Solving the problem of efficiently preparing solvable anyons

The classification of topological phases of quantum matter has recently been extended to allow constant-depth adaptive local quantum circuits. In this setting, a fundamental problem is the classification of topological phases that become equivalent to the trivial phase under adaptive quantum circuits. These are precisely the topological phases with ground states that can be prepared by constant-depth adaptive quantum circuits. In this work we propose such a classification in terms of solvable anyon theories, and conjecture that it is complete. Solvable anyon theories are a vast generalization of solvable groups that includes cyclic nonabelian anyons, and anyons with irrational quantum dimensions such as Ising anyons. We introduce a sequential gauging procedure that can produce a string-net ground state in any topological phase described by solvable anyons via a constant-depth adaptive local quantum circuit. We furthermore introduce a sequential ungauging and regauging procedure to implement string operators of arbitrary length for any solvable anyon theory via constant-depth adaptive local quantum circuits. Our general results are demonstrated for the quantum double of $S_3$ and for several examples that go beyond solvable groups including the doubled Ising theory, and the Drinfeld center of the $\mathbb{Z}_3$ Tambara-Yamagami category.