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THEORY OF COMPUTING, Volume 1 (2005), pp. 47--79 http://www.theoryofcomputing.org

Summary: THEORY OF COMPUTING, Volume 1 (2005), pp. 47--79
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 region---for 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


Source: Aaronson, Scott - Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (MIT)


Collections: Physics; Computer Technologies and Information Sciences