Quantum optical random walk: Quantization rules and quantum simulation of asymptotics

Ellinas, Demosthenes; Smyrnakis, Ioannis

doi:10.1103/physreva.76.022333

Cited by 11 publications

(12 citation statements)

References 36 publications

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“…Quantum walks (QWs) were introduced as a quantization of random walks (RWs) [7]. QWs are determined by a graph, its induced Hilbert space H and a unitary time evolution operator on H. The amplitude of quantum walker is obtained by this unitary iteration to some initial state.…”

Section: Background and Notationmentioning

confidence: 99%

A Characterization of the Graphs to Induce Periodic Grover Walk

Yoshie

2017

Preprint

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Recently, a research on quantum walks has been developed in various areas. In this paper we focus on the periodicity of the Grover walk which is one of quantum walk on the discrete graphs. Then we found some special graphs to induce a periodic Grover walk: At some time k, the quantum state ϕ k returns its initial quantum state ϕ 0 . Our purpose is to characterize graphs which induce a k-periodic Grover walk for a fixed integer k. We do it for k = 2, 3, 4, 5 and obtain a necessary condition for odd k.

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Section: Background and Notationmentioning

confidence: 99%

A Characterization of the Graphs to Induce Periodic Grover Walk

Yoshie

2017

Preprint

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“…OQWs are formulated as a quantum Markov chain on lattices or graphs. Unlike unitary quantum walks (UQWs) [6,8] where dissipation and decoherence effects need to be minimised or eliminated when dealing with quantum systems [10,14], in OQWs, these effects are naturally included into the description of the dynamics of the open quantum "walker". OQWs deal with density matrices instead of pure states.…”

Section: Introductionmentioning

confidence: 99%

A Generalized Quantum Optical Scheme for Implementing Open Quantum Walks

Zungu

Sinayskiy

Petruccione

2020

Electron. Proc. Theor. Comput. Sci.

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Open quantum walks (OQWs) are a new type of quantum walks which are entirely driven by the dissipative interaction with external environments and are formulated as completely positive trace-preserving maps on graphs. A generalized quantum optical scheme for implementing OQWs that includes non-zero temperature of the environment is suggested. In the proposed quantum optical scheme, a two-level atom plays the role of the "walker", and the Fock states of the cavity mode correspond to the lattice sites for the "walker". Using the small unitary rotations approach the effective dynamics of the system is shown to be an OQW. For the chosen set of parameters, an increase in the temperature of the environment causes the system to reach the asymptotic distribution much faster compared to the scheme proposed earlier where the temperature of the environment is zero. For this case the asymptotic distribution is given by a steady Gaussian distribution.

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“…This constitutes a dynamic realization of the coin tossing process. The diagonal elements of the resulting walker's density matrix E cl (ρ w ) =Tr c (AdV cl (ρ c ⊗ ρ w )), realizes the CRW on the integers [5], [6].…”

Section: Introductionmentioning

confidence: 99%

Discrete Randomness in Discrete Time Quantum Walk: Study Via Stochastic Averaging

Ellinas¹,

Bracken²,

Smyrnakis³

2012

Reports on Mathematical Physics

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The role of classical noise in quantum walks (QW) on integers is investigated in the form of discrete dichotomic random variable affecting its reshuffling matrix parametrized as a SU 2)/U (1) coset element. Analysis in terms of quantum statistical moments and generating functions, derived by the completely positive trace preserving (CPTP) map governing evolution, reveals a pronounced eventual transition in walk's diffusion mode, from a quantum ballistic regime with rate O(t) to a classical diffusive regime with rate O( √ t), when condition (strength of noise parameter ) 2 ×(number of steps) = 1, is satisfied. The role of classical randomness is studied showing that the randomized QW, when treated on the stochastic average level by means of an appropriate CPTP averaging map, turns out to be equivalent to a novel quantized classical walk without randomness. This result emphasizes the dual role of quantization/randomization in the context of classical random walk.

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Quantum optical random walk: Quantization rules and quantum simulation of asymptotics

Cited by 11 publications

References 36 publications

A Characterization of the Graphs to Induce Periodic Grover Walk

A Characterization of the Graphs to Induce Periodic Grover Walk

A Generalized Quantum Optical Scheme for Implementing Open Quantum Walks

Discrete Randomness in Discrete Time Quantum Walk: Study Via Stochastic Averaging

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