Description
In the recently introduced unitary circuit games, a competition between two unitary parties, an "entangler" and a "disentangler," can lead to an entanglement phase transition, with a behavior that differes from measurement-induced transitions. In this work, we study unitary circuit games within the framework of free fermion (matchgate) dynamics. First, we examine the game for braiding dynamics, where gates are selected from the intersection of Clifford and matchgates. In this scenario, we find that the disentangler can always control the growth of entanglement, resulting in no volume law phase except for a disentangling probability $p=0$. For generic matchgates, we determine that the optimal strategy for disentangling a state is to choose the unitary that minimizes the Rényi-0 entanglement entropy. We propose an algorithm to identify the optimal disentangling unitary based on the state's correlation matrix. Finally, we employ the Rényi-0 disentangler to compete against random matchgate dynamics in the unitary circuit game.
