Free fermionic Gaussian, also known as matchgate, random circuits exhibit atypical behavior compared to generic interacting systems. They produce anomalously slow entanglement growth, characterized by diffusive scaling $S(t)\sim \sqrt{t}$, and evolve into volume-law entangled states at late times, $S\sim N$, which are highly unstable under measurements. Here, we investigate how doping such circuits with non-Gaussian resources (gates) restores entanglement structures of typical dynamics. We demonstrate that ballistic entanglement growth $S(t)\sim t$ is recovered after injecting an extensive total amount of non-Gaussian gates, which also restores Kardar-Parisi-Zhang fluctuations. When the evolution is perturbed with measurements, we uncover a measurement-induced phase transition between an area-law and a power-law entangled phase, $S\sim N^{\alpha }$, with $\alpha$ controlled by the doping. A genuine volume-law entangled phase is recovered only when non-Gaussian gates are injected at an extensive rate. Our findings bridge the dynamics of free and interacting fermionic systems, identifying non-Gaussianity as a key resource driving the emergence of nonintegrable behavior.

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• Dynamical phase transitions
• Entanglement entropy
• Quantum circuits
• Quantum measurements