scholarly journals Mixing of quantum walk on circulant bunkbeds

2006 ◽  
Vol 6 (4&5) ◽  
pp. 370-381
Author(s):  
P. Lo ◽  
S. Rajaram ◽  
D. Schepens ◽  
D. Sullivan ◽  
C. Tamon ◽  
...  
Keyword(s):  
Continuous Time ◽  
Cartesian Product ◽  
Quantum Walk ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Mixing Dynamics ◽  
Time Quantum ◽  
Explicit Analysis ◽  
Complete Multipartite ◽  
Standard Graph

This paper gives new observations on the mixing dynamics of a continuous-time quantum walk on circulants and their bunkbeds. These bunkbeds are defined through two standard graph operators: the join G + H and the Cartesian product G \cprod H of graphs G and H. Our results include the following: (i) The quantum walk is average uniform mixing on circulants with bounded eigenvalue multiplicity; this extends a known fact about the cycles C_{n}. (ii) Explicit analysis of the probability distribution of the quantum walk on the join of circulants; this explains why complete multipartite graphs are not average uniform mixing, using the fact K_{n} = K_{1} + K_{n-1} and K_{n,\ldots,n} = \overline{K}_{n} + \ldots + \overline{K}_{n}. (iii) The quantum walk on the Cartesian product of a $m$-vertex path P_{m} and a circulant G, namely, P_{m} \cprod G, is average uniform mixing if G is; this highlights a difference between circulants and the hypercubes Q_{n} = P_{2} \cprod Q_{n-1}. Our proofs employ purely elementary arguments based on the spectra of the graphs.

Universal Mixing of Quantum Walk on Graphs

2007 ◽  
Vol 7 (8) ◽  
pp. 738-751
Author(s):  
W. Carlson ◽  
A. Ford ◽  
E. Harris ◽  
J. Rosen ◽  
C. Tamon ◽  
...  
Keyword(s):  
Probability Distributions ◽  
Weighted Graph ◽  
Quantum Walk ◽  
Weighted Graphs ◽  
New Family ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Time Quantum ◽  
Average Probability ◽  
Complete Multipartite

We study the set of probability distributions visited by a continuous-time quantum walk on graphs. An edge-weighted graph $G$ is {\em universal mixing} if the instantaneous or average probability distribution of the quantum walk on $G$ ranges over all probability distributions on the vertices as the weights are varied over non-negative reals. The graph is {\em uniform} mixing if it visits the uniform distribution. Our results include the following: 1) All weighted complete multipartite graphs are instantaneous universal mixing. This is in contrast to the fact that no {\em unweighted} complete multipartite graphs are uniform mixing (except for the four-cycle $K_{2,2}$). 2) For all $n \ge 1$, the weighted claw $K_{1,n}$ is a minimally connected instantaneous universal mixing graph. In fact, as a corollary, the unweighted $K_{1,n}$ is instantaneous uniform mixing. This adds a new family of uniform mixing graphs to a list that so far contains only the hypercubes. 3) Any weighted graph is average almost-uniform mixing unless its spectral type is sublinear in the size of the graph. This provides a nearly tight characterization for average uniform mixing on circulant graphs. 4) No weighted graphs are average universal mixing. This shows that weights do not help to achieve average universal mixing, unlike the instantaneous case. Our proofs exploit the spectra of the underlying weighted graphs and path collapsing arguments.

On mixing in continuous-time quantum walks on some circulant graphs

2003 ◽  
Vol 3 (6) ◽  
pp. 611-618
Author(s):  
A. Ahmadi ◽  
R. Belk ◽  
C. Tamon ◽  
C. Wendler
Keyword(s):  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Complete Graphs ◽  
Circulant Graphs ◽  
Multipartite Graphs ◽  
Time Quantum ◽  
Complete Multipartite ◽  
Cycle Graph ◽  
Mixing Property

Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that the continuous-time quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous exactly uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs of the symmetric group lack this mixing property as well.

Optimal Wirelength of Balanced Complete Multipartite Graphs onto Cartesian Product of {Path, Cycle} and Trees

2021 ◽  
Vol 178 (3) ◽  
pp. 187-202
Author(s):  
Micheal Arockiaraj ◽  
J. Nancy Delaila ◽  
Jessie Abraham
Keyword(s):  
Interconnection Network ◽  
Cartesian Product ◽  
Graph Embedding ◽  
Complete Binary Tree ◽  
Deterministic Algorithms ◽  
Allocation Process ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Complete Multipartite ◽  
Processor Speed

In any interconnection network, task allocation plays a major role in the processor speed as fair distribution leads to enhanced performance. Complete multipartite networks serve well for this purpose as the task can be split into different partites which improves the degree of reliability of the network. Such an allocation process in the network can be done by means of graph embedding. The optimal wirelength of a graph embedding helps in the distribution of deterministic algorithms from the guest graph to other host graphs in order to incorporate its unique deterministic properties on that chosen graph. In this paper, we propose an algorithm to compute the optimal wirelength of balanced complete multipartite graphs onto the Cartesian product of trees with path and cycle. Moreover, we derive the closed formulae for wirelengths in specific trees like (1-rooted) complete binary tree and sibling graphs.

scholarly journals Marking Vertices to Find Graph Isomorphism Mapping Based on Continuous-Time Quantum Walk

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 586 ◽  
Author(s):  
Xin Wang ◽  
Yi Zhang ◽  
Kai Lu ◽  
Xiaoping Wang ◽  
Kai Liu
Keyword(s):  
Polynomial Time ◽  
Continuous Time ◽  
Graph Isomorphism ◽  
Quantum Walk ◽  
Isomorphism Problem ◽  
Quantum Walks ◽  
Regular Graphs ◽  
Structure Preserving ◽  
Topological Structures ◽  
Time Quantum

The isomorphism problem involves judging whether two graphs are topologically the same and producing structure-preserving isomorphism mapping. It is widely used in various areas. Diverse algorithms have been proposed to solve this problem in polynomial time, with the help of quantum walks. Some of these algorithms, however, fail to find the isomorphism mapping. Moreover, most algorithms have very limited performance on regular graphs which are generally difficult to deal with due to their symmetry. We propose IsoMarking to discover an isomorphism mapping effectively, based on the quantum walk which is sensitive to topological structures. Firstly, IsoMarking marks vertices so that it can reduce the harmful influence of symmetry. Secondly, IsoMarking can ascertain whether the current candidate bijection is consistent with existing bijections and eventually obtains qualified mapping. Thirdly, our experiments on 1585 pairs of graphs demonstrate that our algorithm performs significantly better on both ordinary graphs and regular graphs.

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