Spatial Search Algorithms
An interesting problem in the area of algorithms is the spatial search problem, which consists of finding one or more specific points in a physical region that can be modelled by a two-dimensional lattice, so that the vertices are the places one can search and the edges are the directions to which one can move. The quantum version of this problem was analyzed by Benioff in a very concrete way. He imagined a quantum robot that moves to adjacent nodes in a unit time. The position of the robot can be in superposition of a finite number of places. How many steps will the robot take to find a marked node with high probability?
If we consider n consecutive sites in a line, the quantum motion of the robot from one end to the other will take n−1 time units with no possibility of gain in complexity compared to a classical robot. However, if the sites form a two-dimensional lattice with the topology of a torus, the quantum robot can find a marked site more quickly. A direct application of Grover’s algorithm to this problem in two-dimensional lattices does not improve the time complexity compared to searching using a classical robot that moves at random.
The quantum robot will be faster than the classical one if a strategy known as abstract search algorithm is used. In this chapter, we describe how this algorithm works and we analyze its time complexity in details. The two-dimensional lattice is used as a concrete example. At the end, we show that Grover’s algorithm can be seen as a spatial search problem in the complete graph.
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