Quantum Walk on the Generalized Birkhoff Polytope Graph

We study discrete-time quantum walks on generalized Birkhoff polytope graphs (GBPGs), which arise in the solution-set to certain transportation linear programming problems (TLPs). It is known that quantum walks mix at most quadratically faster than random walks on cycles, two-dimensional lattices, hypercubes, and bounded-degree graphs. In contrast, our numerical results show that it is possible to achieve a greater than quadratic quantum speedup for the mixing time on a subclass of GBPG (TLP with two consumers and m suppliers). We analyze two types of initial states. If the walker starts on a single node, the quantum mixing time does not depend on m, even though the graph diameter increases with it. To the best of our knowledge, this is the first example of its kind. If the walker is initially spread over a maximal clique, the quantum mixing time is O(m/ϵ), where ϵ is the threshold used in the mixing times. This result is better than the classical mixing time, which is O(m1.5/ϵ).

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MIXING OF QUANTUM WALKS ON GENERALIZED HYPERCUBES

International Journal of Quantum Information ◽

10.1142/s0219749908004377 ◽

2008 ◽

Vol 06 (06) ◽

pp. 1135-1148 ◽

Cited By ~ 6

Author(s):

ANA BEST ◽

MARKUS KLIEGL ◽

SHAWN MEAD-GLUCHACKI ◽

CHRISTINO TAMON

Keyword(s):

Mixing Time ◽

Quantum Walk ◽

Circulant Matrix ◽

Quantum Walks ◽

Spectral Structure ◽

The Status ◽

Time Quantum ◽

Hamming Graphs ◽

The Rich ◽

The Fourier Transform

We study continuous-time quantum walks on graphs which generalize the hypercube. The only known family of graphs whose quantum walk instantaneously mixes to uniform is the Hamming graphs with small arities. We show that quantum uniform mixing on the hypercube is robust under the addition of perfect matchings but not much else. Our specific results include: • The graph obtained by augmenting the hypercube with an additive matching x ↦ x ⊕ η is instantaneous uniform mixing whenever |η| is even, but with a slower mixing time. This strictly includes the result of Moore and Russell1 on the hypercube. • The class of Hamming graphs H(n,q) is not uniform mixing if and only if q ≥ 5. This is a tight characterization of quantum uniform mixing on Hamming graphs; previously, only the status of H(n,q) with q < 5 was known. • The bunkbed graph [Formula: see text] whose adjacency matrix is I ⊗ Qn + X ⊗ Af, where Af is a [Formula: see text]-circulant matrix defined by a Boolean function f, is not uniform mixing if the Fourier transform of f has support of size smaller than 2n-1. This explains why the hypercube is uniform mixing and why the join of two hypercubes is not. Our work exploits the rich spectral structure of the generalized hypercubes and relies heavily on Fourier analysis of group-circulants.

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Mixing and decoherence in quantum walks on cycles

Quantum Information and Computation ◽

10.26421/qic6.3-3 ◽

2006 ◽

Vol 6 (3) ◽

pp. 263-276 ◽

Cited By ~ 1

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.

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

Entropy ◽

10.3390/e20080586 ◽

2018 ◽

Vol 20 (8) ◽

pp. 586 ◽

Cited By ~ 1

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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A limit theorem for a splitting distribution of a quantum walk

International Journal of Quantum Information ◽

10.1142/s0219749918500235 ◽

2018 ◽

Vol 16 (03) ◽

pp. 1850023

Author(s):

Takuya Machida

Keyword(s):

Limit Theorem ◽

Probability Distribution ◽

Random Walks ◽

Discrete Time ◽

Limit Distribution ◽

Quantum Walk ◽

Quantum Walks ◽

Unitary Evolution ◽

Time Quantum ◽

Long Time

Discrete-time quantum walks are considered a counterpart of random walks and their study has been getting attention since around 2000. In this paper, we focus on a quantum walk which generates a probability distribution splitting to two parts. The quantum walker with two coin states spreads at points, represented by integers, and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.

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The fog on: Generalized teleportation by means of discrete-time quantum walks on N-lines and N-cycles

Modern Physics Letters B ◽

10.1142/s0217984919502701 ◽

2019 ◽

Vol 33 (23) ◽

pp. 1950270 ◽

Cited By ~ 4

Author(s):

Duc Manh Nguyen ◽

Sunghwan Kim

Keyword(s):

Discrete Time ◽

Quantum Teleportation ◽

Quantum Walk ◽

Quantum Walks ◽

One Dimensional ◽

Time Quantum ◽

Infinite Line ◽

The One

The recent paper entitled “Generalized teleportation by means of discrete-time quantum walks on [Formula: see text]-lines and [Formula: see text]-cycles” by Yang et al. [Mod. Phys. Lett. B 33(6) (2019) 1950069] proposed the quantum teleportation by means of discrete-time quantum walks on [Formula: see text]-lines and [Formula: see text]-cycles. However, further investigation shows that the quantum walk over the one-dimensional infinite line can be based over the [Formula: see text]-cycles and cannot be based on [Formula: see text]-lines. The proofs of our claims on quantum walks based on finite lines are also provided in detail.

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CONTINUOUS-TIME QUANTUM WALKS ON TREES IN QUANTUM PROBABILITY THEORY

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025706002354 ◽

2006 ◽

Vol 09 (02) ◽

pp. 287-297 ◽

Cited By ~ 26

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.

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QUANTUM WALKS ON SIERPINSKI GASKETS

International Journal of Quantum Information ◽

10.1142/s021974991350069x ◽

2013 ◽

Vol 11 (08) ◽

pp. 1350069 ◽

Cited By ~ 5

Author(s):

PEDRO CARLOS S. LARA ◽

RENATO PORTUGAL ◽

STEFAN BOETTCHER

Keyword(s):

Mixing Time ◽

Mean Square Displacement ◽

Quantum Walks ◽

Flip Flop ◽

Translational Invariance ◽

Initial Location ◽

Time Quantum ◽

The Mean ◽

Sierpinski Gaskets ◽

Physical Quantities

We analyze discrete-time quantum walks on Sierpinski gaskets using a flip-flop shift operator with the Grover coin. We obtain the scaling of two important physical quantities: The mean-square displacement, and the mixing time as function of the number of points. The Sierpinski gasket is a fractal that lacks translational invariance and the results differ from those described in the literature for ordinary lattices. We find that the diffusion scaling depends on the initial location. Averaged over all initial locations, our simulation obtain an exponent very similar to the classical diffusion.

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Entanglement for quantum walks on the line

Quantum Information and Computation ◽

10.26421/qic11.9-10-9 ◽

2011 ◽

Vol 11 (9&10) ◽

pp. 855-866

Author(s):

Yusuke Ide ◽

Norio Konno ◽

Takuya Machida

Keyword(s):

Information Theory ◽

Shannon Entropy ◽

Quantum Walk ◽

Quantum Walks ◽

Von Neumann Entropy ◽

Initial State ◽

Von Neumann ◽

Time Quantum ◽

The One ◽

Path Counting

The discrete-time quantum walk is a quantum counterpart of the random walk. It is expected that the model plays important roles in the quantum field. In the quantum information theory, entanglement is a key resource. We use the von Neumann entropy to measure the entanglement between the coin and the particle's position of the quantum walks. Also we deal with the Shannon entropy which is an important quantity in the information theory. In this paper, we show limits of the von Neumann entropy and the Shannon entropy of the quantum walks on the one dimensional lattice starting from the origin defined by arbitrary coin and initial state. In order to derive these limits, we use the path counting method which is a combinatorial method for computing probability amplitude.

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Probing the Eigenstates Thermalization Hypothesis with Many-Particle Quantum Walks on Lattices

Open Systems & Information Dynamics ◽

10.1142/s123016121750007x ◽

2017 ◽

Vol 24 (02) ◽

pp. 1750007

Author(s):

Dibwe Pierrot Musumbu ◽

Maria Przybylska ◽

Andrzej J. Maciejewski

Keyword(s):

Thermal State ◽

Particle System ◽

Hamiltonian Dynamics ◽

Quantum Walk ◽

Quantum Walks ◽

Grid Graphs ◽

Time Quantum ◽

Counting Statistics ◽

The Many ◽

The One

We simulate the dynamics of many-particle system of bosons and fermions using discrete time quantum walks on lattices. We present a computational proof of the behaviour of the simulated systems similar to the one observed in Hamiltonian dynamics during quantum thermalization. We record the time evolution of the entropy and the temperature of a specific particle configuration during the entire dynamics and observe how they relax to a state which we call the quantum walk thermal state. This observation is made on two types of lattices while simulating different numbers of particles walking on two grid graphs with 25 vertices. In each case, we observe that the vertices counting statistics, the temperature of the indexed configuration and the dimension of the effective configuration Hilbert space relax simultaneously and remain relaxed for the rest of the many-particle quantum walk.

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Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises

Entropy ◽

10.3390/e22050504 ◽

2020 ◽

Vol 22 (5) ◽

pp. 504

Author(s):

Ce Wang ◽

Caishi Wang

Keyword(s):

Discrete Time ◽

Quantum Information Processing ◽

Quantum Walk ◽

Quantum Walks ◽

Integer Lattice ◽

Initial State ◽

Higher Dimensional ◽

Time Quantum ◽

Gauss Type ◽

Limit Probability

As a discrete-time quantum walk model on the one-dimensional integer lattice Z , the quantum walk recently constructed by Wang and Ye [Caishi Wang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897–1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer d ≥ 2 , by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice Z d , which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.

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