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:1305.1749 · Kodai Mathematical Journal

The generator and quantum Markov semigroup for quantum walks

Chul Ki Ko, Hyun Jae Yoo

Published Jun 1, 2013• 9 citations
arXivHTMLPDF

Abstract

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 to continuous time walks. Then we construct the quantum Markov semigroup for quantum walks and characterize it in an invariant subalgebra. In the meanwhile, we obtain the limit distributions of the quantum walks in one-dimension with a proper scaling, which was obtained by Konno by a different method.

DOI
10.2996/kmj/1372337524

Related papers

Molchanov's Formula and Quantum Walks: A Probabilistic Approach

Hoang Vu

This paper establishes a robust link between quantum dynamics and classical ones by deriving probabilistic representation for both continuous time and discrete time quantum walks. We first adapt Molchanov formula, originally employed in the study of Schrodinger operators on multidimensional integer lattice, to characterize the evolution of continuous time quantum walks. Extending this framework, we develop a probabilistic method to represent discrete time quantum walks on an infinite integer line, bypassing the locality constraints that typically inhibit direct application of Molchanov formula. The validity of our representation is empirically confirmed through a benchmark analysis of the Hadamard walk, demonstrating high fidelity with traditional unitary evolution. Our results suggest that this probabilistic lens offer a powerful alternative for learning multidimensional quantum walks and provides new analytical pathways for investigating quantum systems via classical stochastic processes.

quant-pharXiv:2601.01071•Jan 6, 2026•0 citations

A limit theorem for a splitting distribution of a quantum walk

Takuya Machida

Discrete-time quantum walks are considered a counterpart of random walks and their study has been getting attention since around 2000. In this paper, we focus on a quantum walk which generates a probability distribution splitting to two parts. The quantum walker with two coin states spreads at points, represented by integers, and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.

arXiv:1704.04554•International Journal of Quantum Information•Apr 1, 2018•2 citations

Dilations à la Quantum Probability of Markov Evolutions in Discrete Time

Matteo Gregoratti

We study the classical probability analogue of the unitary dilations of a quantum dynamical semigroup in quantum probability. Given a (not necessarily homogeneous) Markov chain in discrete time in a finite state space E, we introduce a second system, an environment, and a deterministic invertible time-homogeneous global evolution of the system E with this environment such that the original Markov evolution of E is realized by a proper choice of the initial random state of the environment. We also compare these dilations with the unitary dilations of a quantum dynamical semigroup in quantum probability: given a classical Markov semigroup, we show that it can be extended to a quantum dynamical semigroup for which we can find a quantum dilation to a group of $*$-automorphisms admitting an invariant abelian subalgebra where this quantum dilation gives just our classical dilation.

Theory of Probability and Its Applications•Jan 1, 2010•2 citations

The Hamiltonians Generating One-Dimensional Discrete-Time Quantum Walks

Tatsuya Tate

An explicit formula of the Hamiltonians generating one-dimensional discrete-time quantum walks is given. The formula is deduced by using the algebraic structure introduced before. The square of the Hamiltonian turns out to be an operator without, essentially, the ``coin register,'' and hence it can be compared with the one-dimensional continuous-time quantum walk. It is shown that, under a limit with respect to a parameter, which expresses the magnitude of the diagonal components of the unitary matrix defining the discrete-time quantum walks, the one-dimensional continuous-time quantum walk is obtained from operators defined through the Hamiltonians of the one-dimensional discrete-time quantum walks. Thus, this result can be regarded, in one-dimension, as a partial answer to a problem proposed by Ambainis.

arXiv:1306.3557•Interdisciplinary Information Sciences•Jan 1, 2013•1 citation

Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space

Vladimir Mikhailovich Busovikov, Alexander Nikolaevich Pechen, Vsevolod Zhanovich Sakbaev

We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators in the space of functions on a Hilbert space which are square integrable with respect to a shift-invariant measure. We study unitary groups of shift operators in the phase space and averaging of such shifts by Gaussian vectors, which form semigroups of self-adjoint contractions: we find conditions for their strong continuity and establish properties of their generators. Significant differences in their properties allow us to show the absence of the Fourier transform as a unitary transformation that implements the unitary equivalence of these compressive semigroups. Next, we prove the Taylor formula for a certain special subset of smooth functions for shifting to a non-finite vector. It allows us to prove convergence of quantum random walks in the coordinate representation to the evolution of a diffusion process, as well as convergence of quantum random walks in both coordinate and momentum representations to the evolution semigroup of a quantum oscillator in an infinite-dimensional phase space. We find the special essential common domain of generators of semigroups arising in averaging of random shift operators both in position and momentum representations. The invariance of this common domain with respect to both semigroups allows to establish properties of a convex combination of both generators. That convex combination are Hamiltonians of infinite-dimentional quantum oscillators. Thus, we obtain that a Weyl representation of a random walk in an infinite dimensional phase space describes the semigroup of self-adjoint contractions whose generator is the Hamiltonian of an infinite dimensional harmonic oscillator.

arXiv:2406.10699•Jun 15, 2024•1 citation

STRONGLY CONTINUOUS SEMIGROUP ON DECOHERENCE-FREE SUBALGEBRA (DFS) OF QUANTUM MARKOV SEMIGROUP (QMS)

Ezekiel Abiodun Oluwafemi, Micheal O. Ogundiran, U.S. Idiong +1 more

This study examines the strongly continuous semigroup on the decoherence-free subalgebra (DFS) of the QMS. Initially, we derived the infinitesimal generator (IG) for the DFS. A theorem characterizing a strongly continuous semigroup on the decoherence-free subalgebra was established. Strongly continuous semigroups offer a natural framework for modeling the time evolution of quantum systems influenced by a quantum Markov semigroup. When paired with the concept of decoherence-free subalgebras, they enable the modeling of systems that are immune to decoherence due to the specific nature of their interaction with the environment. The dynamics on these subalgebras are governed by the restricted semigroup, which can be analyzed through operator theory and the tools of quantum Markov processes. This approach is particularly relevant for applications in quantum error correction, quantum computing, and quantum control, where isolating parts of a system from decoherence effects is crucial.

FUDMA Journal of Sciences•Feb 15, 2026•0 citations
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 to continuous time walks. Then we construct the quantum Markov semigroup for quantum walks and characterize it in an invariant subalgebra. In the meanwhile, we obtain the limit distributions of the quantum walks in one-dimension with a proper scaling, which was obtained by Konno by a different method.
This paper establishes a robust link between quantum dynamics and classical ones by deriving probabilistic representation for both continuous time and discrete time quantum walks. We first adapt Molchanov formula, originally employed in the study of Schrodinger operators on multidimensional integer lattice, to characterize the evolution of continuous time quantum walks. Extending this framework, we develop a probabilistic method to represent discrete time quantum walks on an infinite integer line, bypassing the locality constraints that typically inhibit direct application of Molchanov formula. The validity of our representation is empirically confirmed through a benchmark analysis of the Hadamard walk, demonstrating high fidelity with traditional unitary evolution. Our results suggest that this probabilistic lens offer a powerful alternative for learning multidimensional quantum walks and provides new analytical pathways for investigating quantum systems via classical stochastic processes.
Discrete-time quantum walks are considered a counterpart of random walks and their study has been getting attention since around 2000. In this paper, we focus on a quantum walk which generates a probability distribution splitting to two parts. The quantum walker with two coin states spreads at points, represented by integers, and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.
We study the classical probability analogue of the unitary dilations of a quantum dynamical semigroup in quantum probability. Given a (not necessarily homogeneous) Markov chain in discrete time in a finite state space E, we introduce a second system, an environment, and a deterministic invertible time-homogeneous global evolution of the system E with this environment such that the original Markov evolution of E is realized by a proper choice of the initial random state of the environment. We also compare these dilations with the unitary dilations of a quantum dynamical semigroup in quantum probability: given a classical Markov semigroup, we show that it can be extended to a quantum dynamical semigroup for which we can find a quantum dilation to a group of ∗*∗-automorphisms admitting an invariant abelian subalgebra where this quantum dilation gives just our classical dilation.
An explicit formula of the Hamiltonians generating one-dimensional discrete-time quantum walks is given. The formula is deduced by using the algebraic structure introduced before. The square of the Hamiltonian turns out to be an operator without, essentially, the ``coin register,'' and hence it can be compared with the one-dimensional continuous-time quantum walk. It is shown that, under a limit with respect to a parameter, which expresses the magnitude of the diagonal components of the unitary matrix defining the discrete-time quantum walks, the one-dimensional continuous-time quantum walk is obtained from operators defined through the Hamiltonians of the one-dimensional discrete-time quantum walks. Thus, this result can be regarded, in one-dimension, as a partial answer to a problem proposed by Ambainis.
We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators in the space of functions on a Hilbert space which are square integrable with respect to a shift-invariant measure. We study unitary groups of shift operators in the phase space and averaging of such shifts by Gaussian vectors, which form semigroups of self-adjoint contractions: we find conditions for their strong continuity and establish properties of their generators. Significant differences in their properties allow us to show the absence of the Fourier transform as a unitary transformation that implements the unitary equivalence of these compressive semigroups. Next, we prove the Taylor formula for a certain special subset of smooth functions for shifting to a non-finite vector. It allows us to prove convergence of quantum random walks in the coordinate representation to the evolution of a diffusion process, as well as convergence of quantum random walks in both coordinate and momentum representations to the evolution semigroup of a quantum oscillator in an infinite-dimensional phase space. We find the special essential common domain of generators of semigroups arising in averaging of random shift operators both in position and momentum representations. The invariance of this common domain with respect to both semigroups allows to establish properties of a convex combination of both generators. That convex combination are Hamiltonians of infinite-dimentional quantum oscillators. Thus, we obtain that a Weyl representation of a random walk in an infinite dimensional phase space describes the semigroup of self-adjoint contractions whose generator is the Hamiltonian of an infinite dimensional harmonic oscillator.
This study examines the strongly continuous semigroup on the decoherence-free subalgebra (DFS) of the QMS. Initially, we derived the infinitesimal generator (IG) for the DFS. A theorem characterizing a strongly continuous semigroup on the decoherence-free subalgebra was established. Strongly continuous semigroups offer a natural framework for modeling the time evolution of quantum systems influenced by a quantum Markov semigroup. When paired with the concept of decoherence-free subalgebras, they enable the modeling of systems that are immune to decoherence due to the specific nature of their interaction with the environment. The dynamics on these subalgebras are governed by the restricted semigroup, which can be analyzed through operator theory and the tools of quantum Markov processes. This approach is particularly relevant for applications in quantum error correction, quantum computing, and quantum control, where isolating parts of a system from decoherence effects is crucial.