  Can Grover's algorithm speed up search of a physical region - for example a
2-D grid of size sqrt(n) by sqrt(n)? The problem is that sqrt(n) time seems to
be needed for each query, just to move amplitude across the grid. Here we show
that this problem can be surmounted, refuting a claim to the contrary by
Benioff. In particular, we show how to search a d-dimensional hypercube in time
O(sqrt n) for d at least 3, or O((sqrt n)(log n)^(3/2)) 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 almost-tight 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(sqrt(n))-qubit communication protocol for the disjointness problem, which
improves an upper bound of Hoyer and de Wolf and matches a lower bound of
Razborov.
