  Measurement-induced phase transition (MIPT) is a novel non-equilibrium phase
transition characterized by entanglement entropy. The scrambling dynamics
induced by random unitary gates can protect information from low-rate
measurements. However, common decoherence noises, such as dephasing, are
detrimental to the volume law phase, posing a significant challenge for
observing MIPT in current noisy intermediate-scale quantum devices. Here, we
demonstrate that incorporating quantum-enhanced operations can effectively
protect MIPT from environmental noise. The conditional entanglement entropy is
associated with a statistical mechanics model wherein noise and
quantum-enhanced operations act as two competing external random fields. Then
we show that an average apparatus-environment exchange symmetry ensures the
conditional entanglement entropy is a valid probe of entanglement. Furthermore,
we provide numerical evidence on a (2+1)-d quantum circuit under dephasing
noise, demonstrating that MIPT can indeed be observed with the aid of
quantum-enhanced operations. This result not only serves as a concrete example
of the power of quantum enhancement in combating noise but also holds
experimental relevance, as the protocol is straightforward to implement in
practice.
