Algorithms for finding the maximum clique based on continuous time quantum walks

2021 ◽  
Vol 21 (1&2) ◽  
pp. 0059-0079
Author(s):  
Xi Li ◽  
Mingyou Wu ◽  
Hanwu Chen ◽  
Zhibao Liu
Keyword(s):  
Random Graphs ◽  
Continuous Time ◽  
Time Complexity ◽  
Maximum Clique ◽  
Quantum Walks ◽  
Recursive Algorithms ◽  
Time Quantum

In this work, the application of continuous time quantum walks (CTQW) to the Maximum Clique (MC) problem was studied. Performing CTQW on graphs can generate distinct periodic probability amplitudes for different vertices. We found that the intensities of the probability amplitudes at some frequencies imply the clique structure of special kinds of graphs. Recursive algorithms with time complexity O(N^6) in classical computers were proposed to determine the maximum clique. We have experimented on random graphs where each edge exists with different probabilities. Although counter examples were not found for random graphs, whether these algorithms are universal is beyond the scope of this work.

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.

Correlation between the continuous-time quantum walk and cliques in graphs and its application

2021 ◽  
pp. 2150009
Author(s):  
Mingyou Wu ◽  
Xi Li ◽  
Zhihao Liu ◽  
Hanwu Chen
Keyword(s):  
Continuous Time ◽  
Maximum Clique ◽  
Quantum Walk ◽  
Approximate Algorithm ◽  
Maximum Clique Problem ◽  
High Approximation ◽  
Transmission Characteristics ◽  
Original Algorithm ◽  
Time Quantum ◽  
General Graphs

The continuous-time quantum walk (CTQW) provides a new approach to problems in graph theory. In this paper, the correlation between the CTQW and cliques in graphs is studied, and an approximate algorithm for the maximum clique problem (MCP) based on the CTQW is given. Via both numerical and theoretical analyses, it is found that the maximum clique is related to the transmission characteristics of the CTQW on some special graphs. For general graphs, the correlation is difficult to describe analytically. Therefore, the transmission characteristics of the CTQW are applied as a vertex selection criterion to a classical MCP algorithm and it is compared with the original algorithm. Numerous simulation on general graphs shows that the new algorithm is more efficient. Furthermore, an approximate MCP algorithm based on the CTQW is introduced, which only requires a very small number of searches with a high approximation ratio.

scholarly journals Continuous-time quantum walks on spatially correlated noisy lattices

2017 ◽  
Vol 96 (4) ◽  
Author(s):  
Matteo A. C. Rossi ◽  
Claudia Benedetti ◽  
Massimo Borrelli ◽  
Sabrina Maniscalco ◽  
Matteo G. A. Paris
Keyword(s):  
Continuous Time ◽  
Quantum Walks ◽  
Spatially Correlated ◽  
Time Quantum

scholarly journals Transport properties of continuous-time quantum walks on Sierpinski fractals

2014 ◽  
Vol 90 (3) ◽  
Author(s):  
Zoltán Darázs ◽  
Anastasiia Anishchenko ◽  
Tamás Kiss ◽  
Alexander Blumen ◽  
Oliver Mülken
Keyword(s):  
Transport Properties ◽  
Continuous Time ◽  
Quantum Walks ◽  
Time Quantum

scholarly journals PERFECT STATE TRANSFER, GRAPH PRODUCTS AND EQUITABLE PARTITIONS

2011 ◽  
Vol 09 (03) ◽  
pp. 823-842 ◽  
Author(s):  
YANG GE ◽  
BENJAMIN GREENBERG ◽  
OSCAR PEREZ ◽  
CHRISTINO TAMON
Keyword(s):  
Continuous Time ◽  
Quantum Walks ◽  
Perfect State ◽  
Graph Products ◽  
State Transfer ◽  
Time Quantum ◽  
Perfect State Transfer

We describe new constructions of graphs which exhibit perfect state transfer on continuous-time quantum walks. Our constructions are based on generalizations of the double cones and variants of the Cartesian graph products (which include the hypercube). We also describe a generalization of the path collapsing argument (which reduces questions about perfect state transfer to simpler weighted multigraphs) for graphs with equitable distance partitions.

Sign in / Sign up

Export Citation Format

Share Document