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Computer Optics

Mathematical modeling of processes in quantum computer elements based on methods of quantum theory to improve their efficiency

Alexander Alexandrovich Biryukov, Марк Александрович Шлеенков

Published Dec 1, 2024• 1 citation
Publisher

Abstract

The paper studies entangled states of two qubits interacting with each other and with an electromagnetic field. The state of the qubits is determined by a statistical density matrix. The degree of entanglement of the state is characterized by the Peres-Gorodeckii (PG) parameter. The statistical density matrix and its evolution are determined in the energy representation within the framework of the path integral formalism. The obtained equations determine the dependence of the PG parameter on the parameters of qubit dipole-dipole interaction and the acting electromagnetic field. The results of numerical calculations are presented in graphs for the PG parameter. It is shown that it is possible to choose parameters corresponding to qubit states with a high degree of entanglement (0.99).

DOI
10.18287/2412-6179-co-1489

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The paper studies entangled states of two qubits interacting with each other and with an electromagnetic field. The state of the qubits is determined by a statistical density matrix. The degree of entanglement of the state is characterized by the Peres-Gorodeckii (PG) parameter. The statistical density matrix and its evolution are determined in the energy representation within the framework of the path integral formalism. The obtained equations determine the dependence of the PG parameter on the parameters of qubit dipole-dipole interaction and the acting electromagnetic field. The results of numerical calculations are presented in graphs for the PG parameter. It is shown that it is possible to choose parameters corresponding to qubit states with a high degree of entanglement (0.99).
In this paper we study the quantum entanglement of two identical qubits which interact with electromagnetic field and each other by time-dependent dipole-dipole interaction. We develop original method in path integral approach for numerical calculation of density matrix and Peres-Horodecki parameter (the measure of qubits entanglement). By the use of obtained equations we investigate the dependence of quantum entanglement on dipole-dipole interaction amplitude and frequency as well as qubits phase difference. The results indicate on possibility of high-entanglement states exiting and long-time non-destructive control of them.
We study the evolution of quantum entanglement in the model of two identical qubits interacting with a single-mode laser field. The density matrix and Peres-Horodecki parameter are calculated within the frameworks of path-integral formalism. The quantum entanglement measure is shown to be strongly dependent upon the phase difference between the laser radiation acting on each cubit. This observation may offer the possibility of quantum entanglement stationary control by varying the distance between the qubits.
Background. The study of entangled states in Jaynes–Cummings models and its multiqubit generalizations and extensions remains an important and relevant topic in modern quantum optics, quantum information science and solid state physics. This is due to the fact that such models naturally describe the dynamics of interaction of logical elements of quantum devices, such as quantum computers or quantum networks (qubits) with microwave fields of resonators used to control the states of qubits. In this regard, it is very important to search for the most efficient schemes of generation, control and monitoring of entangled qubit states within the framework of Jaynes–Cummings type models. Aim. To investigate the entanglement dynamics of a pair of qubits, each of which is locked in a single-mode resonator and interacts non-resonantly with the vacuum field mode, in the presence of detuning between frequencies of transitions in the qubits and frequencies of resonator modes and dipole-dipole interaction of the qubits. Methods. To analyze the dynamics of the considered system, the solution of the time-dependent quantum Schrödinger equation is investigated. The exact solution of the above equation in the case of entangled Bell-type initial qubits states is found. This is used to calculate the criterion of entanglement of a pair of qubits – negativity. Numerical modeling of the time dependence of negativity for different values of the parameters of the considered model has been carried out. Results. It is shown that the presence of detuning, the difference in the qubit-photon couplings and the intensity of the dipole-dipole interaction significantly affect the maximum degree of entanglement of the qubit subsystem during its evolution. It is found that for certain parameters of the model under consideration, the initial Bell entangled states of qubits can be considered as long-lived stable states. Conclusion. The possibility of realization of long-lived and robust two-qubit entangled states in the system under consideration has been established. The obtained results can be used for effective control and manipulation of the degree of entanglement of qubits interacting with microwave fields of resonators.
The work develops an approach to researching the dynamics of qubits that makes it possible to monitor the behavior of the system in the process of quantum computing using computer simulation. The study deals with individual and interacting qubits based on spin ½ in a magnetic field. The goal of the work is to obtain evolutionary equations (equations of motion) for observables that adequately describe the state of the system and provide opportunities for computer modeling of processes. The dynamics of qubits are described using the formalism of the density matrix of pure states. The paper shows derivation of the equations of motion for observables being the average values of operators of physical quantities that describe the state of the system. For spin ½ in magnetic fields, the derivation of the dynamic equation for the spin polarization vector, which is a unit vector in real three-dimensional space, is shown. The shape and parameters of radiofrequency pulses that implement single-qubit operations are presented. In the case of two interacting spins ½, an expression is given for the density matrix and the Hamiltonian of the system for scalar indirect interaction of the magnetic moments of nuclei through the electron shells of atoms in the molecule. Evolution equations are derived for the average dipole moments and quadrupole moment of the system. It is shown that in the case of weak interaction the dynamics of the system can be described using a closed system of equations for the average dipole moments, which in this case coincide with the spin polarization vectors of individual spins. The presentation is accompanied by examples of numerical solutions of the resulting equations describing one-qubit and two-qubit operations. A conclusion is made about the adequacy of the created mathematical model and the possibility of its use for computer modeling of quantum computing processes, the influence of the environment on them and the stabilization of states to prevent or reduce errors in quantum computing.
In this paper the entanglement dynamics for system of two two-level atoms interacting with a mode of thermal electromagnetic field in lossless cavity has been investigated. Initial atomic states are the entangled Bell type states. Using the full set of eigenvectors of the Hamiltonian of the considered model we have derived the exact solution for the density matrix for the whole system. On its basis the reduced atomic density matrix and Peres - Horodetcki parameter have been calculated. Calculating of entanglement parameter shows the possibility of high degree of entanglement even for large intensity of thermal field. Thus there is a possibility of maintenance and control over the degree of entanglement.