 
Summary: THEORY OF COMPUTING, Volume 1 (2005), pp. 4779
http://www.theoryofcomputing.org
Quantum Search of Spatial Regions
Scott Aaronson # Andris Ambainis +
Received: June 13, 2004; published: June ?, 2005.
Abstract: Can Grover's algorithm speed up search of a physical regionfor example a
2D grid of size # n × # n? The problem is that # n time seems to be needed for each
query, just to move amplitude across the grid. Here we show that this problem can be sur
mounted, refuting a claim to the contrary by Benioff. In particular, we show how to search
a ddimensional hypercube in time O( # n) for d # 3, or O( # nlog 5/2 n) for d = 2. More
generally, we introduce a model of quantum query complexity on graphs, motivated by
fundamental physical limits on information storage, particularly the holographic principle
from black hole thermodynamics. Our results in this model include almosttight upper and
lower bounds for many search tasks; a generalized algorithm that works for any graph with
good expansion properties, not just hypercubes; and relationships among several notions of
`locality' for unitary matrices acting on graphs. As an application of our results, we give
an O( # n)qubit communication protocol for the disjointness problem, which improves an
upper bound of Høyer and de Wolf and matches a lower bound of Razborov.
ACM Classification: F.1.2, F.1.3
AMS Classification: 81P68, 68Q10
