2005/04/12 by Shengyu Zhang, Zhang, Shengyu
Physics and Astronomy · #FOS: Physical sciences #Quantum Physics (quant-ph) #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0504085
18 pages, 1 figure. v2: introduction rewritten, references added. v3: a line for grant added. v4: upper bound section rewritten
arxiv created 2005/05/30 · arxiv updated 2009/12/01
The Local Search problem, which finds a local minimum of a black-box function on a given graph, is of both practical and theoretical importance to many areas in computer science and natural sciences. In this paper, we show that for the Boolean hypercube \Bn, the randomized query complexity of Local Search is Θ(2n/2n1/2) and the quantum query complexity is Θ(2n/3n1/6). We also show that for the constant dimensional grid [N1/d]d, the randomized query complexity is Θ(N1/2) for d ≥ 4 and the quantum query complexity is Θ(N1/3) for d ≥ 6. New lower bounds for lower dimensional grids are also given. These improve the previous results by Aaronson [STOC'04], and Santha and Szegedy [STOC'04]. Finally we show for [N1/2]2 a new upper bound of O(N1/4(loglog N)3/2) on the quantum query complexity, which implies that Local Search on grids exhibits different properties at low dimensions.