Entanglement transition and suppression of critical phase of thermofield double state in monitored quantum circuit with unitary $R$ matrix gates

We study quantum circuits with gates composed randomly of identity operators, projectors, or a kind of $R$ matrices which satisfy the Yang-Baxter equation and are unitary and dual-unitary. This enables us to translate the quantum circuit into a topological object with distinguished overcrossings and undercrossings. The circuit corresponds to a classical loop model and is post-selection free when an overcrossing and an undercrossing coincide. The entanglement entropy between the final state and initial state is given by the spanning number of the classical model, and they share the same phase diagram. Whenever an overcrossing and undercrossing differ, the circuit extends beyond the classical model. Considering a specific case with $R$ matrices randomly replaced by SWAP gates, we demonstrate that the topological effect originating from worldline braiding dominates, and only the area-law phase remains in the thermodynamic limit, regardless of how small the replacement probability is. We also find evidence of an altered phase diagram for non-Clifford cases.