Spatial Search Algorithms

  • Renato Portugal
Part of the Quantum Science and Technology book series (QST)


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.


Spatial Search Problem Quantum Robot Classical Robot Benioff Quantum Walk 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Aaronson, S., Ambainis, A.: Quantum search of spatial regions. In: Theory of Computing, pp. 200–209 (2003)Google Scholar
  2. 2.
    Abal, G., Donangelo, R., Marquezino, F.L., Portugal, R.: Spatial search on a honeycomb network. Math. Struct. Comput. Sci. 20(Special Issue 06), 999–1009 (2010)Google Scholar
  3. 8.
    Ambainis, A., Backurs, A., Nahimovs, N., Ozols, R., Rivosh, A.: Search by quantum walks on two-dimensional grid without amplitude amplification. arxiv:1112.3337 (2011)Google Scholar
  4. 10.
    Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 1099–1108 (2005)Google Scholar
  5. 15.
    Benioff, P.: Space searches with a quantum robot. AMS Contemporary Mathematical Series, vol. 305 (2002)Google Scholar
  6. 28.
    Forets, M., Abal, G., Donangelo, R., Portugal, R.: Spatial quantum search in a triangular network. Math. Struct. Comput. Sci. 22(03), 521–531 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 35.
    Hein, B., Tanner, G.: Quantum search algorithms on a regular lattice. Phys. Rev. A 82(1), 012326 (2010)ADSCrossRefGoogle Scholar
  8. 71.
    Shenvi, N., Kempe, J., Whaley, K.B.: A quantum random walk search algorithm. Phys. Rev. A 67(5), 052307 (2003), quant-ph/0210064Google Scholar
  9. 78.
    Tulsi, A.: Faster quantum-walk algorithm for the two-dimensional spatial search. Phys. Rev. A 78(1), 012310 (2008)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Renato Portugal
    • 1
  1. 1.Department of Computer ScienceNational Laboratory of Scientific Computing (LNCC)PetrópolisBrazil

Personalised recommendations