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Quantum State Tomography via Linear Regression Estimation
Bo Qi, Zhibo Hou, Li Li, Daoyi Dong, Guoyong Xiang, Guangcan Guo
TL;DR
Quantum state tomography needs efficient reconstruction and analytically evaluable measurement-set performance, but local-measurement settings impose a scope boundary. The paper uses LRE by converting tomography into linear regression and applying least squares, with extensions to evolving systems and broader linear measurement settings. It reports an analytically measurement-dependent MSE upper bound, identifies measurement sets attaining its minimum, and gives computational complexity O(d^4).
Problem
Quantum state tomography requires efficient reconstruction and analytical evaluation of measurement-set performance, while the local-measurement setting constrains the applicable measurement structure.
Method
LRE converts quantum state tomography into a linear regression problem and applies least squares to estimate the unknown parameters.
Results
The method provides an analytical MSE upper bound dependent on measurement bases, with the minimum reached by mutually unbiased bases and, for local measurements, 2-qubit cube or tetrahedron sets; its complexity is O(d^4).
Takeaways & Limitations
The analytical bound can guide measurement-set selection, and LRE applies whenever measurable quantities are linearly related to density-matrix elements.
Takeaways & Limitations
The local-measurement analysis assumes product-form measurement projectors on the two subsystems.
Abstract
from arXiv · showhide
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 Cramér-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.
20. Hence, the minimum of the MSE
The paper derives MSE upper bounds that depend on the measurement bases and identifies measurement sets attaining the minimum under global or local measurement settings. It also shows how LRE extends to evolving systems and broader reconstruction settings.
- For Werner states measured with cube bases, the MSE varies with q and the number of copies N, while PLRE has larger MSE than LRE.The reported difference indicates that pulling the estimate back to a physical state further reduces estimation error.
- The minimum MSE upper bound can be reached using mutually unbiased measurement bases.
- Under local measurements, the minimum MSE upper bound can be reached by the 2-qubit cube or tetrahedron measurement set.
- LRE can reconstruct states from measurements of an observable evolving under a unitary group by expressing the resulting averages as linear regression equations.
- LRE can be extended to states with prior information or open quantum systems whenever measurable quantities are linearly related to density-matrix elements.
- The MSE upper bound depends explicitly on the chosen measurement bases, enabling analytical comparison of measurement-set optimality.