Discrete-time quantum walk on the Cayley graph of the dihedral group

Dai, Wenjing; Yuan, Jian; Li, Dan

doi:10.1007/s11128-018-2101-9

Cited by 13 publications

(3 citation statements)

References 35 publications

(50 reference statements)

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“…Virtual abelianity allowed them to reduced the problem to an equivalent one on an abelian group with a larger chiral space dimension and use the Fourier method of [20]. More recently, DTCQW has been studied for the Dihedral group D n by [6] Dai et. al.…”

Section: Initial States and Evolutionmentioning

confidence: 99%

Discrete Quantum Walks on the Symmetric Group

Banerjee¹

2022

Preprint

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The theory of random walks on finite graphs is well developed with numerous applications. In quantum walks, the propagation is governed by quantum mechanical rules; generalizing random walks to the quantum setting. They have been successfully applied in the development of quantum algorithms. In particular, to solve problems that can be mapped to searching or property testing on some specific graph. In this paper we investigate the discrete time coined quantum walk (DTCQW) model using tools from non-commutative Fourier analysis. Specifically, we are interested in characterizing the DTCQW on Cayley graphs generated by the symmetric group (S n ) with appropriate generating sets. The lack of commutativity makes it challenging to find an analytical description of the limiting behavior with respect to the spectrum of the walk-operator. We determine certain characteristics of these walks using a path integral approach over the characters of S n .

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Section: Initial States and Evolutionmentioning

confidence: 99%

Discrete Quantum Walks on the Symmetric Group

Banerjee¹

2022

Preprint

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“…Alternate quantum walks, quantum walks with memory, quantum walks on different kinds of graphs are presented in Ref. [2,3,4,5,6] for different purposes. Quantum walk has wide applications in quantum computation and quantum communication, such as database searching, element distinctness, graph isomorphism testing, and quantum network communication.…”

Section: Introductionmentioning

confidence: 99%

Controlled Alternate Quantum Walk based Block Hash Function

Li¹,

Ding²,

Zhou³

et al. 2022

Preprint

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The hash function is an important branch of cryptology. Controlled quantum walk based hash function is a kind of novel hash function, which is safe, flexible, high-efficient, and compatible. All existing controlled quantum walk based hash functions are controlled by one bit message in each step. To process message in batch amounts, in this paper, controlled alternate quantum walk based block hash function is presented by using the time-positiondependent controlled quantum walks on complete graphs with self-loops. The presented hash function accelerate the hash processing dramatically, so it is more high-efficient.

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“…Random walks on groups play an essential role in various fields of natural science, ranging from solid-state physics, polymer chemistry, and biology to mathematics and computer science. Motivated by the immense success of random walk methods in the design of classical algorithms, we consider the DTQW on the Cayley graph of the dihedral group, previously considered by Dai et al [35]. Here, we will further study the three-state DTQW on the Cayley graph of the dihedral group.…”

Section: Introductionmentioning

confidence: 99%

Three-state quantum walk on the Cayley Graph of the Dihedral Group

Liu

Yuan

Dai

et al. 2020

Preprint

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The finite dihedral group generated by one rotation and one reflection is the simplest case of the non-abelian group. Cayley graphs are diagrammatic counterparts of groups. In this paper, much attention is given to the Cayley graph of the dihedral group. Considering the characteristics of the elements in the dihedral group, we propose a model of three-state discretetime quantum walk (DTQW) on the Caylay graph of the dihedral group with Grover coin. We derive analytic expressions for the the position probability distribution and the long-time limit of the return probability starting from the origin. It is shown that the localization effect is governed by the size of the underlying dihedral group, coin operator and initial state. We also numerically investigate the properties of the proposed model via the probability distribution and the time-averaged probability at the designated position. The abundant phenomena of three-state Grover DTQW on the Caylay graph of the dihedral group can help the community to better understand and to develop new quantum algorithms.

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Discrete-time quantum walk on the Cayley graph of the dihedral group

Cited by 13 publications

References 35 publications

Discrete Quantum Walks on the Symmetric Group

Discrete Quantum Walks on the Symmetric Group

Controlled Alternate Quantum Walk based Block Hash Function

Three-state quantum walk on the Cayley Graph of the Dihedral Group

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