Every NAND formula of size N can be evaluated in time N^{1/2+o(1)} on a quantum computer
Quantum Physics
2011-11-09 v3
Abstract
For every NAND formula of size N, there is a bounded-error N^{1/2+o(1)}-time quantum algorithm, based on a coined quantum walk, that evaluates this formula on a black-box input. Balanced, or ``approximately balanced,'' NAND formulas can be evaluated in O(sqrt{N}) queries, which is optimal. It follows that the (2-o(1))-th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.
Keywords
Cite
@article{arxiv.quant-ph/0703015,
title = {Every NAND formula of size N can be evaluated in time N^{1/2+o(1)} on a quantum computer},
author = {Andrew M. Childs and Ben W. Reichardt and Robert Spalek and Shengyu Zhang},
journal= {arXiv preprint arXiv:quant-ph/0703015},
year = {2011}
}
Comments
14 pages, 3 figures, v2: substantially rewritten with clearer analysis, v3: using better discretization we have obtained an optimal algorithm. to appear in FOCS'07