Preparing topological states with finite depth simultaneous commuting gates

We present protocols for preparing two-dimensional abelian and non-abelian topologically ordered states by employing finite depth unitary circuits composed of long-ranged, simultaneous, and mutually commuting two-qubit gates. Our protocols are motivated by recent proposals for circuits in trapped ion systems, which allow each qubit to participate in multiple gates simultaneously. Our circuits are shown to be optimal, in the sense that the number of two-qubit gates and ancilla qubits scales as O\(L²\), where L is the linear size of the system. Examples include the ground states of the toric code, certain Kitaev quantum double models, and string net models. Going beyond two dimensions, we extend our scheme to more general Calderbank-Shor-Steane (CSS) codes. As an application, we present protocols for realizing the three-dimensional Haah's code and X-Cube fracton models.