  Recently, Bravyi and K\"onig have shown that there is a tradeoff between
fault-tolerantly implementable logical gates and geometric locality of
stabilizer codes. They consider locality-preserving operations which are
implemented by a constant depth geometrically local circuit and are thus
fault-tolerant by construction. In particular, they shown that, for local
stabilizer codes in D spatial dimensions, locality preserving gates are
restricted to a set of unitary gates known as the D-th level of the Clifford
hierarchy. In this paper, we elaborate this idea and provide several extensions
and applications of their characterization in various directions. First, we
present a new no-go theorem for self-correcting quantum memory. Namely, we
prove that a three-dimensional stabilizer Hamiltonian with a
locality-preserving implementation of a non-Clifford gate cannot have a
macroscopic energy barrier. Second, we prove that the code distance of a
D-dimensional local stabilizer code with non-trivial locality-preserving m-th
level Clifford logical gate is upper bounded by $O(L^{D+1-m})$. For codes with
non-Clifford gates (m>2), this improves the previous best bound by Bravyi and
Terhal. Third we prove that a qubit loss threshold of codes with non-trivial
transversal m-th level Clifford logical gate is upper bounded by 1/m. As such,
no family of fault-tolerant codes with transversal gates in increasing level of
the Clifford hierarchy may exist. This result applies to arbitrary stabilizer
and subsystem codes, and is not restricted to geometrically-local codes. Fourth
we extend the result of Bravyi and K\"onig to subsystem codes. A technical
difficulty is that, unlike stabilizer codes, the so-called union lemma does not
apply to subsystem codes. This problem is avoided by assuming the presence of
error threshold in a subsystem code, and the same conclusion as Bravyi-K\"onig
is recovered.
