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.

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.

scholarly journals CONTINUOUS-TIME QUANTUM WALKS ON ULTRAMETRIC SPACES

2006 ◽  
Vol 04 (06) ◽  
pp. 1023-1035 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Continuous Time ◽  
Search Algorithm ◽  
Quantum Walk ◽  
Ultrametric Space ◽  
Quantum Walks ◽  
Ultrametric Spaces ◽  
Quantum Search Algorithm ◽  
Time Quantum ◽  
Cycle Graph ◽  
Striking Contrast

We introduce a continuous-time quantum walk on an ultrametric space corresponding to the set of p-adic integers and compute its time-averaged probability distribution. It is shown that localization occurs for any location of the ultrametric space for the walk. This result presents a striking contrast to the classical random walk case. Moreover, we clarify a difference between the ultrametric space and other graphs, such as cycle graph, line, hypercube and complete graph, for the localization of the quantum case. Our quantum walk may be useful for a quantum search algorithm on a tree-like hierarchical structure.

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 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.

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.

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.

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