Measurement-induced entanglement transition in a two-dimensional shallow circuit

We prepare two-dimensional states generated by shallow circuits composed of (1) one layer of the two-qubit controlled-Z (CZ) gate or (2) a few layers of the two-qubit random Clifford gate. After measuring all of the bulk qubits, we study the entanglement structure of the remaining qubits on the one-dimensional boundary. In the first model, we observe that the competition between the bulk X and Z measurements can lead to an entanglement phase transition between an entangled volume law phase and a disentangled area law phase. We numerically evaluate the critical exponents and generalize this idea to other qudit systems with a local Hilbert space dimension larger than 2. In the second model, we observe the entanglement transition by varying the density of the two-qubit gate in each layer. We give an interpretation of this transition in terms of the random bond Ising model in a similar shallow circuit composed of random Haar gates.