Quantum State Tomography via Linear Regression Estimation

TL;DR

This paper introduces a linear regression estimation method for quantum state tomography that is computationally efficient, asymptotically optimal, and guides optimal measurement choices, outperforming traditional maximum-likelihood approaches.

Contribution

The paper proposes a novel linear regression-based approach for quantum state tomography with analytical MSE bounds and improved computational efficiency.

Findings

LRE achieves asymptotic optimality with MSE approaching the Cramér-Rao bound.

LRE has computational complexity O(d^4), faster than maximum-likelihood estimation.

Numerical results confirm LRE's efficiency and accuracy in quantum state reconstruction.

Abstract

A simple yet efficient method of linear regression estimation (LRE) is presented for quantum state tomography. In this method, quantum state reconstruction is converted into a parameter estimation problem of a linear regression model and the least-squares method is employed to estimate the unknown parameters. The asymptotic mean squared error (MSE) bound of the estimate can be given analytically, which can guide one to choose optimal measurement sets. The LRE is asymptotically optimal in the sense that the MSE may achieve the Cram\'{e}r-Rao bound asymptotically. The computational complexity of LRE is O(d^4), where d is the dimension of the quantum state. Numerical examples show that LRE is much faster than maximum-likelihood estimation for quantum state tomography.