Laplacian versus adjacency matrix in quantum walk search

Thomas Wong, Luís Tarrataca, Nikolay Nahimov

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)4029-4048
Number of pages20
JournalQuantum Information Processing
Volume15
Issue number10
DOIs
StatePublished - Oct 1 2016
Externally publishedYes

Fingerprint

Quantum Walk
Laplacian Matrix
Adjacency Matrix
Walk
apexes
matrices
Discrete Laplacian
Graph Laplacian
Hamiltonians
Laplace transformation
Complete Bipartite Graph
Laplace Operator
Regular Graph
Kinetic energy
Mathematical operators
Continuous Time
kinetic energy
Directly proportional
sampling
Sampling

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Statistical and Nonlinear Physics
  • Theoretical Computer Science
  • Signal Processing
  • Modeling and Simulation
  • Electrical and Electronic Engineering

Cite this

Laplacian versus adjacency matrix in quantum walk search. / Wong, Thomas; Tarrataca, Luís; Nahimov, Nikolay.

In: Quantum Information Processing, Vol. 15, No. 10, 01.10.2016, p. 4029-4048.

Research output: Contribution to journalArticle

Wong, Thomas ; Tarrataca, Luís ; Nahimov, Nikolay. / Laplacian versus adjacency matrix in quantum walk search. In: Quantum Information Processing. 2016 ; Vol. 15, No. 10. pp. 4029-4048.
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