Pseudo-Bayesian Quantum Tomography with Rank-adaptation

TL;DR

This paper introduces a new pseudo-Bayesian approach with an adaptive rank prior for quantum state tomography, providing theoretical convergence guarantees and demonstrating strong empirical performance on datasets.

Contribution

It proposes a novel prior and pseudo-Bayesian estimators for quantum states, with theoretical convergence analysis using PAC-Bayesian theorems.

Findings

Derived convergence rates for the posterior mean estimator.

Demonstrated improved empirical performance on simulated and real data.

Provided a practical method for rank-adaptive quantum state estimation.

Abstract

Quantum state tomography, an important task in quantum information processing, aims at reconstructing a state from prepared measurement data. Bayesian methods are recognized to be one of the good and reliable choice in estimating quantum states~\cite{blume2010optimal}. Several numerical works showed that Bayesian estimations are comparable to, and even better than other methods in the problem of $1$-qubit state recovery. However, the problem of choosing prior distribution in the general case of $n$ qubits is not straightforward. More importantly, the statistical performance of Bayesian type estimators have not been studied from a theoretical perspective yet. In this paper, we propose a novel prior for quantum states (density matrices), and we define pseudo-Bayesian estimators of the density matrix. Then, using PAC-Bayesian theorems, we derive rates of convergence for the posterior mean. The numerical performance of these estimators are tested on simulated and real datasets.