Efficient reconstruction of density matrices for high dimensional quantum state tomography

The conventional quantum state tomography (QST) needs large number of measurements to reconstruct the quantum state. Thanks to the compressive sensing (CS) theory, one can recover a pure or nearly pure quantum state with an acceptable accuracy given much fewer number of measurements. However, most existing algorithms for CS based QST are rather slow and difficult to be implemented in practice. To fill the gap between the CS theory and practical QST, this paper firstly applies an improved Alternating Direction Multiplier Method (ADMM) combining with the Iterative Shrinkage-Thresholding Algorithm (ISTA), IST-ADMM for short, aiming at improving the efficiency of QST problem in particular with much lower number of measurements. The IST-ADMM avoids computing the inverse of large-scale matrix, reduces the computational time and required memory space. The computation complexity is reduced from O(d6) for least square (widely used in QST), and O(md4) for Fixed Point-ADMM in our previous work, to IST-ADMM's O(md2). The proposed algorithm makes it practical to reconstruct high dimensional quantum states provided fewer number of measurements. The simulations verify the superiority of the proposed algorithm, where it takes 3.13 minutes to reconstruct an 8-qubit density matrix with 96.17% accuracy, which is faster than many existing and our previous work.