Any AND-OR formula of size N can be evaluated in time N 1/2+0(1) on a quantum computer

SIAM Journal on Computing

Volumn 39, Issue 6, 2010, Pages 2513-2530

  Ambainis, A ^a   Childs, A M ^b   Reichardt, B W ^b   Spalek R ^c   Zhang, S ^d  

^a UNIVERSITY OF LATVIA   (Latvia)

^b UNIVERSITY OF WATERLOO   (Canada)

^c GOOGLE INC   (United States)

^d CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

AND OR trees; Formula evaluation; Quantum computation; Quantum query complexity; Quantum walk

Indexed keywords

AND-OR TREE; FORMULA EVALUATION; QUANTUM COMPUTATION; QUANTUM QUERY COMPLEXITY; QUANTUM WALK;

COMPUTATIONAL LINGUISTICS; QUANTUM COMPUTERS; QUANTUM THEORY;

TREES (MATHEMATICS);

EID: 77952261982     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/080712167     Document Type: Article

References (30)

1. A. AMBAINIS, A. M. CHILDS, F. LE GALL, AND S. TANI, The quantum query complexity of certification, Quantum Information and Computation, to appear.
2. 1/2+0(1) on a quantum computer, in Proceedings of the 48th IEEE FOCS, 2007, pp. 363-372.
3. A. AMBAINIS, Polynomial degree vs. quantum query complexity,J.Comput. System Sci., 72 (2006), pp. 220-238.
4. A. AMBAINIS, A nearly optimal discrete query quantum algorithm for evaluating NAND formulas, arXiv quant-ph/0704.3628, 2007.
5. M. L. BONET AND S. R. BUSS, Size-depth tradeoffs for Boolean formulae, Inform. Process. Lett., 49 (1994), pp. 151-155.
6. R. BEALS, H. BUHRMAN, R. CLEVE, M. MOSCA, AND R. DE WOLF, Quantum lower bounds by polynomials, J. ACM, 48 (2001), pp. 778-797.
7. N. H. BSHOUTY, R. CLEVE, AND W. EBERLY, Size-depth tradeoffs for algebraic formulas, SIAM J. Comput., 24 (1995), pp. 682-705.
8. H. BUHRMAN, R. CLEVE, AND A. WIGDERSON, Quantum vs. classical communication and computation, in Proceedings of the 30th ACM STOC, 1998, pp. 63-68.
9. H. BARNUM AND M. SAKS, A lower bound on the quantum query complexity of readonce functions, J. Comput. System Sci., 69 ( 2004), pp. 244-258.
10. A. M. CHILDS, R. CLEVE, S. P. JORDAN, AND D. YEUNG, Discrete-query quantum algorithm for NAND trees, Theory of Computing, 5 (2009), pp. 119-123.
11. R. CLEVE, A. EKERT, C. MACCHIAVELLO, AND M. MOSCA, Quantum algorithms revisited, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 454 (1969), pp. 339-354.
12. A. M. CHILDS, On the relationship between cont inuous- and discrete-time quantum walk, Commun. Math. Phys., to appear.
13. 1/2+0(1) on a quantum computer, arXiv quant-ph/0703015, 2007.
14. E. FARHI, J. GOLDSTONE, AND S. GUTMANN, A quantum algorithm for the Hamiltonian NAND tree, Theory Comput., 4 (2008), pp. 169-190.
15. L. K. GROVER, A fast quantum mechanical algorithm for database search,Phys. Rev. Lett., 79 (1997), pp. 325-328.
16. L. K. GROVER, Tradeoffs in the quantum search algorithm, arXiv quant-ph/0201152, 2002.
17. P. HøYER, M. MOSCA, AND R. DE WOLF, Quantum search on bounded-error inputs, in Proceedings of the 30th ICALP, 2003, pp. 291-299.
18. Lecture Notes in Comput. Sci. 2719, Springer, New York.
19. T. S. JAYRAM, R. KUMAR, AND D. SIVAKUMAR, Two applications of information complexity, in Proceedings of the 35th ACM STOC, 2003, pp. 673-682.
20. A. R. KLIVANS, R. O'DONNELL, AND R. A. SERVEDIO, Learning intersections and thresholds of halfspaces, J. Comput. System Sci., 68 (2004), pp. 808-840.
21. ō(n1/3) J. Comput. System Sci., 68 (2004), pp. 303-318.
22. A. Y. KITAEV, A. H. SHEN AND M. N. VYALYI, Classical and Quantum Computation, Grad. Stud. Math. 47, American Mathematical Society, Providence, RI 2002.
23. S. LAPLANTE, T. LEE, AND M. SZEGEDY, The quantum adversary method and classical formula size lower bounds, Comput. Complexity, 15 (2006), pp. 163-196.
24. F. MAGNIEZ, A. NAYAK, J. ROLAND, AND M. SANTHA, Search via quantum walk, in Proceedings of the 39th ACM STOC, 2007, pp. 575-584.
25. R. O'DONNELL AND R. A. SERVEDIO, New degree bounds for polynomial threshold functions, in Proceedings of the 35th ACM STOC, 2003, pp. 325-334.
26. B. W. REICHARDT AND R. ŠPALEK, Span-program-based quantum algorithm for evaluating formulas, in Proceedings of the 40th ACM STOC, 2008, pp. 103-112.
27. M. SANTHA, On the Monte Carlo decision tree complexity of read-once formulae, Random Structures Algorithms, 6 (1995), pp. 75-87.
28. M. SNIR, Lower bounds on probabilistic linear de cision trees, Theoret. Comput. Sci., 38 (1985), pp. 69-82.
29. M. SAKS AND A. WIGDERSON, Probabilistic Boolean decision trees and the complexity of evaluating game trees, in Proceedings of the 27th IEEE FOCS, 1986, pp. 29-38.
30. M. SZEGEDY, Quantum speed-up of Markov chain based algorithms, in Proceedings of the 45th IEEE FOCS, 2004, pp. 32-41.