In this work, the task to find a simultaneous low rank approximation of several lowest eigenpairs of a matrix-valued symmetric operator is considered. This problem arises, for instance, in the density matrix renormalization group algorithm (DMRG) for the accurate simulation of quantum chains. The usual approach is to compute the desired set of eigenvectors and then to identify a low rank approximation by truncating the singular value decomposition (SVD). Since SVD truncation is a norm projection, this yields only suboptimal results for the eigenvalues. As an alternative, this article explores a direct trace minimization on the intersection of the Stiefel and a low rank manifold using a Riemannian optimization method, which gives better approximations to the eigenspace. A second algorithm based on alternating optimization is also considered but is less stable. Compared to SVD truncation, the proposed Riemannian method can be seen as a more natural choice for a subsolver in the DMRG algorithm and appears to yield better results when applied to spin chains, for which the singular values of the exact eigenvectors decay only moderately.
A pure or nearly pure quantum state can be described as a low-rank density matrix, which is a positive semidefinite and unit-trace Hermitian. We consider the problem of recovering such a low-rank density matrix contaminated by sparse components, from a small set of linear measurements. This quantum state estimation task can be formulated as a robust principal component analysis (RPCA) problem subject to positive semidefinite and unit-trace Hermitian constraints. We propose an efficient and fast inexact alternating direction method of multipliers (I-ADMM), in which the subproblems are solved inexactly and hence have closed-form solutions. We prove global convergence of the proposed I-ADMM, and the theoretical result provides a guideline for parameter setting. Numerical experiments show that the proposed I-ADMM can recover state density matrices of 5 qubits on a laptop in 0.69 s, with 6 × 10-4accuracy (99.38% fidelity) using 30% compressive sensing measurements, which outperforms existing algorithms.
Density matrices are positively semi-definite Hermitian matrices with unit trace that describe the states of quantum systems. Many quantum systems of physical interest can be represented as high-dimensional low rank density matrices. A popular problem in quantum state tomography (QST) is to estimate the unknown low rank density matrix of a quantum system by conducting Pauli measurements. Our main contribution is twofold. First, we establish the minimax lower bounds in Schatten
p-norms with
1≤p≤+∞ for low rank density matrices estimation by Pauli measurements. In our previous paper [14], these minimax lower bounds are proved under the trace regression model with Gaussian noise and the noise is assumed to have common variance. In this paper, we prove these bounds under the Binomial observation model which meets the actual model in QST. Second, we study the Dantzig estimator (DE) for estimating the unknown low rank density matrix under the Binomial observation model by using Pauli measurements. In our previous papers [14] and [25], we studied the least squares estimator and the projection estimator, where we proved the optimal convergence rates for the least squares estimator in Schatten
p-norms with
1≤p≤2 and, under a stronger condition, the optimal convergence rates for the projection estimator in Schatten
p-norms with
1≤p≤+∞. In this paper, we show that the results of these two distinct estimators can be simultaneously obtained by the Dantzig estimator. Moreover, better convergence rates in Schatten norm distances can be proved for Dantzig estimator under conditions weaker than those needed in [14] and [25]. When the objective function of DE is replaced by the negative von Neumann entropy, we obtain sharp convergence rate in Kullback-Leibler divergence.