Quantum Lower Bounds for Collision and Element Distinctness with Small Range
Year: 2003
Type: preprint
Abstract: We give a general method for proving quantum lower bounds for problems with small range. Namely, we show that, for any symmetric problem defined on functions $f:\\{1, ..., N\\}\\to\\{1, ..., M\\}$, its polynomial degree is the same for all $M\\geq N$. Therefore, if we have a quantum lower bound for some (possibly, quite large) range $M$ which is shown using polynomials method, we immediately get the same lower bound for all ranges $M\\geq N$. In particular, we get $\\Omega(N^{1/3})$ and $\\Omega(N^{2/3})$ quantum lower bounds for collision and element distinctness with small range.
Source: arXiv (Cornell University)
Author: Andris Ambainis
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2969561584