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The Staggered Quantum Walk Model

Renato Portugal, Raqueline A. M. Santos, T. D. Fern, D. N. Gonçalves

Published Jan 1, 2015• 59 citations
OpenAlex

Abstract

There are at least three models of discrete-time quantum walks (QWs) on graphs currently under active development. In this work we focus on the equivalence of two of them, known as Szegedy’s and staggered QWs. We give a formal definition of the staggered model and discuss generalized versions for searching marked vertices. Using this formal definition, we prove that any instance of Szegedy’s model is equivalent to an instance of the staggered model. On the other hand, we show that there are instances of the staggered model that cannot be cast into Szegedy’s framework. Our analysis also works when there are marked vertices. We show that Szegedy’s spatial search algorithms can be converted into search algorithms in staggered QWs. We take advantage of the similarity of those models to define the quantum hitting time in the staggered model and to describe a method to calculate the eigenvalues and eigenvectors of the evolution operator of staggered QWs. 1

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Staggered quantum walks on graphs

Renato Portugal

The staggered quantum-walk model makes it possible to establish an unprecedented connection between discrete-time quantum walks and graph theory. We call attention to the fact that a large subclass of the coined model is included in Szegedy's model, which in its turn is entirely included in the staggered model. In order to compare those three quantum-walk models, we put them in the staggered formalism and show that the Szegedy and coined models are defined on a special subclass of graphs. This inclusion scheme is also true when the searching framework is added. We use graph theory to characterize which staggered quantum walks can be reduced to the Szegedy or coined quantum-walk model. We analyze a staggered-based search that cannot be included in Szegedy's model and show numerically that this search is more efficient than a random-walk-based search.

arXiv:1603.02210•Physical Review A•Jun 27, 2016•44 citations

Staggered quantum walk on hexagonal lattices

Bruno Chagas, Renato Portugal, Stefan Boettcher +1 more

A discrete-time staggered quantum walk was recently introduced as a generalization that allows the unification of other versions, such as the coined and Szegedy's walk. However, it also produces forms of quantum walks not covered by previous versions. To explore their properties, we study here the staggered walk on a hexagonal lattice. Such a walk is defined using a set of overlapping tessellations that cover the graph edges, and each tessellation is a partition of the node set into cliques. The hexagonal lattice requires at least three tessellations. Each tessellation is associated with a local unitary operator and the product of the local operators defines the evolution operator of the staggered walk on the graph. After defining the evolution operator on the hexagonal lattice, we analyze the quantum walk dynamics with the focus on the position standard deviation and localization. We also obtain analytic results for the time complexity of spatial search algorithms with one marked node using cyclic boundary conditions.

arXiv:1806.10249•Physical Review A•Nov 9, 2018•5 citations

Spatial search on Johnson graphs by discrete-time quantum walk

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Abstract The spatial search problem aims to find a marked vertex of a finite graph using a dynamic with two constraints: (1) the walker has no compass and (2) the walker can check whether a vertex is marked only after reaching it. This problem is a generalization of unsorted database search and has many applications to algorithms. Classical algorithms that solve the spatial search problem are based on random walks and the computational complexity is determined by the hitting time. On the other hand, quantum algorithms are based on quantum walks and the computational complexity is determined not only by the number of steps to reach a marked vertex, but also by the success probability, since we need to perform a measurement at the end of the algorithm to determine the walker’s position. In this work, we address the spatial search problem on Johnson graphs using the coined quantum walk model. Since Johnson graphs are vertex- and distance-transitive, we have found an invariant subspace of the Hilbert space, which aids in the calculation of the computational complexity. We have shown that, for every fixed diameter, the asymptotic success probability is 1/2 after taking π N / ( 2 2 ) steps, where N is the number of vertices of the Johnson graph.

arXiv:2112.03744•Journal of Physics A Mathematical and Theoretical•May 12, 2022•9 citations

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Search algorithms based on discrete-time quantum walk (QW) can be considered as alterations of the standard QW: Use a different coin operator that distinguishes target and nontarget vertices, or, mark the target vertices first followed by the standard QW. Two most frequently used marking coins are [Formula: see text] and [Formula: see text] the negative identity operator and the negative Grover diffusion operator. We show that search algorithms corresponding to these four combinations can be reduced to two, denoted as [Formula: see text] and [Formula: see text], and they are equivalent when searching for nonadjacent multiple targets. For adjacent target vertices, numerical simulations show that the performance of the algorithm [Formula: see text] highly depends on the density of the underlying graph, and it outperforms [Formula: see text] when the density is large enough. At last, a generalized stationary state of both search algorithms on the graphs with even-numbered degree is provided.

International Journal of Quantum Information•Feb 3, 2023•0 citations

The spectra of the unitary marix of a 2-tessellable staggered quantum walk on a graph

Norio Konno, Iwao Sato, Etsuo Segawa

Recently, the staggered quantum walk (SQW) on a graph is discussed as a generalization of coined quantum walks on graphs and Szegedy walks. We present a formula for the time evolution matrix of a 2-tessellable SQW on a graph, and so directly give its spectra. Furthermore, we present a formula for the Szegedy matrix of a bipartite graph by the same method, and so give its spectra. As an application, we present a formula for the characteristic polynomial of the modified Szegedy matrix in the quantum search problem on a graph, and give its spectra.

arXiv:1701.08274•Jan 28, 2017•0 citations

Connecting Coined Quantum Walks with Szegedy's Model

Renato Portugal, Etsuo Segawa

We analyze the equivalence between discrete-time coined quantum walks and Szegedy's quantum walks. We characterize a class of flip-flop coined models with generalized Grover coin on a graph Γ that can be directly converted into Szegedy's model on the subdivision graph of Γ and we describe a method to convert one model into the other. This method improves previous results in literature that need to use the staggered model and the concept of line graph, which are avoided here.

arXiv:1612.02448•Interdisciplinary Information Sciences•Jan 1, 2017•9 citations
There are at least three models of discrete-time quantum walks (QWs) on graphs currently under active development. In this work we focus on the equivalence of two of them, known as Szegedy’s and staggered QWs. We give a formal definition of the staggered model and discuss generalized versions for searching marked vertices. Using this formal definition, we prove that any instance of Szegedy’s model is equivalent to an instance of the staggered model. On the other hand, we show that there are instances of the staggered model that cannot be cast into Szegedy’s framework. Our analysis also works when there are marked vertices. We show that Szegedy’s spatial search algorithms can be converted into search algorithms in staggered QWs. We take advantage of the similarity of those models to define the quantum hitting time in the staggered model and to describe a method to calculate the eigenvalues and eigenvectors of the evolution operator of staggered QWs. 1
The staggered quantum-walk model makes it possible to establish an unprecedented connection between discrete-time quantum walks and graph theory. We call attention to the fact that a large subclass of the coined model is included in Szegedy's model, which in its turn is entirely included in the staggered model. In order to compare those three quantum-walk models, we put them in the staggered formalism and show that the Szegedy and coined models are defined on a special subclass of graphs. This inclusion scheme is also true when the searching framework is added. We use graph theory to characterize which staggered quantum walks can be reduced to the Szegedy or coined quantum-walk model. We analyze a staggered-based search that cannot be included in Szegedy's model and show numerically that this search is more efficient than a random-walk-based search.
A discrete-time staggered quantum walk was recently introduced as a generalization that allows the unification of other versions, such as the coined and Szegedy's walk. However, it also produces forms of quantum walks not covered by previous versions. To explore their properties, we study here the staggered walk on a hexagonal lattice. Such a walk is defined using a set of overlapping tessellations that cover the graph edges, and each tessellation is a partition of the node set into cliques. The hexagonal lattice requires at least three tessellations. Each tessellation is associated with a local unitary operator and the product of the local operators defines the evolution operator of the staggered walk on the graph. After defining the evolution operator on the hexagonal lattice, we analyze the quantum walk dynamics with the focus on the position standard deviation and localization. We also obtain analytic results for the time complexity of spatial search algorithms with one marked node using cyclic boundary conditions.
Abstract The spatial search problem aims to find a marked vertex of a finite graph using a dynamic with two constraints: (1) the walker has no compass and (2) the walker can check whether a vertex is marked only after reaching it. This problem is a generalization of unsorted database search and has many applications to algorithms. Classical algorithms that solve the spatial search problem are based on random walks and the computational complexity is determined by the hitting time. On the other hand, quantum algorithms are based on quantum walks and the computational complexity is determined not only by the number of steps to reach a marked vertex, but also by the success probability, since we need to perform a measurement at the end of the algorithm to determine the walker’s position. In this work, we address the spatial search problem on Johnson graphs using the coined quantum walk model. Since Johnson graphs are vertex- and distance-transitive, we have found an invariant subspace of the Hilbert space, which aids in the calculation of the computational complexity. We have shown that, for every fixed diameter, the asymptotic success probability is 1/2 after taking π N / ( 2 2 ) steps, where N is the number of vertices of the Johnson graph.
Search algorithms based on discrete-time quantum walk (QW) can be considered as alterations of the standard QW: Use a different coin operator that distinguishes target and nontarget vertices, or, mark the target vertices first followed by the standard QW. Two most frequently used marking coins are [Formula: see text] and [Formula: see text] the negative identity operator and the negative Grover diffusion operator. We show that search algorithms corresponding to these four combinations can be reduced to two, denoted as [Formula: see text] and [Formula: see text], and they are equivalent when searching for nonadjacent multiple targets. For adjacent target vertices, numerical simulations show that the performance of the algorithm [Formula: see text] highly depends on the density of the underlying graph, and it outperforms [Formula: see text] when the density is large enough. At last, a generalized stationary state of both search algorithms on the graphs with even-numbered degree is provided.
Recently, the staggered quantum walk (SQW) on a graph is discussed as a generalization of coined quantum walks on graphs and Szegedy walks. We present a formula for the time evolution matrix of a 2-tessellable SQW on a graph, and so directly give its spectra. Furthermore, we present a formula for the Szegedy matrix of a bipartite graph by the same method, and so give its spectra. As an application, we present a formula for the characteristic polynomial of the modified Szegedy matrix in the quantum search problem on a graph, and give its spectra.