Span-Program-based quantum algorithm for evaluating formulas

Proceedings of the Annual ACM Symposium on Theory of Computing

Volumn , Issue , 2008, Pages 103-112

  Reichardt, Ben W ^a   Špalek, Robert ^b  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b GOOGLE INC   (United States)

Author keywords

Balanced ternary majority formula; Formula evaluation; Gadget graphs; Quantum adversary bound; Quantum algorithms; Quantum computing; Quantum phase estimation; Quantum walks; Span programs; Spectral analysis

Indexed keywords

COMPUTATION THEORY; EIGENVALUES AND EIGENFUNCTIONS; OPTIMIZATION; QUANTUM COMPUTERS; SPECTRUM ANALYSIS;

BALANCED TERNARY MAJORITY FORMULA; FORMULA EVALUATION; GADGET GRAPHS; QUANTUM ADVERSARY BOUND; QUANTUM ALGORITHMS; QUANTUM COMPUTING; QUANTUM PHASE ESTIMATION; QUANTUM WALK; SPAN PROGRAMS;

QUANTUM THEORY;

EID: 57049181445     PISSN: 07378017     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/1374376.1374394     Document Type: Conference Paper

References (28)

1. E. Allender, R, Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. Computational Complexity, 8:99-126, 1999. Preliminary version in Proc. of 28th ACM STOC, 1996.
2. A. Ambainis. Polynomial degree vs. quantum query complexity. J. Comput. Syst. Scl, 72(2):220-238, 2006. Preliminary version in Proc. of 44th IEEE FOGS, 2003.
3. A. Ambainis. A nearly optimal discrete query quantum algorithm for evaluating NAND formulas. arXiv:0704.3628 [quant-ph], 2007.
4. 1/2+0 on a quantum computer, in Proc. of 48th IEEE FOCS, pages 363-372, 2007.
5. L. Babai, A. Gál, and A. Wigderson. Super-polynomial lower bounds for monotone span programs. Combinatorial, 19(3):301-319, 1999. Preliminary version in Proc. of 28th ACM STOC, 1996.
6. H. Barnum and M. Saks. A lower bound on the quantum query complexity of read-once functions. J. Comput. Syst. Scl, 69(2):244-258, 2004.
7. H. Barnum, M. Saks, and M. Szegedy. Quantum decision trees and semidefinite programming. In Proc. of 18th IEEE Complexity, pages 179-193, 2003.
8. A. Beimel, A. Gál, and M. Paterson. Lower bounds for monotone span programs. Computational Complexity, 6:29-45, 1996. Preliminary version in Proc. of 36th IEEE FOCS, 1995.
9. 1/2+0(1) on a quantum computer. quant-ph/0703015, 2007.
10. R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca. Quantum algorithms revisited. Proc. R. Soc. London A, 454(1969):339-354, 1998.
11. R. Cramer and S. Fehr. Optimal black-box secret sharing over arbitrary Abelian groups. In Proc. CRYPTO 2002, LNCS vol. 2442, pages 272-287. Springer-Verlag, 2002.
12. E. Farhi, J. Goldstone, and S. Gutmann. A quantum algorithm for the Hamiltonian NAND tree. quant-ph/0702144, 2007.
13. A. Gál and P. Pudlak. A note on monotone complexity and the rank of matrices. Information Processing Letters, 87(6):321-326, 2003.
14. L. K. Grover. A fast quantum mechanical algorithm for database search. In Proc. of 28th ACM STOC, pages 212-219, 1996.
15. L. K. Grover. Tradeoffs in the quantum search algorithm, quant-ph/0201152, 2002.
16. P. Høyer, T. Lee, and R,. Špalek. Source codes of semidefinite programs for ADV±. http: //www.ucw http://www.ucw. cz/~robert/papers/adv/:, 2006.
17. P. Høyer, T. Lee, and R. Špalek. Negative weights make adversaries stronger. In Proc. of 39th ACM STOC, pages 526-535, 2007.
18. T. S. Jayram, R. Kumar, and D. Sivakumar. Two applications of information complexity. In Proc. of 35th ACM STOC, pages 673-682, 2003.
19. M. Karchmer and A. Wigderson. On span programs. In Proc. of 8th IEEE Symp. Structure in Complexity Theory, pages 102-111, 1993.
20. S. Laplante and F. Magniez. Lower bounds for randomized and quantum query complexity using Kolmogorov arguments. In Proc. of 19th IEEE Complexity, pages 294-304, 2004.
21. V. Nikov, S. Nikova, and B. Preneel. On the size of monotone span programs. In Proc. SCN 2004, LNCS vol. 3352, pages 249-262, 2005.
22. B. W. Reichardt and R. Špalek. Quantum query complexity of up to 4-bit functions. http://www.ucw.cz/~robert/papers/gadgets/ 2007.
23. B. W. Reichardt and R. Špalek. Span-program-based quantum algorithm for evaluating formulas. arXiv:0710.2630 [quant-ph], 2007.
24. M. Saks and A. Wigderson. Probabilistic Boolean decision trees and the complexity of evaluating game trees. In Proc. of 27th IEEE FOCS, pages 29-38, 1986.
25. M. Santha. On the Monte Carlo decision tree complexity of read-once formulae. Random Structures and Algorithms, 6(1):75-87, 1995. Preliminary version in Proc. of 6th IEEE Structure in Complexity Theory, 1991.
26. M. Snir. Lower bounds on probabilistic linear decision trees. Theoretical Computer Science, 38:69-82, 1985.
27. R.. Špalek and M. Szegedy. All quantum adversary methods are equivalent. Theory of Computing, 2(1):1-18, 2006. Earlier version in ICALP'05.
28. M. Szegedy. Quantum speed-up of Markov chain based algorithms. In Proc. of 45th IEEE FOCS, pages 32-41, 2004.