TY - JOUR

T1 - Equivalence of Szegedy’s and coined quantum walks

AU - Wong, Thomas G.

N1 - Funding Information:
Thanks to Peter H?yer for pointing out Fact 3.2 of [18]. This work was supported by the U.S.?Department of Defense Vannevar Bush Faculty Fellowship of Scott Aaronson.
Publisher Copyright:
© 2017, Springer Science+Business Media, LLC.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2017/9/1

Y1 - 2017/9/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s11128-017-1667-y

DO - 10.1007/s11128-017-1667-y

M3 - Article

AN - SCOPUS:85025704739

VL - 16

JO - Quantum Information Processing

JF - Quantum Information Processing

SN - 1570-0755

IS - 9

M1 - 215

ER -