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 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).
