Pauli Measurements Are Near-Optimal for Single-Qubit Tomography

TL;DR

This paper establishes near-matching lower bounds for the number of copies needed in single-qubit tomography, demonstrating that Pauli measurements are nearly optimal for this task.

Contribution

It provides the first non-trivial lower bounds for multi-qubit tomography and shows that Pauli measurements are close to optimal, nearly settling the complexity question.

Findings

Lower bounds for multi-qubit tomography are established.

Pauli measurements are shown to be nearly optimal.

The results nearly settle the complexity of single-qubit tomography.

Abstract

We provide the first non-trivial lower bounds for single-qubit tomography algorithms and show that at least ${\Omega}\left(\frac{10^N}{\sqrt{N} \varepsilon^2}\right)$ copies are required to learn an $N$-qubit state $\rho\in\mathbb{C}^{d\times d},d=2^N$ to within $\varepsilon$ trace distance. Pauli measurements, the most commonly used single-qubit measurement scheme, have recently been shown to require at most $O\left(\frac{10^N}{\varepsilon^2}\right)$ copies for this problem. Combining these results, we nearly settle the long-standing question of the complexity of single-qubit tomography.