Optimal Sparse Eigenspace and Low-Rank Density Matrix Estimation for Quantum Systems.

Quantum state tomography, which aims to estimate quantum states that are described by density matrices, plays an important role in quantum science and quantum technology. This paper examines the eigenspace estimation and the reconstruction of large low-rank density matrix based on Pauli measurements. Both ordinary principal component analysis (PCA) and iterative thresholding sparse PCA (ITSPCA) estimators of the eigenspace are studied, and their respective convergence rates are established. In particular, we show that the ITSPCA estimator is rate-optimal. We present the reconstruction of the large low-rank density matrix and obtain its optimal convergence rate by using the ITSPCA estimator. A numerical study is carried out to investigate the finite sample performance of the proposed estimators.