Measurement-induced criticality is tomographically optimal

We develop a classical shadow tomography protocol utilizing the randomized measurement scheme based on hybrid quantum circuits, which consist of layers of two-qubit random Clifford unitary gates mixed with single-qubit random Pauli measurements. Unlike conventional protocols that perform all measurements by the end of unitary evolutions, our protocol allows measurements to occur at any space-time position throughout the quantum evolution. We provide a universal classical post-processing strategy to approximately reconstruct the original quantum state from intermittent measurement outcomes given the corresponding random circuit realizations over repeated experiments. We investigated the sample complexity for estimating different observables at different measurement rates of the hybrid quantum circuits. Our result shows that the sample complexity has an optimal scaling at the critical measurement rate when the hybrid circuit undergoes the measurement-induced transition.