Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0903.4508 · Interdisciplinary Information Sciences

Limit Theorems for Discrete-Time Quantum Walks on Trees

Kota Chisaki, Masatoshi Hamada, Norio Konno, Etsuo Segawa

Published Jan 1, 2009• 36 citations
arXivHTMLPDF

Abstract

We consider a discrete-time quantum walk Wt given by the Grover transformation on the homogeneous tree. We reduce Wt to a quantum walk Xt on a half line with a wall at the origin. This paper presents two types of limit theorems for Xt. The first one is Xt as t→∞, which corresponds to a localization in the case of an initial qubit state. The second one is Xt⁄t as t→∞, whose limit density is given by the Konno density function [1–4]. The density appears in various situations of discrete-time cases. The corresponding similar limit theorem was proved in [5] for a continuous-time case on the homogeneous tree.

DOI
10.4036/iis.2009.423

Related papers

Limit theorems for the discrete-time quantum walk on a graph with joined half lines

Kota Chisaki, Norio Konno, Etsuo Segawa

We consider a discrete-time quantum walk W_{t,\kappa} at time t on a graph with joined half lines J_\kappa, which is composed of \kappa half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W_{t,\kappa} can be reduced to the walk on a half line even if the initial state is asymmetric. For W_{t,\kappa}, we obtain two types of limit theorems. The first one is an asymptotic behavior of W_{t,\kappa} which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for W_{t,\kappa}. On each half line, W_{t,\kappa} converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.

arXiv:1009.1306•Quantum Information and Computation•Mar 1, 2012•12 citations

Limit theorems for the discrete-time quantum walk on a graph with joined half lines

ChisakiKota, KonnoNorio, SegawaEtsuo

We consider a discrete-time quantum walk Wt,κ at time t on a graph with joined half lines Jκ, which is composed of κ half lines with the same origin. Our analysis is based on a reduction of the wal...

Quantum Information and Computation•Mar 1, 2012•0 citations

CONTINUOUS-TIME QUANTUM WALKS ON TREES IN QUANTUM PROBABILITY THEORY

Norio Konno

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.

arXiv:quant-ph/0602213•Infinite Dimensional Analysis Quantum Probability and Related Topics•Jun 1, 2006•45 citations

WEAK LIMITS FOR QUANTUM WALKS ON THE HALF-LINE

CHAOBIN LIU, NELSON PETULANTE

For a discrete two-state quantum walk (QW) on the half-line with a general condition at the boundary, we formulate and prove a weak limit theorem describing the terminal behavior of its transition probabilities. In this context, localization is possible even for a walk predicated on the assumption of homogeneity. For the Hadamard walk on the half-line, the weak limit is shown to be independent of the initial coin state and to exhibit no localization.

arXiv:1212.1109•International Journal of Quantum Information•Sep 1, 2013•7 citations

Limit theorems and localization of three-state quantum walks on a line defined by generalized Grover coins

Amrita Mandal, Rohit Sarma Sarkar, Shantanav Chakraborty +1 more

In this article, we undertake a detailed study of the limiting behavior of a three-state discrete-time quantum walk on a one-dimensional lattice with generalized Grover coins. Two limit theorems are proved and consequently we show that the quantum walk exhibits localization at its initial position for a wide range of coin parameters. Finally, we discuss the effect of the coin parameters on the peak velocities of probability distributions of the underlying quantum walks.

arXiv:2204.05625•Physical Review A•Oct 4, 2022•6 citations

Limit distributions of two-dimensional quantum walks

恭平 渡部, Naoki Kobayashi, Makoto Katori +1 more

One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit $t\ensuremath{\rightarrow}\ensuremath{\infty}$ of all joint moments of two components of walker's pseudovelocity, ${X}_{t}/t$ and ${Y}_{t}/t$, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer simulations is also shown.

arXiv:0802.2749•Physical Review A•Jun 19, 2008•91 citations
We consider a discrete-time quantum walk Wt given by the Grover transformation on the homogeneous tree. We reduce Wt to a quantum walk Xt on a half line with a wall at the origin. This paper presents two types of limit theorems for Xt. The first one is Xt as t→∞, which corresponds to a localization in the case of an initial qubit state. The second one is Xt⁄t as t→∞, whose limit density is given by the Konno density function [1–4]. The density appears in various situations of discrete-time cases. The corresponding similar limit theorem was proved in [5] for a continuous-time case on the homogeneous tree.
We consider a discrete-time quantum walk W_{t,\kappa} at time t on a graph with joined half lines J_\kappa, which is composed of \kappa half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W_{t,\kappa} can be reduced to the walk on a half line even if the initial state is asymmetric. For W_{t,\kappa}, we obtain two types of limit theorems. The first one is an asymptotic behavior of W_{t,\kappa} which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for W_{t,\kappa}. On each half line, W_{t,\kappa} converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.
One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit $t\ensuremath{\rightarrow}\ensuremath{\infty}$ of all joint moments of two components of walker's pseudovelocity, Xt/t{X}_{t}/tXt​/t and Yt/t{Y}_{t}/tYt​/t, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer simulations is also shown.