Preparing topological state with finite-depth simultaneous gates

We introduce protocols for preparing abelian and non-abelian topologically ordered states in two dimensions, utilizing finite-depth unitary circuits composed of long-range, simultaneous, and mutually commuting two-qubit gates. These protocols are inspired by recent advancements in trapped ion systems, where each qubit can engage in multiple gates simultaneously.

Our approach obtains optimal scaling of the number of two-qubit gates and additional ancillas. We demonstrate its use for obtaining the ground states of the toric code, certain non-abelian Kitaev quantum double models, and string-net models.

Furthermore, we extend these ideas to higher-dimensional Calderbank-Shor-Steane (CSS) codes. As a key application, we present protocols for realizing the three-dimensional Haah's code and X-Cube fracton models, presenting explicit finite-depth protocols for obtaining ground states of these models.