Pauli Measurements Are Near-Optimal for Single-Qubit Tomography

We provide the first non-trivial lower bounds for single-qubit tomography algorithms and show that at least $Ω\left(\frac{10^N}{\sqrt{N} \varepsilon^2}\right)$ copies are required to learn an $N$-qubit state $ρ\in\mathbb{C}^{d\times d},d=2^N$ 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.