The Wigner-Araki-Yanase theorem shows that conservation laws limit the accuracy of measurement. Here, we generalize the argument to show that conservation laws limit the accuracy of quantum logic operations. A rigorous lower bound is obtained of the error probability of any physical realization of the controlled-NOT gate under the constraint that the computational basis is represented by a component of spin, and that physical implementations obey the angular momentum conservation law. The lower bound is shown to be inversely proportional to the number of ancilla qubits or the strength of the external control field.