Realization of the probability laws in the quantum central limit theorems by a quantum walk

  title={Realization of the probability laws in the quantum central limit theorems by a quantum walk},
  author={T. Machida},
  journal={Quantum Inf. Comput.},
  • T. Machida
  • Published 5 August 2012
  • Physics, Mathematics, Computer Science
  • Quantum Inf. Comput.
Since a limit distribution of a discrete-time quantum walk on the line was derived in 2002, a lot of limit theorems for quantum walks with a localized initial state have been reported. On the other hand, in quantum probability theory, there are four notions of independence (free, monotone, commuting, and boolean independence) and quantum central limit theorems associated to each independence have been investigated. The relation between quantum walks and quantum probability theory is still… 
Quantum Walk in Terms of Quantum Bernoulli Noise and Quantum Central Limit Theorem for Quantum Bernoulli Noise
As a unitary quantum walk with infinitely many internal degrees of freedom, the quantum walk in terms of quantum Bernoulli noise (recently introduced by Wang and Ye) shows a rather classical
Limit theorems of a 3-state quantum walk and its application for discrete uniform measures
  • T. Machida
  • Mathematics, Physics
    Quantum Inf. Comput.
  • 2015
Two long-time limit theorems of a 3-state quantum walk on the line when the walker starts from the origin are presented and discrete uniform limit measures are obtained from the 3- state walk with a delocalized initial state.
The generator and quantum Markov semigroup for quantum walks
The quantum walks in the lattice spaces are represented as unitary evolutions. We find a generator for the evolution and apply it to further understand the walks. We first extend the discrete time
Quantum walks with an anisotropic coin II: scattering theory
We perform the scattering analysis of the evolution operator of quantum walks with an anisotropic coin, and we prove a weak limit theorem for their asymptotic velocity. The quantum walks that we
The uniform measure for discrete-time quantum walks in one dimension
  • N. Konno
  • Mathematics, Computer Science
    Quantum Inf. Process.
  • 2014
We obtain the uniform measure as a stationary measure of the one-dimensional discrete-time quantum walks by solving the corresponding eigenvalue problem. As an application, the uniform probability
The coin operators constructed by QBN Walk and one-dimensional two state quantum walk
This paper constructs coin operators on coin space H by QBN walk and one-dimensional two state quantum walk and obtains some formulas about those coin operators.
Quantum Bayesian networks with application to games displaying Parrondo's paradox
Bayesian networks and their accompanying graphical models are widely used for prediction and analysis across many disciplines. We will reformulate these in terms of linear maps. This reformulation


Quantum Walks for Computer Scientists
The purpose of this lecture is to provide a concise yet comprehensive introduction to quantum walks, an emerging field of quantum computation, is a generalization of random walks into the quantum mechanical world.
Limit measures of inhomogeneous discrete-time quantum walks in one dimension
It is shown that typical spatial homogeneous QWs with ballistic spreading belong to the universality class and it is found that the walk treated here with one defect also belongs to the class.
We obtain some rigorous results on limit theorems for quantum walks driven by many coins introduced by Brun et al. in the long time limit. The results imply that whether the behavior of a particle is
Limit theorems for quantum walks with memory
The weak limit theorem of the rescaled QW is obtained by considering the walk as a 4-state QW without memory and the behavior is strikingly different from the corresponding classical random walk and the usual 2-state quantum walk without memory.
The Heun differential equation and the Gauss differential equation related to quantum walks
The limit theorems of discrete- and continuous-time quantum walks on the line have been intensively studied. We show a relation among limit distributions of quantum walks, Heun differential equations
Crossovers induced by discrete-time quantum walks
This weak convergence theorem gives a phase diagram which maps sufficiently long-time behaviors of the discrete- and continuous-time quantum and random walks.
We consider 2-state quantum walks (QWs) on the line, which are defined by two matrices. One of the matrices operates the walk at only half-time. In the usual QWs, localization does not occur at all.
A new type of limit theorems for the one-dimensional quantum random walk
In this paper we consider the one-dimensional quantum random walk X^{varphi} _n at time n starting from initial qubit state varphi determined by 2 times 2 unitary matrix U. We give a combinatorial
Weak limits for quantum random walks.
A general weak limit theorem for quantum random walks in one and more dimensions is formulated and proved with X(n)/n converges weakly as n--> infinity to a certain distribution which is absolutely continuous and of bounded support.
Quantum walk on the line: Entanglement and nonlocal initial conditions (9 pages)
The conditional shift in the evolution operator of a quantum walk generates entanglement between the coin and position degrees of freedom. This entanglement can be quantified by the von Neumman