  According to an elementary result in quantum computing, any unitary
transformation on a composite system can be generated using 2-local unitaries,
i.e., those that act only on two subsystems. Beside its fundamental importance
in quantum computing, this result can also be regarded as a statement about the
dynamics of systems with local Hamiltonians: although locality puts various
constraints on the short-term dynamics, it does not restrict the possible
unitary evolutions that a composite system with a general local Hamiltonian can
experience after a sufficiently long time. We ask if such universality remains
valid in the presence of conservation laws and global symmetries. In
particular, can k-local symmetric unitaries on a composite system generate all
symmetric unitaries on that system? Interestingly, it turns out that the answer
is negative in the case of continuous symmetries, such as U(1) and SU(2):
generic symmetric unitaries cannot be implemented, even approximately, using
local symmetric unitaries. In fact, the difference between the dimensions of
the manifold of all symmetric unitaries and the submanifold of unitaries
generated by k-local symmetric unitaries, constantly increases with the system
size. On the other hand, we find that this no-go theorem can be circumvented
using ancillary qubits. For instance, any unitary invariant under rotations
around z can be implemented using Hamiltonian XX+YY together with local Z
Hamiltonian on the ancillary qubit. Moreover, any globally energy-conserving
unitary on a composite system can be implemented using a sequence of 2-local
energy-conserving unitaries, provided that one can use a single ancillary qubit
(catalyst).
