According to a fundamental result in quantum control, 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. In the context of quantum thermodynamics this no-go theorem implies that generic energy-conserving unitaries cannot be realized using local energy-conserving unitaries.

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.

References:

1) I. Marvian, Restrictions on realizable unitary operations imposed by symmetry and locality, Nature Physics 18, 283–289 (2022).

2) I. Marvian, H. Liu, and A. Hulse, Qudit circuits with SU(d) symmetry: Locality imposes additional conservation laws, arXiv:2105.12877 (2021).

3) I. Marvian, H. Liu, and A. Hulse, Rotationally-Invariant Circuits: Universality with the exchange interaction and two ancilla qubits, arXiv:2202.01963 (2022).