Quantum algorithms for weighing matrices and quadratic residues

Algorithmica (New York)

Volumn 34, Issue 4, 2002, Pages 413-428

  Van Dam, Wim ^a,b  

^a HEWLETT PACKARD LABORATORIES   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Combinatorics; Quantum algorithms

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; MATRIX ALGEBRA; PROBABILITY; PROBLEM SOLVING;

WEIGHING MATRICES;

QUANTUM THEORY;

EID: 0141991893     PISSN: 01784617     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00453-002-0975-4     Document Type: Article

References (32)

1. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf, Quantum lower bounds by polynomials, Proceedings of the 39th Annual Symposium on Foundations of Computer Science, pages 352-361, IEEE Computer Society Press, Los Alamitos, CA (1998); quant-ph report no. 9802049.
2. Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani, Strengths and weaknesses of quantum computing, SIAM Journal on Computing, Volume 26, No. 5, pages 1510-1523 (1997); quant-ph report no. 9701001.
3. Bruce C. Berndt, Ronald J. Evans, and Kenneth S. Williams, Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, Volume 21, Wiley, New York (1998).
4. Ethan Bernstein and Umesh Vazirani, Quantum complexity theory, SIAM Journal on Computing, Volume 26, No. 5, pages 1411-1473 (1997).
5. André Berthiaume and Gilles Brassard, Oracle quantum computing, Journal of Modern Optics, Volume 41, No. 12, pages 2521-2535 (1994).
6. Michel Boyer, Gilles Brassard, Peter Høyer, and Alain Tapp, Tight bounds on quantum searching, Fortschritte der Physik, Volume 46, No. 4-5, pages 493-505 (1998); quant-ph report no. 9605034.
7. Harry Buhrman and Wim van Dam, Bounded quantum query complexity, Proceedings of the 14th Annual IEEE Conference on Computational Complexity, pages 149-156 (1999); quant-ph report no. 9903035.
8. Richard Cleve, Artur Ekert, Chiara Macchiavello, and Michele Mosca, Quantum algorithms revisited, Proceedings of the Royal Society of London, Series A, Volume 454, pages 339-354 (1998); quant-ph report no. 9708016.
9. Henri Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Springer-Verlag, New York (1993).
10. The CRC Handbook of Combinatorial Designs, edited by Charles J. Colbourn and Jeffrey H. Dinitz, Series on Discrete Mathematics and Its Applications, CRC Press, Boca Raton, FL (1996).
11. Wim van Dam and Sean Hallgren, Efficient quantum algorithms for shifted quadratic characterproblems, quant-ph report no. 0011067.
12. David Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proceedings of the Royal Society of London, Series A, Volume 400, pages 97-117 (1985).
13. David Deutsch and Richard Jozsa, Rapid solution of problems by quantum computation, Proceedings of the Royal Society of London, Series A, Volume 439, pages 553-558 (1992).
14. Lov K. Grover, A fast quantum mechanical algorithm for database search, Proceedings of the 28th Annual ACM Symposium on the Theory of Computing, pages 212-219 (1996); quant-ph report no. 9605043.
15. Jacques Hadamard, Résolution d'une question relative aux déterminants, Bulletin des Sciences Mathématiques, Volume 17, No. 2, pages 241-246 (1893).
16. Sam Hedayat, Neil Sloane, and John Stufken, Orthogonal Arrays: Theory and Applications, Springer-Verlag, New York (1999).
17. Kenneth Ireland and Michael Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, Volume 84, Second edition, Springer-Verlag, New York (1990).
18. Ernst Jacobstahl, Über die Darstellung der Primzahlen der Form 4n 4 + 1 als Summe zweier Quadrate, Journal für die reine und Angewandte Mathematik (Crelle's Journal), Volume 132, pages 238-245 (1907).
19. Richard Jozsa, Characterizing classes of functions computable by quantum parallelism. Proceedings of the Royal Society of London, Series A, Volume 435, pages 563-574 (1991).
20. Hiroshi Kimura, Hadamard matrices and dihedral groups. Designs, Codes, and Cryptography, Volume 9, No. 1, pages 71-77 (1996).
21. Jessie MacWilliams and Neil Sloane, The Theory of Error-Correcting Codes, North-Holland Mathematical Library, Volume 16, Elsevier, New York (1979).
22. Michele Mosca and Artur Ekert, The hidden subgroup problem and eigenvalue estimation on a quantum computer, in Proceedings of the 1 St NASA International Conference on Quantum Computing and Quantum Communications, edited by Colin P. Williams, Lecture Notes in Computer Science, Number 1509, pages 174-188, Springer-Verlag New York (1999); quant-ph report no. 9708019.
23. Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2000).
24. Raymond E.A.C. Paley, On orthogonal matrices, Journal of Mathematics and Physics, Volume 12, pages 311-320 (1933).
25. Daniel R. Simon, On the power of quantum computation, SIAM Journal on Computing, Volume 26, No. 5, pages 1474-1483 (1997).
26. Neil Sloane, A library of Hadamard matrices, http://www.research.att.com/-njas/hadamard/index. html.
27. Jennifer Seberry, A life's work on Hadamard matrices, statistical designs, Bent functions and their application to computer and information security and telecommunications, http://www.cs. uow.edu.au/people/jennie/lifework.html.
28. Jennifer Seberry and Mieko Yamada, Hadamard matrices, sequences, and block designs, in Contemporary Design Theory: A Collection of Surveys, edited by Jeffrey H. Dinitz and Douglas R. Stinson, pages 431-560, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley, New York (1992).
29. Koichi Shinoda and Mieko Yamada, A family of Hadamard matrices of dihedral group type, Discrete Applied Mathematics, Volume 102, No. 1-2, pages 141-150 (2000).
30. Peter W. Shor, Algorithms for quantum computation: discrete logarithms and factoring, SIAM Journal on Computing, Volume 26, No. 5, pages 1484-1509 (1997); quant-ph report no. 9508027.
31. James Joseph Sylvester, Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Volume 34, No. 232, pages 461-475 (1867).
32. Richard J. Turyn, Complex Hadamard matrices, in Combinatorial Structures and Their Applications, edited by Richard K. Guy, Haim Hanani, and Norbert Sauer, pages 435-437, Gordon and Breach, London (1970).