Quantum search of spatial regions

Can Grover’s algorithm speed up search of a physical region—for example a 2-D 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 surmounted, refuting a claim to the contrary by Benioff. In particular, we show how to search a d-dimensional hypercube in time O(√n) for d ≥ 3, or O(√nlog^5/2n) 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(√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.