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 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.

scholarly journals Noise Characterization: Keeping Reduction Based Per-turbed Quantum Walk Search Optimal

2019 ◽  
Vol 198 ◽  
pp. 00001
Author(s):  
Chen-Fu Chiang ◽  
Chang-Yu Hsieh
Keyword(s):  
Invariant Subspace ◽  
Continuous Time ◽  
Quantum Walk ◽  
Coupling Factor ◽  
Necessary Condition ◽  
Subspace Method ◽  
Noise Distribution ◽  
Noisy Environment ◽  
Time Quantum ◽  
Noise Characterization

In a recent work by Novo et al. (Sci. Rep. 5, 13304, 2015), the invariant subspace method was applied to the study of continuous-time quantum walk (CTQW). In this work, we adopt the aforementioned method to investigate the optimality of a perturbed quantum walk search of a marked element in a noisy environment on various graphs. We formulate the necessary condition of the noise distribution in the system such that the invariant subspace method remains effective and efficient. Based on the noise, we further formulate how to set the appropriate coupling factor to preserve the optimality of the quantum walker.

scholarly journals CONTINUOUS-TIME QUANTUM WALKS ON TREES IN QUANTUM PROBABILITY THEORY

2006 ◽  
Vol 09 (02) ◽  
pp. 287-297 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Probability Theory ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Probability ◽  
Quantum Walks ◽  
Homogeneous Tree ◽  
One Dimensional ◽  
Time Quantum ◽  
New Type ◽  
The One

A quantum central limit theorem for a continuous-time quantum walk on a homogeneous tree is derived from quantum probability theory. As a consequence, a new type of limit theorems for another continuous-time walk introduced by the walk is presented. The limit density is similar to that given by a continuous-time quantum walk on the one-dimensional lattice.

scholarly journals NON-UNIFORM MIXING OF QUANTUM WALK ON CYCLES

2007 ◽  
Vol 05 (06) ◽  
pp. 781-793 ◽  
Author(s):  
WILLIAM ADAMCZAK ◽  
KEVIN ANDREW ◽  
LEON BERGEN ◽  
DILLON ETHIER ◽  
PETER HERNBERG ◽  
...  
Keyword(s):  
Random Walk ◽  
Uniform Distribution ◽  
Continuous Time ◽  
Quantum Walk ◽  
Complete Graphs ◽  
Circulant Graph ◽  
Mixing Properties ◽  
Time Quantum ◽  
Average Distribution ◽  
Instantaneous Distribution

A classical lazy random walk on cycles is known to mix with the uniform distribution. In contrast, we show that a continuous-time quantum walk on cycles exhibits strong non-uniform mixing properties. First, we prove that the instantaneous distribution of a quantum walk on most even-length cycles is never uniform. More specifically, we prove that a quantum walk on a cycle Cnis not instantaneous uniform mixing, whenever n satisfies either: (a) n = 2u, for u ≥ 3; or (b) n = 2uq, for u ≥ 1 and q ≡ 3 (mod 4). Second, we prove that the average distribution of a quantum walk on any Abelian circulant graph is never uniform. As a corollary, the average distribution of a quantum walk on any standard circulant graph, such as the cycles, complete graphs, and even hypercubes, is never uniform. Nevertheless, we show that the average distribution of a quantum walk on the cycle Cnis O(1/n)-uniform.

Finding tree symmetries using continuous-time quantum walk

2013 ◽  
Vol 22 (5) ◽  
pp. 050304 ◽  
Author(s):  
Jun-Jie Wu ◽  
Bai-Da Zhang ◽  
Yu-Hua Tang ◽  
Xiao-Gang Qiang ◽  
Hui-Quan Wang
Keyword(s):  
Continuous Time ◽  
Quantum Walk ◽  
Time Quantum

scholarly journals QUANTUM TIMING AND SYNCHRONIZATION PROBLEMS

2004 ◽  
Vol 18 (04n05) ◽  
pp. 623-631 ◽  
Author(s):  
DIEGO DE FALCO ◽  
DARIO TAMASCELLI
Keyword(s):  
Continuous Time ◽  
Quantum Computer ◽  
Linear Chain ◽  
Quantum Walk ◽  
Storage Space ◽  
Input Output ◽  
Output Register ◽  
Additional Amount ◽  
Time Quantum ◽  
Synchronization Problems

Feynman's model of a quantum computer provides an example of a continuous-time quantum walk. Its clocking mechanism is an excitation of a basically linear chain of spins with occasional controlled jumps which allow for motion on a planar graph. The spreading of the wave packet poses limitations on the probability of ever completing the s elementary steps of a computation: an additional amount of storage space δ is needed in order to achieve an assigned completion probability. In this note we study the END instruction, viewed as a measurement of the position of the clocking excitation: a π-pulse indefinitely freezes the contents of the input/output register, with a probability depending only on the ratio δ/s.

scholarly journals Complexity results on graphs with few cliques

2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Bill Rosgen ◽  
Lorna Stewart
Keyword(s):  
Maximum Clique ◽  
Maximum Clique Problem ◽  
Polynomial Time Algorithms ◽  
Helly Property ◽  
Graph Class ◽  
International Audience ◽  
Edge Clique Cover ◽  
Np Complete ◽  
General Graphs ◽  
Clique Problem

Graphs and Algorithms International audience A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs, some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms. Other problems, such as the vertex clique cover and edge clique cover problems remain NP-complete on these classes. Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a given polynomial bound.

scholarly journals Effects of initial states on continuous-time quantum walk in the optical waveguide array

2013 ◽  
Vol 62 (9) ◽  
pp. 090301
Author(s):  
Ren Chun-Nian ◽  
Shi Peng ◽  
Liu Kai ◽  
Li Wen-Dong ◽  
Zhao Jie ◽  
...  
Keyword(s):  
Optical Waveguide ◽  
Continuous Time ◽  
Quantum Walk ◽  
Waveguide Array ◽  
Time Quantum ◽  
Initial States
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