Online Quantum State Tomography via Stochastic Gradient Descent

TL;DR

This paper introduces stochastic gradient descent algorithms for online quantum state tomography, enabling efficient, low-memory, and fast reconstruction of quantum states with theoretical convergence guarantees.

Contribution

It develops and analyzes non-convex mini-batch SGD algorithms for online low-rank quantum state tomography with proven linear convergence and improved computational complexity.

Findings

Achieves linear convergence with proper initialization.

Attains nearly optimal sample complexity.

Outperforms existing non-convex QST algorithms in speed and memory efficiency.

Abstract

We initiate the study of online quantum state tomography (QST), where the matrix representation of an unknown quantum state is reconstructed by sequentially performing a batch of measurements and updating the state estimate using only the measurement statistics from the current round. Motivated by recent advances in non-convex optimization algorithms for solving low-rank QST, we propose non-convex mini-batch stochastic gradient descent (SGD) algorithms to tackle online QST, which leverage the low-rank structure of the unknown quantum state and are well-suited for practical applications. Our main technical contribution is a rigorous convergence analysis of these algorithms. With proper initialization, we demonstrate that the SGD algorithms for online low-rank QST achieve linear convergence both in expectation and with high probability. Our algorithms achieve nearly optimal sample complexity while remaining highly memory-efficient. In particular, their time complexities are better than the state-of-the-art non-convex QST algorithms, in terms of the rank and the logarithm of the dimension of the unknown quantum state.