Evolutionary equations for studying the dynamics of qubits in variable magnetic fields

Abstract

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.