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
The growth of polynomials bounded at equally spaced points
SIAM Journal on Mathematical Analysis
On the degree of Boolean functions as real polynomials
Computational Complexity - Special issue on circuit complexity
A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
SIAM Journal on Computing
Quantum lower bounds by quantum arguments
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Quantum lower bounds by polynomials
Journal of the ACM (JACM)
Quantum computation and quantum information
Quantum computation and quantum information
Concrete Mathematics: A Foundation for Computer Science
Concrete Mathematics: A Foundation for Computer Science
Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
Bounds for Small-Error and Zero-Error Quantum Algorithms
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Towards Proving Strong Direct Product Theorems
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
Quantum lower bounds for the collision and the element distinctness problems
Journal of the ACM (JACM)
Lower Bounds for Randomized and Quantum Query Complexity Using Kolmogorov Arguments
CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
Limitations of Quantum Advice and One-Way Communication
CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Quantum Walk Algorithm for Element Distinctness
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Quantum algorithms for the triangle problem
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On the power of Ambainis lower bounds
Theoretical Computer Science
Quantum search on bounded-error inputs
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
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We give a new version of the adversary method for proving lower bounds on quantum query algorithms. The new method is based on analyzing the eigenspace structure of the problem at hand. We use it to prove a new and optimal strong direct product theorem for 2-sided error quantum algorithms computing k independent instances of a symmetric Boolean function: if the algorithm uses significantly less than k times the number of queries needed for one instance of the function, then its success probability is exponentially small in k. We also use the polynomial method to prove a direct product theorem for 1-sided error algorithms for k threshold functions with a stronger bound on the success probability. Finally, we present a quantum algorithm for evaluating solutions to systems of linear inequalities, and use our direct product theorems to show that the time-space tradeoff of this algorithm is close to optimal.