Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1605.02774 · Physical Review A

Staggered quantum walks with Hamiltonians

Renato Portugal, M. C. de Oliveira, Jalil Khatibi Moqadam

Published Jan 26, 2017• 25 citations
arXivHTMLPDF

Abstract

Quantum walks are recognizably useful for the development of new quantum algorithms, as well as for the investigation of several physical phenomena in quantum systems. Actual implementations of quantum walks face technological difficulties similar to the ones for quantum computers, though. Therefore, there is a strong motivation to develop new quantum-walk models which might be easier to implement. In this work we present an extension of the staggered quantum walk model that is fitted for physical implementations in terms of time-independent Hamiltonians. We demonstrate that this class of quantum walk includes the entire class of staggered quantum walk model, Szegedy's model, and an important subset of the coined model.

DOI
10.1103/physreva.95.012328

Related papers

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

The Staggered Quantum Walk Model

Renato Portugal, Raqueline A. M. Santos, T. D. Fern +1 more

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

•Jan 1, 2015•59 citations

Where to quantum walk

Viv M. Kendon

Quantum versions of random walks have diverse applications that are motivating experimental implementations as well as theoretical studies. However, the main impetus behind this interest is their use in quantum algorithms, which have always employed the quantum walk in the form of a program running on a quantum computer. Recent results showing that quantum walks are "universal for quantum computation" relate entirely to algorithms, and do not imply that a physical quantum walk could provide a new architecture for quantum computers. Nonetheless, quantum walks used to model transport phenomena in spin chains and biomolecules broaden their scope well beyond algorithms, and reopen the question of when a physical implementation might provide useful computational outputs. In this article we determine the conditions under which a physical quantum walk experiment could provide useful results beyond the reach of classical computation.

arXiv:1107.3795•Jul 19, 2011•4 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

Quantum search on the two-dimensional lattice using the staggered model with Hamiltonians

Renato Portugal, Tharso D. Fernandes

Quantum search on the two-dimensional lattice with one marked vertex and cyclic boundary conditions is an important problem in the context of quantum algorithms with an interesting unfolding. It avails to test the ability of quantum walk models to provide efficient algorithms from the theoretical side and means to implement quantum walks in laboratories from the practical side. In this paper, we rigorously prove that the recent-proposed staggered quantum walk model provides an efficient quantum search on the two-dimensional lattice, if the reflection operators associated with the graph tessellations are used as Hamiltonians, which is an important theoretical result for validating the staggered model with Hamiltonians. Numerical results show that on the two-dimensional lattice staggered models without Hamiltonians are not as efficient as the one described in this paper and are, in fact, as slow as classical random-walk-based algorithms.

arXiv:1701.01468•Physical Review A•Apr 26, 2017•19 citations

Efficient Simulation of Szegedy Quantum Walk Formulations and Algorithms

Sergio A. Ortega, Daniel K. Park

Quantum walks provide a versatile framework for quantum algorithms across a wide range of applications. We develop efficient classical simulation methods for Szegedy quantum walks that avoid explicit construction of the full unitary evolution operator. Unlike previous approaches restricted to a particular walk formulation, our framework is built from fundamental update and reflection operators, enabling the simulation of a broader class of Szegedy walk formulations. We further extend these methods to phase-estimation-based algorithms coupled to the walk, including implementations suitable for large sparse graphs. The resulting methods achieve optimal $O(N^2)$ complexity for dense graphs with $N$ nodes. For sparse graphs, the computational cost scales linearly with the number of edges, which is $O(N)$ in many cases. We implement the framework in the Python package SQWLib and illustrate its capabilities through simulations of representative algorithms, including quantum simulated annealing and quantum search on graphs. These results provide a practical tool for studying Szegedy-walk-based algorithms numerically beyond purely analytical treatments.

quant-pharXiv:2606.14226•Jun 15, 2026•0 citations
Quantum walks are recognizably useful for the development of new quantum algorithms, as well as for the investigation of several physical phenomena in quantum systems. Actual implementations of quantum walks face technological difficulties similar to the ones for quantum computers, though. Therefore, there is a strong motivation to develop new quantum-walk models which might be easier to implement. In this work we present an extension of the staggered quantum walk model that is fitted for physical implementations in terms of time-independent Hamiltonians. We demonstrate that this class of quantum walk includes the entire class of staggered quantum walk model, Szegedy's model, and an important subset of the coined model.
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.
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
Quantum versions of random walks have diverse applications that are motivating experimental implementations as well as theoretical studies. However, the main impetus behind this interest is their use in quantum algorithms, which have always employed the quantum walk in the form of a program running on a quantum computer. Recent results showing that quantum walks are "universal for quantum computation" relate entirely to algorithms, and do not imply that a physical quantum walk could provide a new architecture for quantum computers. Nonetheless, quantum walks used to model transport phenomena in spin chains and biomolecules broaden their scope well beyond algorithms, and reopen the question of when a physical implementation might provide useful computational outputs. In this article we determine the conditions under which a physical quantum walk experiment could provide useful results beyond the reach of classical computation.
Quantum search on the two-dimensional lattice with one marked vertex and cyclic boundary conditions is an important problem in the context of quantum algorithms with an interesting unfolding. It avails to test the ability of quantum walk models to provide efficient algorithms from the theoretical side and means to implement quantum walks in laboratories from the practical side. In this paper, we rigorously prove that the recent-proposed staggered quantum walk model provides an efficient quantum search on the two-dimensional lattice, if the reflection operators associated with the graph tessellations are used as Hamiltonians, which is an important theoretical result for validating the staggered model with Hamiltonians. Numerical results show that on the two-dimensional lattice staggered models without Hamiltonians are not as efficient as the one described in this paper and are, in fact, as slow as classical random-walk-based algorithms.
Quantum walks provide a versatile framework for quantum algorithms across a wide range of applications. We develop efficient classical simulation methods for Szegedy quantum walks that avoid explicit construction of the full unitary evolution operator. Unlike previous approaches restricted to a particular walk formulation, our framework is built from fundamental update and reflection operators, enabling the simulation of a broader class of Szegedy walk formulations. We further extend these methods to phase-estimation-based algorithms coupled to the walk, including implementations suitable for large sparse graphs. The resulting methods achieve optimal O(N2)O(N^2)O(N2) complexity for dense graphs with NNN nodes. For sparse graphs, the computational cost scales linearly with the number of edges, which is O(N)O(N)O(N) in many cases. We implement the framework in the Python package SQWLib and illustrate its capabilities through simulations of representative algorithms, including quantum simulated annealing and quantum search on graphs. These results provide a practical tool for studying Szegedy-walk-based algorithms numerically beyond purely analytical treatments.