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:1103.3899 · Physical Review A

Limit theorems for quantum walks associated with Hadamard matrices

Clement Boateng Ampadu

Published Jul 20, 2011• 13 citations
arXivHTMLPDF

Abstract

We study a one-parameter family of discrete-time quantum walk models on $\mathbb{Z}$ and ${\mathbb{Z}}^{2}$ associated with the Hadamard walk. Weak convergence in the long-time limit of all moments of the walker's pseudovelocity on $\mathbb{Z}$ and ${\mathbb{Z}}^{2}$ is proved. Symmetrization on $\mathbb{Z}$ and ${\mathbb{Z}}^{2}$ is theoretically investigated, leading to the resolution of the Konno-Namiki-Soshi conjecture in the special case of symmetrization of the unbiased Hadamard walk on $\mathbb{Z}$. A necessary condition for the existence of a phenomenon known as localization is given.

DOI
10.1103/physreva.84.012324

Related papers

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

Large Qudit Limit of One-dimensional Quantum Walks

Mitsunori Sato, Naoki Kobayashi, Makoto Katori +1 more

We study a series of one-dimensional discrete-time quantum-walk models labeled by half integers $j=1/2, 1, 3/2, ...$, introduced by Miyazaki {\it et al.}, each of which the walker's wave function has $2j+1$ components and hopping range at each time step is $2j$. In long-time limit the density functions of pseudovelocity-distributions are generally given by superposition of appropriately scaled Konno's density function. Since Konno's density function has a finite open support and it diverges at the boundaries of support, limit distribution of pseudovelocities in the $(2j+1)$-component model can have $2j+1$ pikes, when $2j+1$ is even. When $j$ becomes very large, however, we found that these pikes vanish and a universal and monotone convex structure appears around the origin in limit distributions. We discuss a possible route from quantum walks to classical diffusion associated with the $j \to \infty$ limit.

arXiv:0802.1997•Feb 14, 2008•4 citations

Limit theorem for continuous-time quantum walk on the line

Norio Konno

Concerning a discrete-time quantum walk ${X}_{t}^{(d)}$ with a symmetric distribution on the line, whose evolution is described by the Hadamard transformation, it was proved by the author that the following weak limit theorem holds: ${X}_{t}^{(d)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}(1\ensuremath{-}{x}^{2})\sqrt{1\ensuremath{-}2{x}^{2}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. The present paper shows that a similar type of weak limit theorem is satisfied for a continuous-time quantum walk ${X}_{t}^{(c)}$ on the line as follows: ${X}_{t}^{(c)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}\sqrt{1\ensuremath{-}{x}^{2}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: ${Y}_{t}∕\sqrt{t}\ensuremath{\rightarrow}{e}^{\ensuremath{-}{x}^{2}∕2}dx∕\sqrt{2\ensuremath{\pi}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. The work deals also with the issue of the relationship between discrete and continuous-time quantum walks. This topic, subject of a long debate in the previous literature, is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases.

arXiv:quant-ph/0408140•Physical Review E•Aug 12, 2005•125 citations

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

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

Probability Distributions and Weak Limit Theorems of Quaternionic Quantum Walks in One Dimension

Kei Saito

The discrete-time quantum walk (QW) is determined by a unitary matrix whose components are complex numbers. Konno (2015) extended the QW to the quaternionic quantum walk (QQW) whose components are quaternions and presented some properties of the QQW. Furthermore, Konno (2015) presented the question of whether or not the dynamics of a QQW is exactly the same as that of the corresponding QW. We give an answer to the problem by calculating the probability distribution and the weak limit density function of some classes of the QQW.

arXiv:1710.01482•Interdisciplinary Information Sciences•Jan 1, 2018•2 citations
We study a one-parameter family of discrete-time quantum walk models on Z\mathbb{Z}Z and Z2{\mathbb{Z}}^{2}Z2 associated with the Hadamard walk. Weak convergence in the long-time limit of all moments of the walker's pseudovelocity on Z\mathbb{Z}Z and Z2{\mathbb{Z}}^{2}Z2 is proved. Symmetrization on Z\mathbb{Z}Z and Z2{\mathbb{Z}}^{2}Z2 is theoretically investigated, leading to the resolution of the Konno-Namiki-Soshi conjecture in the special case of symmetrization of the unbiased Hadamard walk on Z\mathbb{Z}Z. A necessary condition for the existence of a phenomenon known as localization is given.
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.
We study a series of one-dimensional discrete-time quantum-walk models labeled by half integers j=1/2,1,3/2,...j=1/2, 1, 3/2, ...j=1/2,1,3/2,..., introduced by Miyazaki {\it et al.}, each of which the walker's wave function has 2j+12j+12j+1 components and hopping range at each time step is 2j2j2j. In long-time limit the density functions of pseudovelocity-distributions are generally given by superposition of appropriately scaled Konno's density function. Since Konno's density function has a finite open support and it diverges at the boundaries of support, limit distribution of pseudovelocities in the (2j+1)(2j+1)(2j+1)-component model can have 2j+12j+12j+1 pikes, when 2j+12j+12j+1 is even. When jjj becomes very large, however, we found that these pikes vanish and a universal and monotone convex structure appears around the origin in limit distributions. We discuss a possible route from quantum walks to classical diffusion associated with the j→∞j \to \inftyj→∞ limit.
Concerning a discrete-time quantum walk Xt(d){X}_{t}^{(d)}Xt(d)​ with a symmetric distribution on the line, whose evolution is described by the Hadamard transformation, it was proved by the author that the following weak limit theorem holds: ${X}_{t}^{(d)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}(1\ensuremath{-}{x}^{2})\sqrt{1\ensuremath{-}2{x}^{2}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. The present paper shows that a similar type of weak limit theorem is satisfied for a continuous-time quantum walk Xt(c){X}_{t}^{(c)}Xt(c)​ on the line as follows: ${X}_{t}^{(c)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}\sqrt{1\ensuremath{-}{x}^{2}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: ${Y}_{t}∕\sqrt{t}\ensuremath{\rightarrow}{e}^{\ensuremath{-}{x}^{2}∕2}dx∕\sqrt{2\ensuremath{\pi}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. The work deals also with the issue of the relationship between discrete and continuous-time quantum walks. This topic, subject of a long debate in the previous literature, is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases.
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.
The discrete-time quantum walk (QW) is determined by a unitary matrix whose components are complex numbers. Konno (2015) extended the QW to the quaternionic quantum walk (QQW) whose components are quaternions and presented some properties of the QQW. Furthermore, Konno (2015) presented the question of whether or not the dynamics of a QQW is exactly the same as that of the corresponding QW. We give an answer to the problem by calculating the probability distribution and the weak limit density function of some classes of the QQW.