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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.