Restrictions on realizable unitary operations imposed by symmetry and locality

Understand

According to a fundamental result in quantum computing, any unitary transformation on a composite system can be generated using so-called 2-local unitaries that act only on two subsystems.

• Beyond its 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.
• Here we show that this universality does not remain valid in the presence of conservation laws and global continuous symmetries such as U(1) and SU(2).
• In particular, we show that generic symmetric unitaries cannot be implemented, even approximately, using local symmetric unitaries.