Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThree-state quantum walk on the Cayley Graph of the Dihedral Group
88
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 discrete-time 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.
rate research
Read More
The finite dihedral group generated by one rotation and one flip 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 conduct the model of discrete-time quantum walk on the Cayley graph of the dihedral group by special coding mode. This construction makes Fourier transformation can be used to carry out spectral analysis of the dihedral quantum walk, i.e. the non-abelian case. Furthermore, the relation between quantum walk without memory on the Cayley graph of the dihedral group and quantum walk with memory on a cycle is discussed, so that we can explore the potential of quantum walks without and with memory. Here, the numerical simulation is carried out to verify the theoretical analysis results and other properties of the proposed model are further studied.
Dukes (2014) and Konno, Shimizu, and Takei (2017) studied the periodicity for 2-state quantum walks whose coin operator is the Hadamard matrix on cycle graph C_N with N vertices. The present paper treats the periodicity for 3-state quantum walks on C_N. Our results follow from a new method based on cyclotomic field. This method shows a necessary condition for the coin operator of quantum walks to have the finite period. Moreover, we reveal the period T_N of two kinds of typical quantum walks, the Grover and Fourier walks. We prove that both walks do not have any finite period except for N=3, in which case T_3=6 (Grover), =12 (Fourier).
A microscopic derivation of an open quantum walk on a two node graph is presented. It is shown that for the considered microscopic model of the system-bath interaction the resulting quantum master equation takes the form of a generalized master equation. The explicit form of the quantum coin operators is derived. The formalism is demonstrated for the example of a two-level system walking on a two-node graph.
We investigate the behavior of coherence in scattering quantum walk search on complete graph under the condition that the total number of vertices of the graph is greatly larger than the marked number of vertices we are searching, $N gg v$. We find that the consumption of coherence represents the increase of the success probability for the searching,also the consumption of coherence is related to the efficiency of the algorithm represented by oracle queries.If no coherence is consumed, the efficiency of the algorithm will be the same as the classical blind search, implying that coherence is responsible for the speed up in this quantum algorithm over its classical counterpart. In case the initial state is incoherent, still $N gg v$ is assumed,the probability of success for searching will not change with time, indicating that this quantum search algorithm loses its power.We then conclude that the coherence plays an essential role and is responsible for the speed up in this quantum algorithm.
Quantum Stochastic Walks (QSW) allow for a generalization of both quantum and classical random walks by describing the dynamic evolution of an open quantum system on a network, with nodes corresponding to quantum states of a fixed basis. We consider the problem of quantum state discrimination on such a system, and we solve it by optimizing the network topology weights. Finally, we test it on different quantum network topologies and compare it with optimal theoretical bounds.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
