# Efficient Bayesian credible-region certification for quantum-state tomography

## TL;DR

This paper introduces an efficient method for certifying Bayesian credible regions in quantum-state tomography by reformulating the theory, employing accelerated Monte Carlo sampling, and deriving analytical approximations to improve computational efficiency.

## Contribution

It presents a reformulated Bayesian credible-region theory, an accelerated hit-and-run Monte Carlo sampling method, and analytical formulas for size and credibility estimation without heavy sampling.

## Key findings

- Region-average quantities effectively characterize credible-region properties.
- Accelerated hit-and-run sampling significantly reduces computational complexity.
- Analytical approximations accurately estimate size and credibility without Monte Carlo.

## Abstract

Standard Bayesian credible-region theory for constructing an error region on the unique estimator of an unknown state in general quantum-state tomography to calculate its size and credibility relies on heavy Monte~Carlo sampling of the state space followed by sample rejection. This conventional method typically gives negligible yield for very small error regions originating from large datasets. We propose an operational reformulated theory to compute both size and credibility from region-average quantities that in principle convey information about behavior of these two properties as the credible-region changes. We next suggest the accelerated hit-and-run Monte~Carlo sampling, customized to the construction of Bayesian error-regions, to efficiently compute region-average quantities, and provide its complexity estimates for quantum states. Finally by understanding size as the region-average distance between two states in the region (measured for instance with either the Hilbert-Schmidt, trace-class or Bures distance), we derive approximation formulas to analytically estimate both distance-induced size and credibility under the pseudo-Bloch parametrization without resorting to any Monte~Carlo computation.