Quantum Information Processing

, Volume 15, Issue 10, pp 4029–4048 | Cite as

Laplacian versus adjacency matrix in quantum walk search

  • Thomas G. WongEmail author
  • Luís Tarrataca
  • Nikolay Nahimov


A quantum particle evolving by Schrödinger’s equation contains, from the kinetic energy of the particle, a term in its Hamiltonian proportional to Laplace’s operator. In discrete space, this is replaced by the discrete or graph Laplacian, which gives rise to a continuous-time quantum walk. Besides this natural definition, some quantum walk algorithms instead use the adjacency matrix to effect the walk. While this is equivalent to the Laplacian for regular graphs, it is different for non-regular graphs and is thus an inequivalent quantum walk. We algorithmically explore this distinction by analyzing search on the complete bipartite graph with multiple marked vertices, using both the Laplacian and adjacency matrix. The two walks differ qualitatively and quantitatively in their required jumping rate, runtime, sampling of marked vertices, and in what constitutes a natural initial state. Thus the choice of the Laplacian or adjacency matrix to effect the walk has important algorithmic consequences.


Quantum walk Continuous time Spatial search Laplacian Adjacency matrix 



TW and NN were supported by the European Union Seventh Framework Programme (FP7/2007-2013) under the QALGO (Grant Agreement No. 600700) project, and the ERC Advanced Grant MQC. LT was supported by CNPq CSF/BJT grant reference 301181/2014-4.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Thomas G. Wong
    • 1
    Email author
  • Luís Tarrataca
    • 2
  • Nikolay Nahimov
    • 1
  1. 1.Faculty of ComputingUniversity of LatviaRīgaLatvia
  2. 2.Laboratório Nacional de Computação CientíficaPetrópolisBrazil

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