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arXiv:2311.15068

Modelling of q-deformed harmonic oscilator on quantum computer

M. I. Samar, V. M. Tkachuk

Published Nov 25, 2023• 1 citation
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Abstract

We present a method of a quantum simulation of a quantum harmonic oscillator in a special case of the deformed commutation relation, which corresponds to the so-called q-deformed oscillator on an IBM quantum computer. Using the method of detection of energy levels of a spin system on a quantum computer by probe spin evolution we obtain the energy levels of both the q-deformed quantum harmonic and anharmonic oscillators.

DOI
10.48550/arxiv.2311.15068

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We present a method of a quantum simulation of a quantum harmonic oscillator in a special case of the deformed commutation relation, which corresponds to the so-called q-deformed oscillator on an IBM quantum computer. Using the method of detection of energy levels of a spin system on a quantum computer by probe spin evolution we obtain the energy levels of both the q-deformed quantum harmonic and anharmonic oscillators.
The interpretation of the q-deformed 1-D quantum harmonic oscillator is investigated for two definitions of q-deformation.This investigation is achieved by using Zaslavskii's method to obtain the Heisenberg equations of motion (quantum Liouville equations) in the undeformed phase space.These quantum Liouville equations exhibit a non-commutative geometry produce from the existence of the dilatation operator which is inherent in the q-deformation process.The classical limits of these equations are obtained by applying a special classical limiting condition to produce the classical Liouville equations of the q-deformed oscillator.These classical Liouville equations are solved by using the method of characteristics in order to obtain the classical probability distribution functions for this system.The 2-D and 3-D behaviors of these functions were then investigated using a computer visualization method.The results of the mathematical derivations together with the computer visualization method show that the classical limit of the quantum Liouville equations for the q-deformed 1-D quantum harmonic oscillator are statistical in nature where the nonlinearity parameter for the q-deformed oscillator is connected with h .This result conforms to that obtained by Ghosh et al. for the undeformed 1-D quantum harmonic oscillator.The obtained classical probability distribution functions exhibit whorl shapes that evolve with time in phase space that are similar to the shapes obtained for the 1-D classical q-deformed oscillator.These whorl shapes in phase space are similar to those introduced by Milburn for the 1-D classical anharmonic oscillator.This similarity results from the fact that the anharmonicity itself represents a kind of deformation with a frequency that is a function of amplitude.
A q-deformed version of standard quantum mechanics in the coordinate Schrodinger picture is obtained by replacing the ordinary coordinate derivative by the so-called q-discrete derivative as the representative of the momentum operator. The chosen q-discrete derivative is symmetric with respect to the exchange of q and q -1 . Under the usually adopted assumptions a q-deformed Schrodinger equation is derived for a harmonic oscillator. The complete set of eigenfunctions can be explicitly constructed as special q-functions and the corresponding energy eigenvalues are identical to those obtained by Biedenharn in his pioneering work (1989). This q-deformed oscillator exhibits a rich novel structure including dynamical symmetry and in the limit q to 1 it reveals some hitherto unknown features of the harmonic oscillator eigenfunctions.
The Liouville equation for the q-deformed 1-D classical harmonic oscillator is derived for two definitions of q-deformation.This derivation is achieved by using two different representations for the q-deformed Hamiltonian of this oscillator corresponding to undeformed and deformed phase spaces.The resulting Liouville equation is solved by using the method of characteristics in order to obtain the classical probability distribution function for this system.The 2-D and 3-D behaviors of this function are then investigated using a computer visualization method.The results are compared with those for the classical anharmonic oscillator.This comparison reveals that there are some similarities between these two models, where the results for the q-deformed oscillator exhibit similar whorl shapes that evolve with time as for the anharmonic oscillator.It is concluded that studying the Liouville dynamics gives more details about the physical nature of q-deformation than using the equation of motion method.It is also concluded that this result could have reflections on the interpretation of the quantized version of this q-deformed oscillator.
Finding energy levels of a quantum system is an important problem that can be solved using quantum computers. We showed that transition energies of a quantum system are related with the evolution of mean value of a physical quantity. In particular case when the operator of a physical quantity anticommutes with the Hamiltonian of a system we found that studies of the evolution of the mean value of this physical quantity give a possibility to determine the energy levels of the system and their degeneracy. On the basis of the results the energy levels of spin systems were detected on IBMs quantum computers ibmq manila and ibmq melbourne. It is worth noting that spin systems are one of the most suitable systems for modeling them on a quantum computer. We considered a spin in magnetic field, spin chain with Ising interaction, spin square-lattice with isotropic and spatially anisotropic Ising interactions. The results obtained on the quantum devices correspond to the analytical one. It is important to note that finding the maximal or the minimal energy of the Ising model on square lattice with spatially anisotropic interaction is not a trivial problem. Therefore, the proposed algorithm for obtaining the energy levels opens a possibility to achieve the quantum supremacy in solving this problem on many qubit quantum computers.
The energy level structure of the q deformed harmonic oscillator in a finite dimensional Hilbert space is studied.The Coherent states of the q deformed harmonic oscillator in a finite dimensional Hilbert space are constructed.It is shown that the energy level structure and the dynamic action are very different from that of usual harmonic oscillator due to the finiteness of the dimension of Hilbert space and the deformation parameter′s influence,and the coherent states are different from the Glauber coherent states.