Open Quantum Random Walks with Decoherence on Coins with n Degrees of Freedom

Xiong, Sheng; Yang, Wei-Shih

doi:10.1007/s10955-013-0772-2

Cited by 14 publications

(8 citation statements)

References 9 publications

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“…The diverse dynamical behaviour of OQWs has been extensively studied [10][11][12][16][17][18][19][20][21]. The asymptotic analysis of OQWs leads to a central limit theorem [22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Microscopic derivation of open quantum walks

Sinayskiy

Petruccione

2015

Phys. Rev. A

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Open Quantum Walks (OQWs) are exclusively driven by dissipation and are formulated as completely positive trace preserving (CPTP) maps on underlying graphs. The microscopic derivation of discrete and continuous in time OQWs is presented. It is assumed that connected nodes are weakly interacting via a common bath. The resulting reduced master equation of the quantum walker on the lattice is in the generalised master equation form. The time discretisation of the generalised master equation leads to the OQWs formalism. The explicit form of the transition operators establishes a connection between dynamical properties of the OQWs and thermodynamical characteristics of the environment. The derivation is demonstrated for the examples of the OQW on a circle of nodes and on a finite chain of nodes. For both examples a transition between diffusive and ballistic quantum trajectories is observed and found to be related to the temperature of the bath.

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“…The diverse dynamical behaviour of OQWs has been extensively studied [10][11][12][16][17][18][19][20][21]. The asymptotic analysis of OQWs leads to a central limit theorem [22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Microscopic derivation of open quantum walks

Sinayskiy

Petruccione

2015

Phys. Rev. A

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“…The next lemma follows a similar argument about the convergence of convex combinations of operators in [LP11b] and [LP11a], as well as in [XY13]: Proof. If x ∈ V can be written with generalized eigenvectors of A as…”

Section: Chapter 4 Convergence Of Convex Combination Operators 41 Eimentioning

confidence: 85%

A Perron–Frobenius Type of Theorem for Quantum Operations

2017

Self Cite

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Professor Wei-Shih Yang, Chair Quantum random walks are a generalization of classical Markovian random walks to a quantum mechanical or quantum computing setting. Quantum walks have promising applications but are complicated by quantum decoherence. We prove that the long-time limiting behavior of the class of quantum operations which are the convex combination of norm one operators is governed by the eigenvectors with norm one eigenvalues which are shared by the operators. This class includes all operations formed by a coherent operation with positive probability of orthogonal measurement at each step. We also prove that any operation that has range contained in a low enough dimension subspace of the space of density operators has limiting behavior isomorphic to an associated Markov chain. A particular class of such operations are coherent operations followed by an orthogonal measurement.

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“…More recently, Attal et al 23 showed the quantum version of central limit theorem for open quantum random walks. Subsequently, Xiong and Yang 24 extended their results and showed the scaling limit of a partially open quantum random walk converges to a convex combination of Gaussian distributions.…”

Section: Behavior Of Quantum Walks On Z Dmentioning

confidence: 93%

The localization of quantum random walks on sierpinski gaskets

Zhao,

Yang

2021

Preprint

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We consider the discrete time quantum random walks on a Sierpinski gasket. We study the hitting probability as the level of fractal goes to infinity in terms of their localization exponents β w , total variation exponents δ w and relative entropy exponents η w . We define and solve the amplitude Green functions recursively when the level of the fractal graph goes to infinity. We obtain exact recursive formulas for the amplitude Green functions, based on which the hitting probabilities and expectation of the first-passage time are calculated. Using the recursive formula with the aid of Monte Carlo integration, we evaluate their numerical values. We also show that when the level of the fractal graph goes to infinity, with probability 1, the quantum random walks will return to origin, i.e., the quantum walks on Sierpinski gasket are recurrent.

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Open Quantum Random Walks with Decoherence on Coins with n Degrees of Freedom

Cited by 14 publications

References 9 publications

Microscopic derivation of open quantum walks

Microscopic derivation of open quantum walks

A Perron–Frobenius Type of Theorem for Quantum Operations

The localization of quantum random walks on sierpinski gaskets

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