085.
228. Serre, J.-P., 1977, Linear Representations of Finite Groups, Graduate Texts in Mathematics Vol. 42 (Springer, New York).
229. Shenvi, N., J. Kempe, and K. B. Whaley, 2003, Phys. Rev. A 67, 052307. 10.1103/PhysRevA.67.052307
230. Shi, Y., 2003, Quantum Inf. Comput. 3, 84.
231. Shor, P. W., 1996, IEEE Symposium on Foundations of Computer Science (IEEE, Los Alamitos, CA), pp. 56-65;
232. e-print arXiv:quant-ph/9605011.
233. Shor, P. W., 1997, SIAM J. Comput. 26, 1484, preliminary version in FOCS 1994. 10.1137/S0097539795293172
234. Shor, P. W., and S. P. Jordan, 2008, Quantum Inf. Comput. 8, 681.
235. Shoup, V., 2005, A Computational Introduction to Number Theory and Algebra (Cambridge University Press, Cambridge).
236. Simon, D. R., 1997, SIAM J. Comput. 26, 1474, preliminary version in FOCS 1994. 10.1137/S0097539796298637
237. Solovay, R., 2000, Lie groups and quantum circuits, http://www.msri.org/ publications/ln/msri/2000/qcomputing/solovay/1/.
238. Spielman, D. A., 1996, ACM Symposium on Theory of Computing (ACM, New York), pp. 576-584.
239. Stigler, S. M., 1980, Trans. N. Y. Acad. Sci. 39, 147.
240. Szegedy, M., 2004, IEEE Symposium on Foundations of Computer Science (IEEE, Los Alamitos, CA), pp. 32-41;
241. e-print arXiv:quant-ph/0401053.
242. Terras, A., 1999, Fourier Analysis On Finite Groups and Applications, London Mathematical Society Student Texts Vol. 43 (Cambridge University Press, Cambridge).
243. Thiel, C., 1995, Ph.D. thesis, Universität des Saarlandes, Saarbrücken, Germany.
244. van Emde Boas, P., 1981, University of Amsterdam Department of Mathematics Technical Report No. 8104.
245. van Dam, W., 2002, Algorithmica 34, 413. 10.1007/s00453-002-0975-4
246. van Dam, W., 2004, e-print arXiv:quant-ph/0405081.
247. van Dam, W., G. M. D'Ariano, A. Ekert, C. Macchiavello, and M. Mosca, 2007, J. Phys. A 40, 7971. 10.1088/1751-8113/40/28/S07
248. van Dam, W., S. Hallgren, and L. Ip, 2006, SIAM J. Comput. 36, 763. 10.1137/S009753970343141X
249. van Dam, W., M. Mosca, and U. Vazirani, 2001, IEEE Symposium on Foundations of Computer Science (IEEE, Los Alamitos, CA), pp. 279-287;
250. e-print arXiv:quant-ph/0206003.
251. van Dam, W., and G. Seroussi, 2002, e-print arXiv:quant-ph/0207131.
252. van Dam, W., and U. Vazirani, 2003, http://www.cs.berkeley.edu/~vazirani/ pubs/qao.ps
253. Vollmer, U., 2000, Proceedings of the Fourth International Symposium on Algorithmic Number Theory, Lecture Notes in Computer Science Vol. 1838 (Springer, Berlin), pp. 581-594.
254. von zur Gathen, J., M. Karpinski, and I. Shparlinski, 1997, Comput. Complexity 6, 64. 10.1007/BF01202042
255. Watrous, J., 2001a, ACM Symposium on Theory of Computing (ACM, New York), pp. 60-67.
256. Watrous, J., 2001b, J. Comput. Syst. Sci. 62, 376. 10.1006/jcss.2000.1732
257. Watrous, J., 2009, in Encyclopedia of Complexity and Systems Science (Springer, New York);
258. e-print arXiv:0804.3401.
259. Wiesner, S., 1996, e-print arXiv:quant-ph/9603028.
260. Witten, E., 1989, Commun. Math. Phys. 121, 351. 10.1007/BF01217730
261. Wocjan, P., and J. Yard, 2008, Quantum Inf. Comput. 8, 147.
262. Yao, A. C.-C., 1993, IEEE Symposium on Foundations of Computer Science (IEEE, Los Alamitos, CA), pp. 352-361.
263. Yuen, H. P., R. S. Kennedy, and M. Lax, 1975, IEEE Trans. Inf. Theory 21, 125. 10.1109/TIT.1975.1055351
264. Zalka, C., 1998, Proc. R. Soc. London, Ser. A 454, 313. 10.1098/rspa.1998.0162