Measurement-Induced Criticality is Tomographically Optimal

TL;DR

This paper introduces a classical shadow tomography protocol using hybrid quantum circuits with measurements at any point during evolution, revealing optimal sample complexity at the measurement-induced transition.

Contribution

It presents a novel tomography protocol that allows measurements during quantum evolution and demonstrates optimal sample complexity at critical measurement rates.

Findings

Sample complexity scales optimally at the measurement-induced transition.

Protocol enables state reconstruction from intermittent measurements.

Measurement timing flexibility improves tomography efficiency.

Abstract

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 unitary gates mixed with single-qubit random projective measurements. Unlike conventional protocols that perform all measurements by the end of unitary evolutions, our protocol allows measurements to occur at any spacetime 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.