Bayesian Quantum State Tomography with Python's PyMC

TL;DR

This paper demonstrates how to use Python's PyMC probabilistic programming package to perform Bayesian quantum state tomography efficiently, providing a practical alternative to traditional maximum likelihood estimation methods.

Contribution

It introduces a method to apply PyMC for Bayesian QST, simplifying the complex integration process and enabling quick, automated inference with error estimates.

Findings

Bayesian QST with PyMC is computationally feasible.

The method provides error estimates alongside state reconstructions.

It offers a practical alternative to MLE in data-starved scenarios.

Abstract

Quantum state tomography (QST) is typically performed from a frequentist viewpoint using maximum likelihood estimation (MLE) which seeks to find the best plausible state consistent with the data by maximizing a likelihood function / distribution. The likelihood function holds an implicit assumption that there is suitable data to infer frequency. In data-starved experiments, this may or may not be a feasible assumption. Moreover, MLE returns no error estimates on the final solution and users are forced to rely on alternative approaches involving either additional measurements or simulated data. Alternatively, Bayesian methods can return a solution with error estimates consistent with the data's uncertainty, but at the expense of a difficult integration over the likelihood distribution. The integration usually requires computational methods with appropriately chosen step sizes in a somewhat complicated problem formulation. This additional complexity serves as a strong deterrent from using Bayesian methods despite the advantages. Probabilistic programming is becoming a common alternative with growing computational power and the development of robust automated integration techniques such as Markov-Chain Monte Carlo (MCMC). Here, we show how to use Python-3's open source PyMC probabilistic programming package to transform an otherwise complicated QST optimization problem into a simple form that can be quickly optimized with efficient under-the-hood MCMC samplers.