Theory of Computing ------------------- Title : Quantum Search of Spatial Regions Authors : Scott Aaronson and Andris Ambainis Volume : 1 Number : 4 Pages : 47-79 URL : https://theoryofcomputing.org/articles/v001a004 Abstract -------- Can Grover's 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 Benioff. In particular, we show how to search a d-dimensional hypercube in time O(\sqrt{n}) for d\ge 3, or O(\sqrt{n}\log^{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 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.