In recent investigations, it has been found that conservation laws generally lead to precision limits on quantum computing. Lower bounds of the error probability have been obtained for various logic operations. We show the generalization of the previous method to the problem for any self-adjoint gate on a single qubit and for conservation laws of the angular momentum in any direction. This approach provides a geometrical relation between conservation laws and precision limits for self-adjoint gates. However, the problem of obtaining the precision limit to realizing the quantum NOT gate has eluded a solution from these approaches when the conservation law of the angular momentum in the same direction as the computational basis is assumed. We will present a new method for this problem based on analyzing the trace distance between the output state from the realization under consideration and the one from the ideal gate.