Methods for Finding Analytic Solutions for Time Dependent Two-Level Quantum Systems and Its Generalizations

TL;DR

This paper reviews two methods for analytically solving time-dependent two-level quantum systems, extends the transfer matrix approach to more complex systems, and discusses challenges in practical applications.

Contribution

It introduces and compares two analytic solution methods for two-level systems and generalizes the transfer matrix method to higher-dimensional quantum systems.

Findings

Analytic solutions for driven two-level systems are obtained.

Transfer matrix method is extended to qutrits and two-qubit systems.

Challenges in applying these methods to real-world systems are discussed.

Abstract

Two-level systems are one of the most important quantum systems and they form the basis of quantum computers. We briefly look at the traditional approach to two-level systems with an external driving field as well as those subjected to noise. This project is aimed at studying two specific methods for obtaining analytic solutions for two-level systems. One of the methods enables us to obtain analytic solutions for driven time-dependent two-level systems while the other attempts to give exact solution of qubit decoherence using a transfer matrix method. A thorough study of both papers is done and results are reproduced. The latter method is generalized for a qutrit system as well as a two qubit system subjected to noise. A general method is formally derived for an N-dimensional quantum system and the difficulties in applying the method in real life systems is discussed.