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International Journal of Quantum Information

The continuous-time quantum walk on some graphs based on the view of quantum probability

Qi Han, Yaxin Kou, Ning Bai, Huan Wang

Published Apr 29, 2022• 6 citations
Publisher

Abstract

In this paper, continuous-time quantum walk is discussed based on the view of quantum probability, i.e. the quantum decomposition of the adjacency matrix A of graph. Regard adjacency matrix A as Hamiltonian which is a real symmetric matrix with elements 0 or 1, so we regard [Formula: see text] as an unbiased evolution operator, which is related to the calculation of probability amplitude. Combining the quantum decomposition and spectral distribution [Formula: see text] of adjacency matrix A, we calculate the probability amplitude reaching each stratum in continuous-time quantum walk on complete bipartite graphs, finite two-dimensional lattices, binary tree, [Formula: see text]-ary tree and [Formula: see text]-fold star power [Formula: see text]. Of course, this method is also suitable for studying some other graphs, such as growing graphs, hypercube graphs and so on, in addition, the applicability of this method is also explained.

DOI
10.1142/s0219749922500150

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In this paper, continuous-time quantum walk is discussed based on the view of quantum probability, i.e. the quantum decomposition of the adjacency matrix A of graph. Regard adjacency matrix A as Hamiltonian which is a real symmetric matrix with elements 0 or 1, so we regard [Formula: see text] as an unbiased evolution operator, which is related to the calculation of probability amplitude. Combining the quantum decomposition and spectral distribution [Formula: see text] of adjacency matrix A, we calculate the probability amplitude reaching each stratum in continuous-time quantum walk on complete bipartite graphs, finite two-dimensional lattices, binary tree, [Formula: see text]-ary tree and [Formula: see text]-fold star power [Formula: see text]. Of course, this method is also suitable for studying some other graphs, such as growing graphs, hypercube graphs and so on, in addition, the applicability of this method is also explained.
In this paper, continuous-time quantum walk on hypercube is discussed in view of Cartesian product structure. We find that the [Formula: see text]-fold Cartesian power of the complete graph [Formula: see text] is the [Formula: see text]-dimensional hypercube, which give us new ideas for the study of quantum walk on hypercube. Combining the product structure, the spectral distribution of the graph and the quantum decomposition of the adjacency matrix, the probability amplitudes of the continuous-time quantum walker’s position at time [Formula: see text] are given, and it is discussed that the probability distribution for the continuous-time case is uniform when [Formula: see text]. The application of this product structure greatly improves the study of quantum walk on complex graphs, which has far-reaching influence and great significance.
In the present paper, we study the continuous-time quantum walk on quotient graphs. On such graphs, there is a straightforward reduction of the problem to a subspace that can be considerably smaller than the original one. Along the lines of reductions, by using the idea of calculation of the probability amplitudes for continuous-time quantum walk in terms of the spectral distribution associated with the adjacency matrix of graphs [Jafarizadeh and Salimi (Ann. Phys.322 (2007))], we show that the continuous-time quantum walk on original graph Γ induces a continuous-time quantum walk on quotient graph ΓH. Finally, for example, we investigate the continuous-time quantum walk on some quotient Cayley graphs.
This paper investigates continuous-time quantum walks on directed bipartite graphs based on a graph's adjacency matrix. We prove that on bipartite graphs, probability transport between the two node partitions can be completely suppressed by tuning a model parameter $\ensuremath{\alpha}$. We provide analytic solutions to the quantum walks for the star and circulant graph classes that are valid for an arbitrary value of the number of nodes NNN, time ttt, and the model parameter $\ensuremath{\alpha}$. We discuss quantitative and qualitative aspects of quantum walks based on directed graphs and their undirected counterparts. Numerical simulations of quantum walks on circulant graphs show complex interference phenomena and how complete suppression of transport is achieved near $\ensuremath{\alpha}=\ensuremath{\pi}/2$. By proving two mirror symmetries around $\ensuremath{\alpha}=0$ and $\ensuremath{\pi}/2$ we show that these quantum walks have a period of $\ensuremath{\pi}$ in $\ensuremath{\alpha}$. We show that undirected edges lose their effect on the quantum walk at $\ensuremath{\alpha}=\ensuremath{\pi}/2$ and present non-bipartite graphs that exhibit suppression of transport. Finally, we analytically compute the Hamiltonians of quantum walks on the directed ring graph.
Abstract. The adjacency matrix of a graph [Formula: see text] is the Hamiltonian for a continuous-time quantum walk on the vertices of [Formula: see text]. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the behavior of the walk is typically not periodic. In this paper, we develop a theory to exactly study any quantum walk generated by an integral Hamiltonian, and we put emphasis on those with irrational eigenvalues—what we call irrational quantum walks. As a result, we provide exact methods to compute the average of the mixing matrices, and to decide whether pretty good (or almost perfect) state transfer occurs in a given graph. We also use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix and discuss possible applications of these results. Throughout the paper, we emphasize the interplay between different fields of mathematics applied to the study of quantum walks.
Let [Formula: see text] be a nonnegative integer and [Formula: see text] the power set of the set [Formula: see text]. Then there is an adjacency relation in [Formula: see text] such that [Formula: see text] together with the relation forms a regular graph. In this paper, we propose a model of continuous-time magnetic quantum walk (MQW) on the graph [Formula: see text], and investigate its properties from a viewpoint of probability and quantum information. We first introduce a magnetic Laplacian [Formula: see text] on the graph [Formula: see text] and examine its spectrum. And then, with [Formula: see text] as the Hamiltonian, we construct our model of continuous-time MQW on the graph [Formula: see text]. We find that the model has probability distributions that are completely independent of the magnetic potential at all times. And we show that it has perfect state transfer at time [Formula: see text] when the magnetic potential satisfies some mild conditions. Some other interesting results are also obtained.