  Recently, universally valid uncertainty relations have been established to
set a precision limit for any instruments given a disturbance constraint in a
form more general than the one originally proposed by Heisenberg. One of them
leads to a quantitative generalization of the Wigner-Araki-Yanase theorem on
the precision limit of measurements under conservation laws. Applying this, a
rigorous lower bound is obtained for the gate error probability of physical
implementations of Hadamard gates on a standard qubit of a spin 1/2 system by
interactions with control fields or ancilla systems obeying the angular
momentum conservation law.
