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:1909.13075

Asymptotic Behavior and Limiting Distribution of Quantum Walk on Cycles with General U(2) Coin by Reduced Characteristic Matrix Method

Majid Moradi, Mostafa Annabestani

Published Sep 28, 2019• 0 citations
arXivHTMLPDF

Abstract

We calculated reduced density characteristic matrix (RDCM) for quantum walk on cycles (QWC) to study asymptotic properties of the most general form of quantum walk on cycles with general $U(2)$ coin operator. As an example, entanglement temperature for general initial state has been calculated and compared to previous results. Also, we have modified RDCM to derive analytical expression for general form of limiting distribution (LD).

DOI
10.48550/arxiv.1909.13075

Related papers

Asymptotic reduced density matrix of discrete-time quantum walks

Mostafa Annabestani

Abstract In this article we show that for any quantum walker with m -dimensional coin subspace, we have m 2 × m 2 specific constant matrix C where it completely determines the asymptotic reduced density matrix of the walker. We show that for any initial state with P 0 projector, the reduced density matrix can be obtained by T r 1 P 0 ⊗ I C or equivalently T r 2 I ⊗ P 0 C . It is worth mentioning that the characteristic matrix C is independent of the initial state and just depends on the coin operator, so by finding this matrix for a specific type of quantum walks (QW), the long-time behavior of it, such as local state of the coin after a long-time walk and asymptotic entanglement between the coin and position will be completely known for any initial state. We have found the characteristic matrix C for the general coin operator, U 2 , as well as the exact form of this matrix for the local initial state.

arXiv:1909.09214•Journal of Physics A Mathematical and Theoretical•Feb 28, 2020•0 citations

Asymptotic evolution of quantum walks on the N -cycle subject to decoherence on both the coin and position degrees of freedom

Chaobin Liu, Nelson Petulante

Consider a discrete-time quantum walk on the $N$-cycle subject to decoherence both on the coin and the position degrees of freedom. By examining the evolution of the density matrix of the system, we derive some conclusions about the asymptotic behavior of the system. When $N$ is odd, the density matrix of the system tends, in the long run, to the maximally mixed state, independent of the initial state. When $N$ is even, although the behavior of the system is not necessarily asymptotically stationary, in this case too an explicit formulation is obtained of the asymptotic dynamics of the system. Moreover, this approach enables us to specify the limiting behavior of the mutual information, viewed as a measure of quantum entanglement between subsystems (coin and walker). In particular, our results provide efficient theoretical confirmation of the findings of previous authors, who have arrived at their results through extensive numerical simulations. Our results can be attributed to an important theorem which, for a generalized random unitary operation, explicitly identifies the structure of all of its eigenspaces corresponding to eigenvalues of unit modulus.

arXiv:1104.2653•Physical Review A•Jul 18, 2011•21 citations

Asymptotic entanglement in 2D quantum walks

Mostafa Annabestani, Mohammad Reza Abolhasani, Gonzalo Abal

The evolution operator of a discrete-time quantum walk involves a conditional shift in position space which entangles the 'coin' and position degrees of freedom of the walker. After several steps, the coin-position entanglement (CPE) converges to a well-defined value which depends on the initial state. In this work we provide an analytical method which allows for the exact calculation of the asymptotic reduced density operator and the corresponding CPE for a discrete-time quantum walk on a two-dimensional lattice. We use the von Neumann entropy of the reduced density operator as an entanglement measure. The method is applied to the case of a Hadamard walk for which the dependence of the resulting CPE on initial conditions is obtained. Initial states leading to the maximum or minimum CPE are identified and the relation between the coin or position entanglement present in the initial state of the walker and the final level of CPE is discussed. The CPE obtained from separable initial states satisfies an additivity property in terms of CPE of the corresponding one-dimensional cases. Non-local initial conditions are also considered and we find that the extreme case of an initial uniform position distribution leads to the largest CPE variation.

arXiv:0901.1188•Journal of Physics A Mathematical and Theoretical•Jan 27, 2010•45 citations

Entanglement in coined quantum walks on regular graphs

Ivens Carneiro, Meng Loo, Xibai Xu +3 more

Quantum walks, both discrete (coined) and continuous time, form the basis of several recent quantum algorithms. Here we use numerical simulations to study the properties of discrete, coined quantum walks. We investigate the variation in the entanglement between the coin and the position of the particle by calculating the entropy of the reduced density matrix of the coin. We consider both dynamical evolution and asymptotic limits for coins of dimensions from two to eight on regular graphs. For low coin dimensions, quantum walks which spread faster (as measured by the mean square deviation of their distribution from uniform) also exhibit faster convergence towards the asymptotic value of the entanglement between the coin and particle's position. For high-dimensional coins, the DFT coin operator is more efficient at spreading than the Grover coin. We study the entanglement of the coin on regular finite graphs such as cycles, and also show that on complete bipartite graphs, a quantum walk with a Grover coin is always periodic with period four. We generalize the 'glued trees' graph used by Childs et al (2003 Proc. STOC , pp 59–68) to higher branching rate (fan out) and verify that the scaling with branching rate and with tree depth is polynomial.

arXiv:quant-ph/0504042•New Journal of Physics•Jul 12, 2005•147 citations

Asymptotic evolution of quantum walks with random coin

André Ahlbrecht, Holger Vogts, Albert H. Werner +1 more

We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., nonrandom) case, we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other. For example, a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense, with any lattice dimension and coin dimension. We find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time, namely as the distribution of the discrete time analog of the group velocity. In the random case, we let a Markov chain (control process) pick in each step one of finitely many unitary walks, in the sense described above. In ballistic order, we find a nonrandom drift which depends only on the mean of the control process and not on the initial state. In diffusive scaling, the limiting distribution is asymptotically Gaussian, with a covariance matrix (diffusion matrix) depending on momentum. The diffusion matrix depends not only on the mean but also on the transition rates of the control process. In the nonrandom limit, i.e., when the coins chosen are all very close or the transition rates of the control process are small, leading to long intervals of ballistic evolution, the diffusion matrix diverges. Our method is based on spatial Fourier transforms, and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum.

arXiv:1009.2019•Journal of Mathematical Physics•Apr 1, 2011•119 citations

Temperature and entanglement of the three-state quantum walk

Luísa Toledo Tude, M. C. de Oliveira

Abstract In the present work, the evolution of a three state quantum walk without decoherence is investigated. Despite being a closed quantum system under unitary evolution, its Hilbert space can be divided in two subspaces, which enables the analysis of the subsystems (the coin or the walker) as an open system in contact with a reservoir. We calculate the asymptotic reduced density matrix of the coin space of the three-state quantum walk in an infinite line, and use that result to analyze the entanglement between the chirality, and position space. We calculate the von Neumann entropy and the entanglement temperature per mean energy of the system in the asymptotic limit.

arXiv:2012.07904•Quantum Science and Technology•Apr 25, 2022•3 citations
We calculated reduced density characteristic matrix (RDCM) for quantum walk on cycles (QWC) to study asymptotic properties of the most general form of quantum walk on cycles with general U(2)U(2)U(2) coin operator. As an example, entanglement temperature for general initial state has been calculated and compared to previous results. Also, we have modified RDCM to derive analytical expression for general form of limiting distribution (LD).
Abstract In this article we show that for any quantum walker with m -dimensional coin subspace, we have m 2 × m 2 specific constant matrix C where it completely determines the asymptotic reduced density matrix of the walker. We show that for any initial state with P 0 projector, the reduced density matrix can be obtained by T r 1 P 0 ⊗ I C or equivalently T r 2 I ⊗ P 0 C . It is worth mentioning that the characteristic matrix C is independent of the initial state and just depends on the coin operator, so by finding this matrix for a specific type of quantum walks (QW), the long-time behavior of it, such as local state of the coin after a long-time walk and asymptotic entanglement between the coin and position will be completely known for any initial state. We have found the characteristic matrix C for the general coin operator, U 2 , as well as the exact form of this matrix for the local initial state.
Consider a discrete-time quantum walk on the NNN-cycle subject to decoherence both on the coin and the position degrees of freedom. By examining the evolution of the density matrix of the system, we derive some conclusions about the asymptotic behavior of the system. When NNN is odd, the density matrix of the system tends, in the long run, to the maximally mixed state, independent of the initial state. When NNN is even, although the behavior of the system is not necessarily asymptotically stationary, in this case too an explicit formulation is obtained of the asymptotic dynamics of the system. Moreover, this approach enables us to specify the limiting behavior of the mutual information, viewed as a measure of quantum entanglement between subsystems (coin and walker). In particular, our results provide efficient theoretical confirmation of the findings of previous authors, who have arrived at their results through extensive numerical simulations. Our results can be attributed to an important theorem which, for a generalized random unitary operation, explicitly identifies the structure of all of its eigenspaces corresponding to eigenvalues of unit modulus.
The evolution operator of a discrete-time quantum walk involves a conditional shift in position space which entangles the 'coin' and position degrees of freedom of the walker. After several steps, the coin-position entanglement (CPE) converges to a well-defined value which depends on the initial state. In this work we provide an analytical method which allows for the exact calculation of the asymptotic reduced density operator and the corresponding CPE for a discrete-time quantum walk on a two-dimensional lattice. We use the von Neumann entropy of the reduced density operator as an entanglement measure. The method is applied to the case of a Hadamard walk for which the dependence of the resulting CPE on initial conditions is obtained. Initial states leading to the maximum or minimum CPE are identified and the relation between the coin or position entanglement present in the initial state of the walker and the final level of CPE is discussed. The CPE obtained from separable initial states satisfies an additivity property in terms of CPE of the corresponding one-dimensional cases. Non-local initial conditions are also considered and we find that the extreme case of an initial uniform position distribution leads to the largest CPE variation.
Quantum walks, both discrete (coined) and continuous time, form the basis of several recent quantum algorithms. Here we use numerical simulations to study the properties of discrete, coined quantum walks. We investigate the variation in the entanglement between the coin and the position of the particle by calculating the entropy of the reduced density matrix of the coin. We consider both dynamical evolution and asymptotic limits for coins of dimensions from two to eight on regular graphs. For low coin dimensions, quantum walks which spread faster (as measured by the mean square deviation of their distribution from uniform) also exhibit faster convergence towards the asymptotic value of the entanglement between the coin and particle's position. For high-dimensional coins, the DFT coin operator is more efficient at spreading than the Grover coin. We study the entanglement of the coin on regular finite graphs such as cycles, and also show that on complete bipartite graphs, a quantum walk with a Grover coin is always periodic with period four. We generalize the 'glued trees' graph used by Childs et al (2003 Proc. STOC , pp 59–68) to higher branching rate (fan out) and verify that the scaling with branching rate and with tree depth is polynomial.
We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., nonrandom) case, we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other. For example, a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense, with any lattice dimension and coin dimension. We find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time, namely as the distribution of the discrete time analog of the group velocity. In the random case, we let a Markov chain (control process) pick in each step one of finitely many unitary walks, in the sense described above. In ballistic order, we find a nonrandom drift which depends only on the mean of the control process and not on the initial state. In diffusive scaling, the limiting distribution is asymptotically Gaussian, with a covariance matrix (diffusion matrix) depending on momentum. The diffusion matrix depends not only on the mean but also on the transition rates of the control process. In the nonrandom limit, i.e., when the coins chosen are all very close or the transition rates of the control process are small, leading to long intervals of ballistic evolution, the diffusion matrix diverges. Our method is based on spatial Fourier transforms, and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum.
Abstract In the present work, the evolution of a three state quantum walk without decoherence is investigated. Despite being a closed quantum system under unitary evolution, its Hilbert space can be divided in two subspaces, which enables the analysis of the subsystems (the coin or the walker) as an open system in contact with a reservoir. We calculate the asymptotic reduced density matrix of the coin space of the three-state quantum walk in an infinite line, and use that result to analyze the entanglement between the chirality, and position space. We calculate the von Neumann entropy and the entanglement temperature per mean energy of the system in the asymptotic limit.