scholarly journals Quantum walks and elliptic integrals

2010 ◽  
Vol 20 (6) ◽  
pp. 1091-1098 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Random Walk ◽  
Generating Function ◽  
Elliptic Integral ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Elliptic Integrals ◽  
Two Dimensional ◽  
One Dimensional ◽  
Similar Expression ◽  
Return Probability

Pólya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.

scholarly journals Persistence of topological phases in non-Hermitian quantum walks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Vikash Mittal ◽  
Aswathy Raj ◽  
Sanjib Dey ◽  
Sandeep K. Goyal
Keyword(s):  
Quantum Walk ◽  
Quantum Walks ◽  
Topological Phases ◽  
Topological Order ◽  
Topological Phase ◽  
Two Dimensional ◽  
One Dimensional ◽  
Topological Phase Transition ◽  
Topological States ◽  
Time Quantum

AbstractDiscrete-time quantum walks are known to exhibit exotic topological states and phases. Physical realization of quantum walks in a lossy environment may destroy these phases. We investigate the behaviour of topological states in quantum walks in the presence of a lossy environment. The environmental effects in the quantum walk dynamics are addressed using the non-Hermitian Hamiltonian approach. We show that the topological phases of the quantum walks are robust against moderate losses. The topological order in one-dimensional split-step quantum walk persists as long as the Hamiltonian respects exact $${{\mathcal {P}}}{{\mathcal {T}}}$$ P T -symmetry. Although the topological nature persists in two-dimensional quantum walks as well, the $${{\mathcal {P}}}{{\mathcal {T}}}$$ P T -symmetry has no role to play there. Furthermore, we observe topological phase transition in two-dimensional quantum walks that is induced by losses in the system.

scholarly journals One-dimensional quantum walks via generating function and the CGMV method

2014 ◽  
Vol 14 (13&14) ◽  
pp. 1165-1186
Author(s):  
Norio Konno ◽  
Etsuo Segawa
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Generating Function ◽  
Power Law ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Linear Scaling ◽  
One Dimensional ◽  
Power Law Decay ◽  
Quantum Coin

We treat quantum walk (QW) on the line whose quantum coin at each vertex tends to the identity as the distance goes to infinity. We obtain a limit theorem that this QW exhibits localization with not an exponential but a ``power-law" decay around the origin and a ``strongly" ballistic spreading called bottom localization in this paper. This limit theorem implies the weak convergence with linear scaling whose density has two delta measures at $x=0$ (the origin) and $x=1$ (the bottom) without continuous parts.

Return probability of quantum walks with final-time dependence

2011 ◽  
Vol 11 (9&10) ◽  
pp. 761-773
Author(s):  
Yusuke Ide ◽  
Norio Konno ◽  
Takuya Machida ◽  
Etsuo Segawa
Keyword(s):  
Random Walk ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Time Dependent ◽  
One Dimension ◽  
Final Time ◽  
Counting Approach ◽  
Return Probability ◽  
Time Quantum ◽  
Path Counting

We analyze final-time dependent discrete-time quantum walks in one dimension. We compute asymptotics of the return probability of the quantum walk by a path counting approach. Moreover, we discuss a relation between the quantum walk and the corresponding final-time dependent classical random walk.

scholarly journals One dimensional quantum walks with memory

2010 ◽  
Vol 10 (5&6) ◽  
pp. 509-524
Author(s):  
M. Mc Gettrick
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Quantum Walks ◽  
One Dimensional ◽  
With Memory ◽  
Previous Step

We investigate the quantum versions of a one-dimensional random walk, whose corresponding Markov Chain is of order 2. This corresponds to the walk having a memory of one previous step. We derive the amplitudes and probabilities for these walks, and point out how they differ from both classical random walks, and quantum walks without memory.

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.

LATTICE GAS WAVE PROPAGATION MODELS INCLUDING DISSIPATION

1995 ◽  
Vol 03 (01) ◽  
pp. 69-93 ◽  
Author(s):  
YASUSHI SUDO ◽  
VICTOR W. SPARROW
Keyword(s):  
Wave Propagation ◽  
Random Walk ◽  
Diffusion Equation ◽  
Lattice Gas ◽  
Sound Propagation ◽  
Two Dimensional ◽  
Propagation Models ◽  
One Dimensional ◽  
Lattice Gas Models ◽  
Wave Models

New lattice gas models for one-dimensional (1D) and two-dimensional (2D) sound propagation have been recently proposed by the authors. These models were dissipationless and deterministic. In this paper, it will be shown how dissipation effects can be included into these lattice gas wave models. To simulate these dissipation effects, the lattice gas particles are assumed to take a random walk. The resulting models combine the authors' lattice gas wave models with published lattice gas models for the diffusion equation. The formulations are stable and consistent.

scholarly journals Efficient circuits for quantum walks

2010 ◽  
Vol 10 (5&6) ◽  
pp. 420-434
Author(s):  
C.-F. Chiang ◽  
D. Nagaj ◽  
P. Wocjan
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Probability Distributions ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Problem Size ◽  
Integrable Probability ◽  
The Difference ◽  
General Method

We present an efficient general method for realizing a quantum walk operator corresponding to an arbitrary sparse classical random walk. Our approach is based on Grover and Rudolph's method for preparing coherent versions of efficiently integrable probability distributions \cite{GroverRudolph}. This method is intended for use in quantum walk algorithms with polynomial speedups, whose complexity is usually measured in terms of how many times we have to apply a step of a quantum walk \cite{Szegedy}, compared to the number of necessary classical Markov chain steps. We consider a finer notion of complexity including the number of elementary gates it takes to implement each step of the quantum walk with some desired accuracy. The difference in complexity for various implementation approaches is that our method scales linearly in the sparsity parameter and poly-logarithmically with the inverse of the desired precision. The best previously known general methods either scale quadratically in the sparsity parameter, or polynomially in the inverse precision. Our approach is especially relevant for implementing quantum walks corresponding to classical random walks like those used in the classical algorithms for approximating permanents \cite{Vigoda, Vazirani} and sampling from binary contingency tables \cite{Stefankovi}. In those algorithms, the sparsity parameter grows with the problem size, while maintaining high precision is required.

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.

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