Locality and Conservation Laws: How, in the presence of symmetry, locality restricts realizable unitaries
Abstract
According to an elementary result in quantum computing, any unitary transformation on a composite system can be generated using 2local 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 shortterm 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 klocal 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 klocal symmetric unitaries, constantly increases with the system size. On the other hand, we find that this nogo 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 energyconserving unitary on a composite system can be implemented using a sequence of 2local energyconserving unitaries, provided that one can use a single ancillary qubit (catalyst).
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 DOI:
 10.48550/arXiv.2003.05524
 arXiv:
 arXiv:2003.05524
 Bibcode:
 2020arXiv200305524M
 Keywords:

 Quantum Physics;
 Condensed Matter  Statistical Mechanics;
 High Energy Physics  Theory;
 Mathematical Physics;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 8 pages+ 24 pages of Supplementary Material, 4 figures. V2 contains several new results (Discussion on approximate implementation of symmetric unitaries and a protocol for implementing energyconserving unitaries with a single ancillary qubit)