arXiv:1208.1005 · Quantum Information and Computation

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

Takuya Machida

Published May 1, 2013• 8 citations

Abstract

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 unknown. As random walks are fundamental models in the Kolmogorov probability theory, can the quantum walks play an important role in quantum probability theory? To discuss this problem, we focus on a discrete-time 2-state quantum walk with a non-localized initial state and present a limit theorem. By using our limit theorem, we generate probability laws in the quantum central limit theorems from the quantum walk.

DOI: 10.26421/qic13.5-6-4

Related papers

Limit distributions in a quantum walk and quantum probability theory

Takuya Machida

Aug 5, 20120 citations

Limit theorems for the interference terms of discrete-time quantum walks on the line

Takuya Machida

The probability distributions of discrete-time quantum walks have been often investigated, and many interesting properties of them have been discovered. The probability that the walker can be find at a position is defined by diagonal elements of the density matrix. On the other hand, although off-diagonal parts of the density matrices have an important role to quantify quantumness, they have not received attention in quantum walks. We focus on the off-diagonal parts of the density matrices for discrete-time quantum walks on the line and derive limit theorems for them.

arXiv:1112.4633Quantum Information and ComputationMay 1, 20133 citations

Quantum central limit theorem and statistical hypothesis testing in discrete quantum walk

Yucheng Hu, Nan Wu, Fangmin Song +1 more

Discrete quantum walk is one of the de facto models of quantum computation and as an efficient tool to develop quantum search algorithms. Although the theoretical model of quantum walks is straightforward, there are many complex scenarios such as coherence decay and/or decoherence in the implementations. It is hard to test experimentally if quantum walk works, or it just decays into a version of classic random walk. We propose a quantum central limit theorem (QCLT) for discrete quantum walks and conduct the statistical hypothesis testing for the standard or decayed walker probability distribution for imperfect quantum walks based on the QCLT. A reliable statistical analysis result is obtained for the imperfect distribution by the experimental quantum walk study.

Apr 24, 20200 citations

Central Limit Theorems for Open Quantum Random Walks and Quantum Measurement Records

Stéphane Attal, Nadine Guillotin‐Plantard, Christophe Sabot

Open Quantum Random Walks, as developed in \cite{APSS}, are a quantum generalization of Markov chains on finite graphs or on lattices. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random walk). Typically, in the case of Open Quantum Random Walks on lattices, their distribution seems to always converge to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous Open Quantum Random Walks on $\ZZ^d$ we prove such a Central Limit Theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on $\ZZ^d$ times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a Central Limit Theorem with explicit drift and explicit covariance matrix. The interesting point which appears with our construction and result is that it applies actually to a wider setup: it provides a Central Limit Theorem for the sequence of recordings of the quantum trajectories associated to any completely positive map. This is what we show and develop as an application of our result. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block-diagonal in a common basis.

arXiv:1206.1472Jun 7, 201214 citations

Limit theorems of a 3-state quantum walk and its application for discrete uniform measures

Takuya Machida

We present two long-time limit theorems of a 3-state quantum walk on the line when the walker starts from the origin. One is a limit measure which is obtained from the probability distribution of the walk at a long-time limit, and the other is a convergence in distribution for the walker’s position in a rescaled space by time. In addition, as an application of the walk, we obtain discrete uniform limit measures from the 3-state walk with a delocalized initial state.

arXiv:1404.1522Quantum Information and ComputationApr 1, 201525 citations

Central Limit Theorems for Open Quantum Random Walks ∗

Stéphane Attal, Nadine Guillotin‐Plantard, Christophe Sabot +2 more

Open Quantum Random Walks are the exact quantum generalization of Markov chains on ﬁnite graphs or on nets. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random walk). Typically, in the case of Open Quantum Random Walks on nets, their distribution seems to always converges to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous Open Quantum Random Walks on Z^d we prove such a Central Limit Theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on Z^d times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a Central Limit Theorem with explicit drift and explicit covariance matrix. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block-diagonal in a common basis.

Jan 1, 20120 citations