Dartmouth Events

Abstract: 

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. In this talk, I show that this universality does not hold in the presence of conservation laws and global continuous symmetries: generic symmetric unitaries on a composite system cannot be implemented, even approximately, using local symmetric unitaries on the subsystems. I also argue that in some cases this no-go theorem can be circumvented using ancilla qubits. For instance, any rotationally-invariant unitary on qubits can be realized using the Heisenberg exchange interaction, which is 2-local and rotationally-invariant, provided that the qubits in the system interact with a pair of ancilla qubits.  Finally, I briefly present some results on qudit systems with SU(d) symmetry, which reveal a surprising distinction between the case of d=2 and d>2.