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Control of Spin-1/2 Quantum System Using Lyapunov Methods to Achieve Quantum NOT Gate Operation

Theoretical study on the control of spin-1/2 quantum system is an important matter for the development of quantum control theory. Lyapunov methods play vital role in quantum control because of their concise mathematical rigor. Previously, using the Lyapunov methods, it has been shown that signum function control law can be employed to control spin-1/2 quantum system to achieve NOT quantum gate operation. This paper demonstrates that other forms of control law can be used to control spin-1/2 quantum system to achieve the same NOT quantum gate operation. Two examples of those other forms are provided, namely hyperbolic tangent (tanh) function and proportional multiplier. It is also explained why signum function is not the only function that can be used to control spin-1/2 quantum system. This paper also shows the advantage of using fully-differentiable control law instead of signum function, which is not fully-differentiable. Control law design and stability analysis are conducted using Lyapunov methods. The numerical simulation results show that tanh and proportional multiplier function can also control spin-1/2 quantum system and achieve the NOT quantum gate operation.