Quantum Lower Bounds by Polynomials

We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0, 1}/sup N/ in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i. e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T/sup 6/) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.

Authors

• Robert Beals
• Harry Buhrman
• Richard Cleve
• Michele Mosca
• Ronald de Wolf

Keywords

• Polynomials
• Quantum computing
• Postal services
• Hip
• Computational modeling
• Mathematical model
• Mathematics
• US Department of Transportation
• Computer science
• Laboratories
• Lower Bound
• Upper Bound
• Decision Tree
• Complex Functions
• Classification Algorithms
• Part Of Function
• Complex Class
• Symmetric Function
• Exact Set
• Boolean Function
• Quantum Algorithms
• Quantum Network
• Polynomial Relationship
• Basic Conditions
• Network State
• Polynomial Of Degree
• Probability 1
• Unitary Transformation
• Counting Algorithm
• Hamming Weight
• Proof Let
• Complex Quantum
• Exact Search