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.

Physically-motivated dynamical algorithms for the graph isomorphism problem

2005 ◽  
Vol 5 (6) ◽  
pp. 492-506
Author(s):  
S.-Y. Shiau ◽  
R. Joynt ◽  
S.N. Coppersmith
Keyword(s):  
Polynomial Time ◽  
Graph Isomorphism ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Isomorphism Problem ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Original Graph ◽  
Graph Isomorphism Problem ◽  
Strongly Regular

The graph isomorphism problem (GI) plays a central role in the theory of computational complexity and has importance in physics and chemistry as well \cite{kobler93,fortin96}. No polynomial-time algorithm for solving GI is known. We investigate classical and quantum physics-based polynomial-time algorithms for solving the graph isomorphism problem in which the graph structure is reflected in the behavior of a dynamical system. We show that a classical dynamical algorithm proposed by Gudkov and Nussinov \cite{gudkov02} as well as its simplest quantum generalization fail to distinguish pairs of non-isomorphic strongly regular graphs. However, by combining the algorithm of Gudkov and Nussinov with a construction proposed by Rudolph \cite{rudolph02} in which one examines a graph describing the dynamics of two particles on the original graph, we find an algorithm that successfully distinguishes all pairs of non-isomorphic strongly regular graphs that we tested with up to 29 vertices.

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.

Accelerating Stochastic Dynamics Simulation With Continuous-Time Quantum Walks

2016 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Path Integral ◽  
Continuous Time ◽  
Stochastic Dynamics ◽  
Diffusion Processes ◽  
Dimensional Space ◽  
Planck Equation ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Dynamics Simulation ◽  
Time Quantum

Stochastic diffusion is a general phenomenon observed in various national and engineering systems. It is typically modeled by either stochastic differential equation (SDE) or Fokker-Planck equation (FPE), which are equivalent approaches. Path integral is an accurate and effective method to solve FPEs. Yet, computational efficiency is the common challenge for path integral and other numerical methods, include time and space complexities. Previously, one-dimensional continuous-time quantum walk was used to simulate diffusion. By combining quantum diffusion and random diffusion, the new approach can accelerate the simulation with longer time steps than those in path integral. It was demonstrated that simulation can be dozens or even hundreds of times faster. In this paper, a new generic quantum operator is proposed to simulate drift-diffusion processes in high-dimensional space, which combines quantum walks on graphs with traditional path integral approaches. Probability amplitudes are computed efficiently by spectral analysis. The efficiency of the new method is demonstrated with stochastic resonance problems.

scholarly journals Unstructured search by random and quantum walk

2022 ◽  
Vol 22 (1&2) ◽  
pp. 53-85
Author(s):  
Thomas G. Wong
Keyword(s):  
Random Walks ◽  
Continuous Time ◽  
Quantum Computer ◽  
Success Probability ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Search Problem ◽  
Large N ◽  
Time Quantum ◽  
Classical Computer

The task of finding an entry in an unsorted list of $N$ elements famously takes $O(N)$ queries to an oracle for a classical computer and $O(\sqrt{N})$ queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large $N$, the random walks converge to the same evolution, both taking $N \ln(1/\epsilon)$ time to reach a success probability of $1-\epsilon$. In contrast, the discrete-time quantum walk asymptotically takes $\pi\sqrt{N}/2\sqrt{2}$ timesteps to reach a success probability of $1/2$, while the continuous-time quantum walk takes $\pi\sqrt{N}/2$ time to reach a success probability of $1$.

Mixing and decoherence in quantum walks on cycles

2006 ◽  
Vol 6 (3) ◽  
pp. 263-276 ◽  
Author(s):  
L. Fedichkin ◽  
D. Solenov ◽  
C. Tamon
Keyword(s):  
Continuous Time ◽  
Continuous Monitoring ◽  
Mixing Time ◽  
Numerical Data ◽  
Quantum Walk ◽  
Physical Review ◽  
Quantum Walks ◽  
Mixing Times ◽  
Analytical Treatment ◽  
Time Quantum

We prove analytical results showing that decoherence can be useful for mixing time in a continuous-time quantum walk on finite cycles. This complements the numerical observations by Kendon and Tregenna (Physical Review A 67 (2003), 042315) of a similar phenomenon for discrete-time quantum walks. Our analytical treatment of continuous-time quantum walks includes a continuous monitoring of all vertices that induces the decoherence process. We identify the dynamics of the probability distribution and observe how mixing times undergo the transition from quantum to classical behavior as our decoherence parameter grows from zero to infinity. Our results show that, for small rates of decoherence, the mixing time improves linearly with decoherence, whereas for large rates of decoherence, the mixing time deteriorates linearly towards the classical limit. In the middle region of decoherence rates, our numerical data confirms the existence of a unique optimal rate for which the mixing time is minimized.

scholarly journals STUDY OF CONTINUOUS-TIME QUANTUM WALKS ON QUOTIENT GRAPHS VIA QUANTUM PROBABILITY THEORY

2008 ◽  
Vol 06 (04) ◽  
pp. 945-957 ◽  
Author(s):  
S. SALIMI
Keyword(s):  
Probability Theory ◽  
Adjacency Matrix ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Probability ◽  
Quantum Walks ◽  
Original Graph ◽  
Quotient Graph ◽  
Quotient Graphs ◽  
Time Quantum

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.

scholarly journals General methods and properties to evaluate continuum limits of the 1D discrete time quantum walk

2020 ◽  
Vol 19 (10) ◽  
Author(s):  
Michael Manighalam ◽  
Mark Kon
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Systems ◽  
Quantum Walks ◽  
Time Limit ◽  
Continuous Limit ◽  
Continuous Space ◽  
Time Quantum ◽  
Space Limit

Abstract Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and nonrelativistic regimes (Molfetta GD, Arrighi P. A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in Molfetta and Arrighi (A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions (“coins”) admit nontrivial continuum limits. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. Finally, we demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk when the coin is allowed to transition through the continuous limit process.

scholarly journals Detecting factorization effect in continuous-time quantum walks

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Jin-Hui Zhu ◽  
Li-Hua Lu ◽  
You-Quan Li
Keyword(s):  
Continuous Time ◽  
Main Chain ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Side Chain ◽  
Nearest Neighbour ◽  
Time Quantum ◽  
Effect Of Disorder ◽  
Terminal Site ◽  
Hopping Strength

AbstractWe consider an continuous-time quantum walk on triple graphs with a potential well in one of the terminal sites on the main chain and study its dynamical properties in the open system by introducing a ‘spontaneous damping’ process between the terminal site and its nearest neighbour one. Calculating the stable probability of the terminal site and the time evolution of probabilities of the other sites on the main chain, we show that the system can manifest the factorization and coherence protection phenomena when the length and position of the side chain satisfy some concrete conditions. Furthermore we investigate the effect of disorder on the hopping strength and show that the existence of the disorder can destroy the coherence protection phenomena and increase the stable probability of the terminal site on the main chain.

scholarly journals Uniform Mixing and Association Schemes

10.37236/4745 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Chris Godsil ◽  
Natalie Mullin ◽  
Aidan Roy
Keyword(s):  
Continuous Time ◽  
Hadamard Matrices ◽  
Quantum Walks ◽  
Association Schemes ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Paley Graph ◽  
Time Quantum ◽  
Strongly Regular ◽  
Distance Regular Graphs

We consider continuous-time quantum walks on distance-regular graphs. Using results about the existence of complex Hadamard matrices in association schemes, we determine which of these graphs have quantum walks that admit uniform mixing.First we apply a result due to Chan to show that the only strongly regular graphs that admit instantaneous uniform mixing are the Paley graph of order nine and certain graphs corresponding to regular symmetric Hadamard matrices with constant diagonal. Next we prove that if uniform mixing occurs on a bipartite graph $X$ with $n$ vertices, then $n$ is divisible by four. We also prove that if $X$ is bipartite and regular, then $n$ is the sum of two integer squares. Our work on bipartite graphs implies that uniform mixing does not occur on $C_{2m}$ for $m \geq 3$. Using a result of Haagerup, we show that uniform mixing does not occur on $C_p$ for any prime $p$ such that $p \geq 5$. In contrast to this result, we see that $\epsilon$-uniform mixing occurs on $C_p$ for all primes $p$.

scholarly journals Transport Efficiency of Continuous-Time Quantum Walks on Graphs

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 85
Author(s):  
Luca Razzoli ◽  
Matteo G. A. Paris ◽  
Paolo Bordone
Keyword(s):  
Transport Properties ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Transport Processes ◽  
Initial State ◽  
Transport Efficiency ◽  
Harvesting Systems ◽  
Time Quantum ◽  
Modeling Transport

Continuous-time quantum walk describes the propagation of a quantum particle (or an excitation) evolving continuously in time on a graph. As such, it provides a natural framework for modeling transport processes, e.g., in light-harvesting systems. In particular, the transport properties strongly depend on the initial state and specific features of the graph under investigation. In this paper, we address the role of graph topology, and investigate the transport properties of graphs with different regularity, symmetry, and connectivity. We neglect disorder and decoherence, and assume a single trap vertex that is accountable for the loss processes. In particular, for each graph, we analytically determine the subspace of states having maximum transport efficiency. Our results provide a set of benchmarks for environment-assisted quantum transport, and suggest that connectivity is a poor indicator for transport efficiency. Indeed, we observe some specific correlations between transport efficiency and connectivity for certain graphs, but, in general, they are uncorrelated.

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