Quantum Walks and Search Algorithms pp 145-163 | Cite as

# Spatial Search Algorithms

## Abstract

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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