  Whether a given target state can be prepared by starting with a simple
product state and acting with a finite-depth quantum circuit is a key question
in condensed matter physics and quantum information science. It underpins
classifications of topological phases, as well as the understanding of
topological quantum codes, and has obvious relevance for device
implementations. Traditionally, this question assumes that the quantum circuit
is made up of unitary gates that are geometrically local. Inspired by the
advent of noisy intermediate-scale quantum devices, we reconsider this question
with $k$-local gates, i.e. gates that act on no more than $k$ degrees of
freedom, but are not restricted to be geometrically local. First, we construct
explicit finite-depth circuits of symmetric $k$-local gates which create
symmetry-protected topological (SPT) states from an initial a product state.
Our construction applies both to SPT states protected by global symmetries and
subsystem symmetries, but not to those with higher-form symmetries, which we
conjecture remain nontrivial. Next, we show how to implement arbitrary
translationally invariant quantum cellular automata (QCA) in any dimension
using finite-depth circuits of $k$-local gates. These results imply that the
topological classifications of SPT phases and QCA both collapse to a single
trivial phase in the presence of $k$-local interactions. We furthermore argue
that SPT phases are fragile to generic $k$-local symmetric perturbations. We
conclude by discussing the implications for other phases, such as fracton
phases, and surveying future directions. Our analysis opens a new
experimentally motivated conceptual direction examining the stability of phases
and the feasibility of state preparation without the assumption of geometric
locality.
