  We explore the relationship between approximate symmetries of a gapped
Hamiltonian and the structure of its ground space. We start by showing that
approximate symmetry operators---unitary operators whose commutators with the
Hamiltonian have norms that are sufficiently small---which possess certain
mutual commutation relations can be restricted to the ground space with low
distortion. We generalize the Stone-von Neumann theorem to matrices that
approximately satisfy the canonical (Heisenberg-Weyl-type) commutation
relations, and use this to show that approximate symmetry operators can certify
the degeneracy of the ground space even though they only approximately form a
group. Importantly, the notions of "approximate" and "small" are all
independent of the dimension of the ambient Hilbert space, and depend only on
the degeneracy in the ground space. Our analysis additionally holds for any
gapped band of sufficiently small width in the excited spectrum of the
Hamiltonian, and we discuss applications of these ideas to topological quantum
phases of matter and topological quantum error correcting codes. Finally, in
our analysis we also provide an exponential improvement upon bounds concerning
the existence of shared approximate eigenvectors of approximately commuting
operators which may be of independent interest.
