  We define several quantitative measures of the robustness of a quantum gate
against noise. Exact analytic expressions for the robustness against
depolarizing noise are obtained for all unitary quantum gates, and it is found
that the controlled-not is the most robust two-qubit quantum gate, in the sense
that it is the quantum gate which can tolerate the most depolarizing noise and
still generate entanglement. Our results enable us to place several analytic
upper bounds on the value of the threshold for quantum computation, with the
best bound in the most pessimistic error model being 0.5.
