Neural Network Decoders for Detecting Measurement Induced Phase Transitions in Random Clifford Circuits*

Random unitary circuits with intermittent projective measurements can host a phase transition between a  pure phase and a mixed phase where the pure phase has a non-extensive in circuit size entanglement entropy and the mixed phase has an extensive in size entanglement entropy. In these circuits the mixed phase can be considered as a familiy of quantum error codes where the unitary time dynamics protects quantum entanglement from projective measurements that play the role of errors. Recently, it has been shown that these phase transitions can be locally probed via entangling reference qubits to the circuit whose purification dynamics determines the phase of the quantum circuit. In this work we study these phase transitions in random Clifford stabilizer circuits and we design a neural network (NN) decoder that by using the measurement outcomes of the qubits in the circuit can learn the state of the reference qubit. Using the dynamics of the reference qubit we demonstrate that for a given circuit our NN decoder can detect the phase of the quantum circuit without measuring the entanglement entropy which can be used in noisy intermediate-scale quantum devices.