Online Riemannian Gradient Descent for Quantum State Tomography with Matrix Product Operators

TL;DR

This paper introduces an online Riemannian gradient descent algorithm for quantum state tomography using matrix product operators, achieving efficient, scalable reconstruction with theoretical guarantees and validated by numerical experiments.

Contribution

It develops a novel online Riemannian gradient descent method for MPO-based quantum state tomography with proven convergence and improved sample complexity bounds.

Findings

oRGD converges linearly to the target MPO with proper initialization.

The measurement complexity scales quadratically with system size.

Numerical experiments confirm the method's effectiveness and scalability.

Abstract

Matrix product operators (MPOs) provide a scalable approach for quantum state tomography (QST) by offering a compact representation of many-body mixed states with limited entanglement, using only a number of parameters that scales polynomially with the system size. In this paper, we study QST for quantum density matrices that can be represented by MPOs. We first derive an equivalent characterization of Hermiticity in terms of the MPO core tensors and show that the coefficient tensor of an MPO under the Pauli or generalized Gell-Mann basis admits a real-valued low tensor-train (TT) rank structure. This establishes an explicit connection between MPO-based QST and noisy low-rank tensor completion. Motivated by this formulation, we develop an online Riemannian gradient descent (oRGD) algorithm that sequentially incorporates measurement data during the reconstruction process. With a proper initialization, we prove that oRGD converges linearly to the target MPO and succeeds with a number of distinct measurement settings that scales quadratically with the system size. As a byproduct, our analysis also yields a significantly improved sample complexity bound for the low TT rank tensor completion task. Furthermore, we propose a tailored spectral initialization method and establish its theoretical guarantee. Numerical experiments on several classes of quantum states validate the effectiveness and scalability of the proposed method.