Fast state tomography with optimal error bounds

TL;DR

This paper provides a rigorous, non-asymptotic confidence region for the projected least squares method in quantum state tomography, achieving near-optimal sample complexity across various measurement schemes.

Contribution

It introduces a non-asymptotic confidence region for PLS in quantum tomography, with analysis applicable to multiple measurement types and matching fundamental lower bounds in certain cases.

Findings

Sample complexity matches the best known convergence guarantees.

Confidence regions are valid for a variety of measurements including 2-designs.

Numerical simulations confirm theoretical results.

Abstract

Projected least squares (PLS) is an intuitive and numerically cheap technique for quantum state tomography. The method first computes the least-squares estimator (or a linear inversion estimator) and then projects the initial estimate onto the space of states. The main result of this paper equips this point estimator with a rigorous, non-asymptotic confidence region expressed in terms of the trace distance. The analysis holds for a variety of measurements, including 2-designs and Pauli measurements. The sample complexity of the estimator is comparable to the strongest convergence guarantees available in the literature and -- in the case of measuring the uniform POVM -- saturates fundamental lower bounds.The results are derived by reinterpreting the least-squares estimator as a sum of random matrices and applying a matrix-valued concentration inequality. The theory is supported by numerical simulations for mutually unbiased bases, Pauli observables, and Pauli basis measurements.