Locality and Conservation Laws: How, in the presence of symmetry, locality restricts realizable unitaries.

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 which 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 SO(3): unless there are interactions which act non-trivially on every subsystem in the system, some symmetric unitaries cannot be implemented using symmetric Hamiltonians. In fact, the difference between the dimensions of the Lie algebra of all symmetric Hamiltonians and its subalgebra generated by k-local symmetric Hamiltonians with a fixed k, constantly increases with the system size (i.e., the number of subsystems). On the other hand, in the case of group U(1), we find that this no-go theorem can be circumvented if one is allowed to use a pair of ancillary qubits. In particular, any unitary which is invariant under rotations around z, can be implemented using Hamiltonians XX+YY and local Z on qubits. We discuss some implications of these results in the context of quantum thermodynamics and quantum computing.