The Phase–Space Approach to time Evolution of Quantum States in Confined Systems: the Spectral Split–Operator Method

Kołaczek, Damian; Spisak, B. J.; Wołoszyn, M.

doi:10.2478/amcs-2019-0032

Cited by 5 publications

(3 citation statements)

References 43 publications

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“…is the time increment. Calculation-efficient form of the time evolution operator can be derived by applying the symmetric Strang splitting formula 89 – 91 Using the partial Fourier transforms in the first or second variable defined as follows, one can obtain a formula for a single step of the time evolution of the WDF in the form where the auxiliary function is defined as the central difference of the potential energies, namely Since Eq. ( 4 ) is defined for , in order to perform numerical computation the phase space is limited to the box of size with periodic boundary conditions imposed by the numerical method.…”

Section: Methodsmentioning

confidence: 99%

Dynamical entropic measure of nonclassicality of phase-dependent family of Schrödinger cat states

Kalka,

Spisak,

Woźniak

et al. 2023

Sci Rep

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The phase-space approach based on the Wigner distribution function is used to study the quantum dynamics of the three families of the Schrödinger cat states identified as the even, odd, and Yurke–Stoler states. The considered states are formed by the superposition of two Gaussian wave packets localized on opposite sides of a smooth barrier in a dispersive medium and moving towards each other. The process generated by this dynamics is analyzed regarding the influence of the barrier parameters on the nonclassical properties of these states in the phase space below and above the barrier regime. The performed analysis employs entropic measure resulting from the Wigner–Rényi entropy for the fixed Rényi index. The universal relation of this entropy for the Rényi index equal one half with the nonclassicality parameter understood as a measure of the negative part of the Wigner distribution function is proved. This relation is confirmed in the series of numerical simulations for the considered states. Furthermore, the obtained results allowed the determination of the lower bound of the Wigner–Rényi entropy for the Rényi index greater than or equal to one half.

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Section: Methodsmentioning

confidence: 99%

Dynamical entropic measure of nonclassicality of phase-dependent family of Schrödinger cat states

Kalka,

Spisak,

Woźniak

et al. 2023

Sci Rep

View full text Add to dashboard Cite

show abstract

“…Although this basic splitting formula is sufficient for our needs, let us note that the high-order variants of the splitting formula are more precise, but also unavoidably more complicated when applied to the Moyal equation, as has been discussed in Ref. 68 . Going back to the symmetric Strang splitting formula ( 18 ), it can be noted that each operator is unitarily equivalent to some multiplication operator owing to the adequate Fourier transform, specifically and where the symbol denotes the ordinary Fourier transform for the x variable with dual variable .…”

Section: Theoretical Frameworkmentioning

confidence: 99%

Phase-space studies of backscattering diffraction of defective Schrödinger cat states

Kołaczek

Spisak

Wołoszyn

2021

Sci Rep

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The coherent superposition of two well separated Gaussian wavepackets, with defects caused by their imperfect preparation, is considered within the phase-space approach based on the Wigner distribution function. This generic state is called the defective Schrödinger cat state due to this imperfection which significantly modifies the interference term. Propagation of this state in the phase space is described by the Moyal equation which is solved for the case of a dispersive medium with a Gaussian barrier in the above-barrier reflection regime. Formally, this regime constitutes conditions for backscattering diffraction phenomena. Dynamical quantumness and the degree of localization in the phase space of the considered state as a function of its imperfection are the subject of the performed analysis. The obtained results allow concluding that backscattering communication based on the defective Schrödinger cat states appears to be feasible with existing experimental capabilities.

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“…Artificial neural networks and other tools of machine learning still evolve, and play a more and more important role in data processing. Today, the whole world is on the verge of a quantum revolution where data are going to be encoded as quantum states and processed with the use of laws of quantum mechanics (Nielsen and Chuang, 2010;Kołaczek et al, 2019) and machine learning methods are also developed for quantum computational systems. Researchers deal with the different kinds of quantum machine learning (Biamonte et al, 2017;Schuld et al, 2014;, e.g., quantum neural networks (Narayanan and Menneer, 2000;Zoufal et al, 2019), quantum kNN * Corresponding author methods (Wiebe et al, 2015), quantum self-organized maps (Weigang, 1998), or the quantum k-means method (Veenman and Reinders, 2005).…”

Section: Introductionmentioning

confidence: 99%

Basic quantum circuits for classification and approximation tasks

Wiśniewska¹,

Sawerwain²,

Obuchowicz³

2020

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We discuss a quantum circuit construction designed for classification. The circuit is built of regularly placed elementary quantum gates, which implies the simplicity of the presented solution. The realization of the classification task is possible after the procedure of supervised learning which constitutes parameter optimization of Pauli gates. The process of learning can be performed by a physical quantum machine but also by simulation of quantum computation on a classical computer. The parameters of Pauli gates are selected by calculating changes in the gradient for different sets of these parameters. The proposed solution was successfully tested in binary classification and estimation of basic non-linear function values, e.g., the sine, the cosine, and the tangent. In both the cases, the circuit construction uses one or more identical unitary operations, and contains only two qubits and three quantum gates. This simplicity is a great advantage because it enables the practical implementation on quantum machines easily accessible in the nearest future.

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The Phase–Space Approach to time Evolution of Quantum States in Confined Systems: the Spectral Split–Operator Method

Cited by 5 publications

References 43 publications

Dynamical entropic measure of nonclassicality of phase-dependent family of Schrödinger cat states

Dynamical entropic measure of nonclassicality of phase-dependent family of Schrödinger cat states

Phase-space studies of backscattering diffraction of defective Schrödinger cat states

Basic quantum circuits for classification and approximation tasks

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