Localization and discrete probability function of Szegedy’s quantum search one-dimensional cycle with self-loops

2020 ◽  
Vol 20 (15&16) ◽  
pp. 1281-1303
Author(s):  
Mengke Xu ◽  
Zhihao Liu ◽  
Hanwu Chen ◽  
Sihao Zheng
Keyword(s):  
Probability Function ◽  
Quantum Walk ◽  
Quantum Search ◽  
Discrete Probability ◽  
One Dimensional ◽  
Average Probability ◽  
Analytic Expression ◽  
Initial States ◽  
The One ◽  
Lebesgue Theorem

We study the localization and the discrete probability function of a quantum search on the one-dimensional (1D) cycle with self-loops for n vertices and m marked vertices. First, unmarked vertices have no localization since the quantum search on unmarked vertices behaves like the 1D three-state quantum walk (3QW) and localization does not occur with nonlocal initial states on a 3QW, according to residue calculations and the Riemann-Lebesgue theorem. Second, we show that localization does occur on the marked vertices and derive an analytic expression for localization by the degenerate 1-eigenvalues contributing to marked vertices. Therefore localization can contribute to a quantum search. Furthermore, we emphasize that localization comes from the self-loops. Third, using the localization of a quantum search, the asymptotic average probability distribution (AAPD) and the discrete probability function (DPF) of a quantum search are obtained. The DPF shows that Szegedy’s quantum search on the 1D cycle with self-loops spreads ballistically.

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

2019 ◽  
Vol 33 (23) ◽  
pp. 1950270 ◽  
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.

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 Two component quantum walk in one-dimensional lattice with hopping imbalance

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Mrinal Kanti Giri ◽  
Suman Mondal ◽  
Bhanu Pratap Das ◽  
Tapan Mishra
Keyword(s):  
Interaction Strength ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Fast Particle ◽  
Particle Correlation ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Slow Particle ◽  
Two Component ◽  
Initial States

AbstractWe investigate the two-component quantum walk in one-dimensional lattice. We show that the inter-component interaction strength together with the hopping imbalance between the components exhibit distinct features in the quantum walk for different initial states. When the walkers are initially on the same site, both the slow and fast particles perform independent particle quantum walks when the interaction between them is weak. However, stronger inter-particle interactions result in quantum walks by the repulsively bound pair formed between the two particles. For different initial states when the walkers are on different sites initially, the quantum walk performed by the slow particle is almost independent of that of the fast particle, which exhibits reflected and transmitted components across the particle with large hopping strength for weak interactions. Beyond a critical value of the interaction strength, the wave function of the fast particle ceases to penetrate through the slow particle signalling a spatial phase separation. However, when the two particles are initially at the two opposite edges of the lattice, then the interaction facilitates the complete reflection of both of them from each other. We analyze the above mentioned features by examining various physical quantities such as the on-site density evolution, two-particle correlation functions and transmission coefficients.

The one-dimensional Hulthén potential in the quantum phase space representation

Open Physics ◽  
2014 ◽  
Vol 12 (2) ◽  
Author(s):  
Jerzy Stanek
Keyword(s):  
Phase Space ◽  
Wigner Function ◽  
Quantum Phase ◽  
Space Representation ◽  
Hulthén Potential ◽  
One Dimensional ◽  
Hulthen Potential ◽  
Analytic Expression ◽  
The One ◽  
Phase Space Representation

AbstractThe analytic expression of the Wigner function for bound eigenstates of the Hulthén potential in quantum phase space is obtained and presented by plotting this function for a few quantum states. In addition, the correct marginal distributions of the Wigner function in spherical coordinates are determined analytically.

scholarly journals Three routes to the exact asymptotics for the one-dimensional quantum walk

2003 ◽  
Vol 36 (33) ◽  
pp. 8775-8795 ◽  
Author(s):  
Hilary A Carteret ◽  
Mourad E H Ismail ◽  
Bruce Richmond
Keyword(s):  
Quantum Walk ◽  
Exact Asymptotics ◽  
One Dimensional ◽  
The One

scholarly journals Role of Symmetry in Quantum Search via Continuous-Time Quantum Walk

SPIN ◽  
2021 ◽  
pp. 2140002
Author(s):  
Yunkai Wang ◽  
Shengjun Wu
Keyword(s):  
Invariant Subspace ◽  
Continuous Time ◽  
Quantum Walk ◽  
Dimensional Subspace ◽  
Quantum Search ◽  
Trivial Representation ◽  
Time Quantum ◽  
The One ◽  
Dimensionality Reduction Method

For quantum search via the continuous-time quantum walk, the evolution of the whole system is usually limited in a small subspace. In this paper, we discuss how the symmetries of the graphs are related to the existence of such an invariant subspace, which also suggests a dimensionality reduction method based on group representation theory. We observe that in the one-dimensional subspace spanned by each desired basis state which assembles the identically evolving original basis states, we always get a trivial representation of the symmetry group. So, we could find the desired basis by exploiting the projection operator of the trivial representation. Besides being technical guidance in this type of problem, this discussion also suggests that all the symmetries are used up in the invariant subspace and the asymmetric part of the Hamiltonian is very important for the purpose of quantum search.

scholarly journals Quantum search on Hanoi network

2019 ◽  
Vol 17 (07) ◽  
pp. 1950060
Author(s):  
Pulak Ranjan Giri ◽  
Vladimir Korepin
Keyword(s):  
Quantum Walk ◽  
Quantum Walks ◽  
Periodic Lattice ◽  
Quantum Search ◽  
Main Structure ◽  
Running Time ◽  
One Dimensional ◽  
Target State ◽  
Amplitude Amplification ◽  
Walk Algorithm

Hanoi network (HN) has a one-dimensional periodic lattice as its main structure with additional long-range edges, which allow having efficient quantum walk algorithm that can find a target state on the network faster than the exhaustive classical search. In this paper, we use regular quantum walks and lackadaisical quantum walks, respectively, to search for a target state. From the curve fitting of the numerical results for HN of degrees three and four, we find that their running time for the regular quantum walks are followed by amplitude amplification scales as [Formula: see text] and [Formula: see text], respectively. For the search by lackadaisical quantum walks, the running time scales are as [Formula: see text] and [Formula: see text], respectively.

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