On the degree of polynomials that approximate symmetric Boolean functions (preliminary version)
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
On the Power of Quantum Computation
SIAM Journal on Computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Quantum vs. classical communication and computation
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
The quantum query complexity of approximating the median and related statistics
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Quantum lower bounds by polynomials
Journal of the ACM (JACM)
Quantum lower bound for the collision problem
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
Quantum Lower Bounds for the Collision and the Element Distinctness Problems
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Quantum lower bounds by quantum arguments
Journal of Computer and System Sciences - Special issue on STOC 2000
Quantum Oracle Interrogation: Getting All Information for Almost Half the Price
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Quantum Algorithms for Element Distinctness
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Polynomial Degree vs. Quantum Query Complexity
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Lower bounds for local search by quantum arguments
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Graph Properties and Circular Functions: How Low Can Quantum Query Complexity Go?
CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
Lower Bounds for Randomized and Quantum Query Complexity Using Kolmogorov Arguments
CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
A lower bound on the quantum query complexity of read-once functions
Journal of Computer and System Sciences
Quantum Walk Algorithm for Element Distinctness
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
All quantum adversary methods are equivalent
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
New upper and lower bounds for randomized and quantum local search
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Negative weights make adversaries stronger
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Quantum Separation of Local Search and Fixed Point Computation
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Quantum algorithms for matching and network flows
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
Reflections for quantum query algorithms
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Quantum counterfeit coin problems
Theoretical Computer Science
Reconstructing strings from substrings with quantum queries
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Adversary lower bound for the k-sum problem
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Hi-index | 5.23 |
The polynomial method and the Ambainis lower bound (or Alb, for short) method are two main quantum lower bound techniques. While recently Ambainis showed that the polynomial method is not tight, the present paper aims at studying the power and limitation of Alb's. We first use known Alb's to derive Ω(n1.5) lower bounds for BIPARTITENESS, BIPARTITENESS MATCHING and GRAPH MATCHING, in which the lower bound for BIPARTITENESS improves the previous Ω(n) one. We then show that all the three known Ambainis lower bounds have a limitation √N min{C0(f), C1(f)}, where C0(f) and C1(f) are the 0- and 1-certificate complexities, respectively. This implies that for many problems such as TRIANGLE, k-CLIQUE, BIPARTITENESS and BIPARTITE/GRAPH MATCHING which draw wide interest and whose quantum query complexities are still open, the best known lower bounds cannot be further improved by using Ambainis techniques. Another consequence is that all the Ambainis lower bounds are not tight. For total functions, this upper bound for Alb's can be further improved to min{√C0(f)C1(f), √NċCI(f)}, where CI(f) is the size of max intersection of a 0- and a 1-certificate set. Again this implies that Alb's cannot improve the best known lower bound for some specific problems such as AND-OR TREE, whose precise quantum query complexity is still open. Finally, we generalize the three known Alb's and give a new Alb style lower bound method, which may be easier to use for some problems.