Can Grover?'s quantum search algorithm speed up search of a physical region — for example a 2-D grid of size \sqrt n \times \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 Benio. In particular, we show how to search a d-dimensional hypercube in time 0(\sqrt n ) for d \geqslant 3, or 0(\sqrt {n\log ^3 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 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 0(\sqrt {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.