  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 from the
commutation relation between the noise operator and the conserved quantity or
from the recently developed universal uncertainty principle for the
noise-disturbance trade-off in general measurements. However, the problem of
obtaining the precision limit to realizing the quantum NOT gate has eluded a
solution from these approaches. Here, we develop 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. Using the
mathematical apparatus of orthogonal polynomials, we obtain a general lower
bound on the error probability for the realization of the quantum NOT gate in
terms of the number of qubits in the control system under the conservation of
the total angular momentum of the computational qubit plus the the control
system along the direction used to encode the computational basis. The lower
bound turns out to be more stringent than one might expect from previous
results. The new method is expected to lead to more accurate estimates for
physical realizations of various types of quantum computations under
conservation laws, and to contribute to related problems such as the accuracy
of programmable quantum processors.
