Quantum information and quantum error correction are built extensively around stabilizer states, which provide a simple way to describe quantum states. This formalism also enables efficient classical simulation of Clifford circuits. In this talk, I will focus on the dynamics of the stabilizer group itself, while ignoring the signs of individual stabilizers. Although this description does not specify the full quantum state, it nevertheless retains key information about entanglement and the logical degrees of freedom that survive measurement.

I will first introduce a general normal form for an arbitrary monitored Clifford circuit, reducing its dynamics to an effective unitary evolution together with stabilizer measurements. I will then apply this perspective to time-periodic, or Floquet-like, monitored circuits. In these systems, persistent spatial structures can cause entanglement and purification to behave very differently, leading to new measurement-induced critical behavior.