Optimal Estimation of Eigenspace of Large Density Matrices of Quantum Systems Based on Pauli Measurements
Tony Cai, Donggyu Kim, and Yazhen Wang
Quantum state tomography, which aims to reconstruct quantum states described by density matrices, is becoming increasingly important in many scientific studies involving quantum systems. This paper considers the reconstruction of high-dimensional low-rank density matrices based on Pauli measurements. In particular it focuses on estimation of eigenspace for a large low-rank density matrix. Both ordinary principal component analysis (PCA) and iterative thresholding sparse PCA (ITSPCA) are studied and optimal rates of convergence are established. Minimax lower bounds for eigenspace estimation under the spectral and Frobenius norms are derived. It is shown that the convergence rates of the ITSPCA algorithm matches the minimax lower bounds and the procedure is thus rate-optimal. With these PCA estimators, we reconstruct the large low-rank density matrix and obtain the optimal convergence rate. A simulation study is carried out to investigate the finite sample performance of the proposed estimators of the density matrices.