  Quantum circuits with gates (local unitaries) respecting a global symmetry
have broad applications in quantum information science and related fields, such
as condensed matter theory and thermodynamics. However, despite their
widespread use, fundamental properties of such circuits are not
well-understood. Recently, it was found that generic unitaries respecting a
global symmetry cannot be realized, even approximately, using gates that
respect the same symmetry. This observation raises important open questions:
What unitary transformations can be realized with k-local gates that respect a
global symmetry? In other words, in the presence of a global symmetry, how does
the locality of interactions constrain the possible time evolution of a
composite system? In this work, we address these questions for the case of
Abelian (commutative) symmetries and develop constructive methods for
synthesizing circuits with such symmetries. Remarkably, as a corollary, we find
that, while the locality of interactions still imposes additional constraints
on realizable unitaries, certain restrictions observed in the case of
non-Abelian symmetries do not apply to circuits with Abelian symmetries. For
instance, in circuits with a general non-Abelian symmetry such as SU($d$), the
unitary realized in a subspace with one irreducible representation (charge) of
the symmetry dictates the realized unitaries in multiple other sectors with
inequivalent representations of the symmetry. Furthermore, in certain sectors,
rather than all unitaries respecting the symmetry, the realizable unitaries are
the symplectic or orthogonal subgroups of this group. We prove that none of
these restrictions appears in the case of Abelian symmetries. This result
suggests that global non-Abelian symmetries may affect the thermalization of
quantum systems in ways not possible under Abelian symmetries.
