Equivalence of Szegedy’s and coined quantum walks

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Szegedy’s quantum walk is a quantization of a classical random walk or Markov chain, where the walk occurs on the edges of the bipartite double cover of the original graph. To search, one can simply quantize a Markov chain with absorbing vertices. Recently, Santos proposed two alternative search algorithms that instead utilize the sign-flip oracle in Grover’s algorithm rather than absorbing vertices. In this paper, we show that these two algorithms are exactly equivalent to two algorithms involving coined quantum walks, which are walks on the vertices of the original graph with an internal degree of freedom. The first scheme is equivalent to a coined quantum walk with one walk step per query of Grover’s oracle, and the second is equivalent to a coined quantum walk with two walk steps per query of Grover’s oracle. These equivalences lie outside the previously known equivalence of Szegedy’s quantum walk with absorbing vertices and the coined quantum walk with the negative identity operator as the coin for marked vertices, whose precise relationships we also investigate.

Original languageEnglish (US)
Article number215
JournalQuantum Information Processing
Volume16
Issue number9
DOIs
StatePublished - Sep 1 2017

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Quantum Walk
equivalence
apexes
Equivalence
Walk
Absorbing
Markov processes
Markov chains
Markov chain
Query
random walk
Flip
Graph in graph theory
degrees of freedom
Search Algorithm
Quantization
Random walk
operators
Degree of freedom
Cover

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

Equivalence of Szegedy’s and coined quantum walks. / Wong, Thomas.

In: Quantum Information Processing, Vol. 16, No. 9, 215, 01.09.2017.

Research output: Contribution to journalArticle

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