A Comparative Study on the Convergence Rate of Two Online Quantum State Reconstruction Algorithms

TL;DR

This paper compares the convergence rates of two online quantum state reconstruction algorithms, OPG-ADMM and KF-QSE, analyzing their theoretical convergence properties and validating results through numerical simulations on a 4-bit quantum system.

Contribution

The paper provides a theoretical analysis of the convergence rates of OPG-ADMM and KF-QSE algorithms and verifies these through numerical experiments.

Findings

Both algorithms' convergence orders are derived mathematically.

Numerical simulations confirm the theoretical convergence rates.

The algorithms effectively reconstruct 4-bit quantum states online.

Abstract

In this paper, the convergence rates of two algorithms for the online quantum states reconstruction with Gaussian measurement noise in continuous weak measurement are studied, one is the online proximal gradient-based alternating direction method of multipliers (OPG-ADMM) algorithm, and another is Kalman fitering-based quantum state estimation (KF-QSE) algorithm. For the OPG-ADMM algorithm, by defining the loss function of the optimization function and the constraint condition in the times T tracking process, the convergence rate theorem of the two loss functions is obtained and proved. Then, the convergence order of the normalized distance of the density matrix under the OPG-ADMM algorithm is derived from the conclusion of the theorem. For the KF-QSE algorithm, after defining the loss function of the optimization function, the theorem of the convergence order of the loss function is investigated. Then, the convergence order of the normalized distance of the KF-QSE algorithm is deduced from the conclusion of the theorem. Finally, in the numerical simulation experiments, we use the normalized distance of density matrix as the indicator and use two algorithms for online reconstruction of the 4-bit quantum system. The derived performance of algorithm convergence rates are verified by the comparison and analysis of the results.