  Certain quantum codes allow logic operations to be performed on the encoded
data, such that a multitude of errors introduced by faulty gates can be
corrected. An important class of such operations are {\em transversal}, acting
bitwise between corresponding qubits in each code block, thus allowing error
propagation to be carefully limited. If any quantum operation could be
implemented using a set of such gates, the set would be {\em universal}; codes
with such a universal, transversal gate set have been widely desired for
efficient fault-tolerant quantum computation. We study the structure of
GF(4)-additive quantum codes and prove that no universal set of transversal
logic operations exists for these codes. This result strongly supports the idea
that additional primitive operations, based for example on quantum
teleportation, are necessary to achieve universal fault-tolerant computation on
additive codes.
