Quantum state tomography is an essential task in quantum information technology. It aims to reconstruct a quantum state from repeated measurements of copies of the state. While reconstructing the full density matrix requires exponentially many samples in many-body systems, predicting a collection of (possibly exponentially many) properties of the quantum system can be efficiently achieved with only a polynomial number of samples under the name of shadow tomography. Huang, Kueng, Preskill further propose a more experiment-friendly shadow tomography scheme, called the classical shadow tomography, which reduces the data acquisition and classical post-processing complexity while retaining the superior polynomial sample complexity.
However, the original proposal was limited to two measurement schemes: the single-qubit (local) Pauli measurement, which is well suited for predicting local operators but inefficient for large operators; and the global Clifford measurement, which is efficient for low-rank operators but infeasible on near-term quantum devices due to the extensive gate overhead. It is desired to go beyond these two limits and develop more flexible measurement schemes for classical shadow tomography.
In recent work arXiv:2209.02093, we developed a scalable classical shadow tomography approach for generic randomized measurements implemented with finite-depth local Clifford random unitary circuits, which interpolates between the limits of Pauli and Clifford measurements. For more details, see my talk Scalable Classical Shadow Tomography with Shallow Circuits and Quantum Dynamics - Yi-Zhuang You, UC San Diego at the KITP Program of Quantum Many-Body Dynamics and Noisy Intermediate-Scale Quantum Systems.