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
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
Strengths and Weaknesses of Quantum 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
Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
The Query Complexity of Order-Finding
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Quantum Lower Bounds by Polynomials
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Fault-tolerant quantum computation
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Polynomial degree vs. quantum query complexity
Journal of Computer and System Sciences - Special issue on FOCS 2003
On the power of Ambainis lower bounds
Theoretical Computer Science
Span-program-based quantum algorithm for evaluating formulas
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Quantum Algorithms for Evaluating Min-Max Trees
Theory of Quantum Computation, Communication, and Cryptography
Quantum random walks - new method for designing quantum algorithms
SOFSEM'08 Proceedings of the 34th conference on Current trends in theory and practice of computer science
On directional vs. undirectional randomized decision tree complexity for read-once formulas
CATS '10 Proceedings of the Sixteenth Symposium on Computing: the Australasian Theory - Volume 109
Lower bounds on quantum query complexity for read-once decision trees with parity nodes
CATS '09 Proceedings of the Fifteenth Australasian Symposium on Computing: The Australasian Theory - Volume 94
New developments in quantum algorithms
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Any AND-OR Formula of Size $N$ Can Be Evaluated in Time $N^{1/2+o(1)}$ on a Quantum Computer
SIAM Journal on Computing
The quantum query complexity of certification
Quantum Information & Computation
All quantum adversary methods are equivalent
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Faster quantum algorithm for evaluating game trees
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Reflections for quantum query algorithms
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
The quantum query complexity of read-many formulas
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Hi-index | 0.00 |
We establish a lower bound of Ω(√n) on the bounded-error quantum query complexity of read-once Boolean functions. The result is proved via an inductive argument, together with an extension of a lower bound method of Ambainis. Ambainis' method involves viewing a quantum computation as a mapping from inputs to quantum states (unit vectors in a complex inner-product space) which changes as the computation proceeds. Initially the mapping is constant (the state is independent of the input). If the computation evalutes the function f then at the end of the computation the two states associated with any f-distinguished pair of inputs (having different f values) are nearly orthogonal. Thus the inner product of their associated states must have changed from 1 to nearly 0. For any set of f-distinguished pairs of inputs, the sum of the inner products of the corresponding pairs of states must decrease significantly during the computation, By deriving an upper bound on the decrease in such a sum, during a single step, for a carefully selected set of input pairs, one can obtain a lower bound on the number of steps. We extend Ambainis' bound by considering general weighted sums of f-distinguished pairs. We then prove our result for read-once functions by induction on the number of variables, where the induction step involves a careful choice of weights depending on f to optimize the lower bound attained.