Equivalence of Szegedy's and Coined Quantum Walks

TL;DR

This paper establishes the exact equivalence between Szegedy's quantum walks and coined quantum walks, including new insights into their relationships and alternative search algorithms using Grover's oracle.

Contribution

The paper proves the precise equivalence between Szegedy's and coined quantum walks, extending understanding of their relationships and connecting alternative search algorithms to these models.

Findings

Szegedy's quantum walk is equivalent to a specific coined quantum walk.

Two alternative search algorithms are shown to be equivalent to coined quantum walks with different step counts.

The relationships between Szegedy's walk, coined walks, and previous models are clarified.

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.