Comparison of Some Methods of Quantum State Estimation

Abstract

In the paper the Bayesian and the least squares methods of quantum state tomography are compared for a single qubit. The quality of the estimates are compared by computer simulation when the true state is either mixed or pure. The fidelity and the Hilbert-Schmidt distance are used to quantify the error.It was found that in the regime of low measurement number the Bayesian method outperforms the least squares estimation. Both methods are quite sensitive to the degree of mixedness of the state to be estimated, that is, their performance can be quite bad near pure states.

An efficient adaptive MCMC algorithm for Pseudo-Bayesian quantum tomography

“…Starting receiving attention in recent years, Bayesian QST had been shown as a promising method in this problem (Blume-Kohout 2010; Bužek et al 1998;Baier et al 2007;Lukens et al 2020). Through Bayes' theorem, experimental uncertainty is explicitly accounted in Bayesian estimation.…”

“…Bayesian methods have been recognized as a powerful paradigm for quantum state tomography (Blume-Kohout 2010), that deal with uncertainty in meaningful and informative ways and are the most accurate approach with respect to the expected error (operational divergence) even with finite samples. Several studies have been conducted: for example, the papers Bužek et al (1998) and Baier et al (2007) performed numerical comparisons between Bayesian estimations with other methods on simulated data; algorithms for computing Bayesian estimators have been discussed in Kravtsov et al (2013), Ferrie (2014), Kueng and Ferrie (2015), Schmied (2016) and Lukens et al (2020).…”

Pseudo-Bayesian quantum tomography with rank-adaptation

“…In the single qubit state estimation n = 1, the representation of the quantum constraints is explicit [5,38]. Thus, one can place a prior distribution on the polar reparametrization of the density.…”

“…The papers [12,5] compare Bayesian methods to other methods on simulated data. More recently, [30,19,31,38] discuss efficient algorithms for computing Bayesian estimators.…”