Neural Networks Based on the Eigenstates of the Quantum Harmonic Oscillator

2014 ◽  
pp. 221-249
Author(s):  
Gerasimos G. Rigatos
Keyword(s):  
Neural Networks ◽  
Harmonic Oscillator ◽  
Quantum Harmonic Oscillator

scholarly journals Feed-Forward Neural Networks Based on the Eigenstates of the Quantum Harmonic Oscillator

2006 ◽  
Vol 10 (4) ◽  
pp. 567-577
Author(s):  
Gerasimos Rigatos ◽  
Keyword(s):  
Neural Networks ◽  
Harmonic Oscillator ◽  
Hermite Polynomials ◽  
Basis Functions ◽  
System Modelling ◽  
General Category ◽  
Quantum Harmonic Oscillator ◽  
Feed Forward ◽  
Change Of Scale ◽  
Feed Forward Neural Networks

The paper introduces feed-forward neural networks where the hidden units employ orthogonal Hermite polynomials for their activation functions. The proposed neural networks have some interesting properties: (i) the basis functions are invariant under the Fourier transform, subject only to a change of scale, (ii) the basis functions are the eigenstates of the quantum harmonic oscillator, and stem from the solution of Schrödinger’s diffusion equation. The proposed feed-forward neural networks belong to the general category of nonparametric estimators and can be used for function approximation, system modelling and image processing.

Attractors and Energy Spectrum of Neural Structures Based on the Model of the Quantum Harmonic Oscillator

2009 ◽  
pp. 376-410
Author(s):  
G.G. Rigatos ◽  
S.G. Tzafestas
Keyword(s):  
Neural Networks ◽  
Quantum Mechanics ◽  
Brownian Motion ◽  
Harmonic Oscillator ◽  
Quantum Physics ◽  
Neural Model ◽  
Neural Computation ◽  
Quantum Harmonic Oscillator ◽  
Brain Functioning ◽  
Computing Machines

Neural computation based on principles of quantum mechanics can provide improved models of memory processes and brain functioning and is of primary importance for the realization of quantum computing machines. To this end, this chapter studies neural structures with weights that follow the model of the quantum harmonic oscillator. The proposed neural networks have stochastic weights which are calculated from the solution of Schrödingers equation under the assumption of a parabolic (harmonic) potential. These weights correspond to diffusing particles, which interact with each other as the theory of Brownian motion (Wiener process) predicts. The learning of the stochastic weights (convergence of the diffusing particles to an equilibrium) is analyzed. In the case of associative memories the proposed neural model results in an exponential increase of patterns storage capacity (number of attractors). It is also shown that conventional neural networks and learning algorithms based on error gradient can be conceived as a subset of the proposed quantum neural structures. Thus, the complementarity between classical and quantum physics is also validated in the field of neural computation.

Neural Structures Using the Eigenstates of a Quantum Harmonic Oscillator

2006 ◽  
Vol 13 (01) ◽  
pp. 27-41 ◽  
Author(s):  
Gerasimos Rigatos ◽  
Spyros Tzafestas
Keyword(s):  
Neural Networks ◽  
Harmonic Oscillator ◽  
Hermite Polynomials ◽  
Feedforward Neural Networks ◽  
Basis Functions ◽  
System Modelling ◽  
Quantum Harmonic Oscillator ◽  
Wave Nature ◽  
Change Of Scale ◽  
The Fourier Transform

The main result of the paper is the use of orthogonal Hermite polynomials as the basis functions of feedforward neural networks. The proposed neural networks have some interesting properties: (i) the basis functions are invariant under the Fourier transform, subject only to a change of scale, (ii) the basis functions are the eigenstates of the quantum harmonic oscillator, and stem from the solution of Schrödinger's diffusion equation. The proposed feed-forward neural networks demonstrate the particle-wave nature of information and can be used in nonparametric estimation. Possible applications of the proposed neural networks include function approximation, image processing and system modelling.

scholarly journals Quadratic open quantum harmonic oscillator

2020 ◽  
Vol 110 (7) ◽  
pp. 1759-1782
Author(s):  
Ameur Dhahri ◽  
Franco Fagnola ◽  
Hyun Jae Yoo
Keyword(s):  
Harmonic Oscillator ◽  
Quantum Harmonic Oscillator ◽  
Open Quantum

scholarly journals Fast magnetohydrodynamic oscillation of longitudinally inhomogeneous prominence threads: an analogue with quantum harmonic oscillator

2014 ◽  
Vol 565 ◽  
pp. A35 ◽  
Author(s):  
S. N. Lomineishvili ◽  
T. V. Zaqarashvili ◽  
I. Zhelyazkov ◽  
A. G. Tevzadze
Keyword(s):  
Harmonic Oscillator ◽  
Quantum Harmonic Oscillator

scholarly journals The quantum harmonic oscillator as a Zariski geometry

2014 ◽  
Vol 165 (6) ◽  
pp. 1149-1168 ◽  
Author(s):  
Vinesh Solanki ◽  
Dmitry Sustretov ◽  
Boris Zilber
Keyword(s):  
Harmonic Oscillator ◽  
Quantum Harmonic Oscillator

scholarly journals Reading a Qubit Quantum State with a Quantum Meter: Time Unfolding of Quantum Darwinism and Quantum Information Flux

2019 ◽  
Vol 26 (04) ◽  
pp. 1950023
Author(s):  
Salvatore Lorenzo ◽  
Mauro Paternostro ◽  
G. Massimo Palma
Keyword(s):  
Quantum Information ◽  
Harmonic Oscillator ◽  
Quantum System ◽  
Quantum State ◽  
Common Theme ◽  
Quantum Harmonic Oscillator ◽  
Collision Model ◽  
Single Qubit ◽  
Quantum Darwinism

Quantum non-Markovianity and quantum Darwinism are two phenomena linked by a common theme: the flux of quantum information between a quantum system and the quantum environment it interacts with. In this work, making use of a quantum collision model, a formalism initiated by Sudarshan and his school, we will analyse the efficiency with which the information about a single qubit gained by a quantum harmonic oscillator, acting as a meter, is transferred to a bosonic environment. We will show how, in some regimes, such quantum information flux is inefficient, leading to the simultaneous emergence of non-Markovian and non-darwinistic behaviours.

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