A note on tomography of states with low stabilizer rank

In this note, we consider tomography of $n$-qubit quantum states $\mathrm{\mid }\psi ⟩$ that are promised to have stabilizer rank $k$ i.e., admit a stabilizer decomposition of $\mathrm{\mid }\psi ⟩={\sum }_{i=1}^k{c}_i\mathrm{\mid }{\phi }_i⟩$ where each $\mathrm{\mid }{\phi }_i⟩$ is a stabilizer state. We give a tomography protocol for such states up to trace distance $\epsilon >0$ running in $\mathsf{\text{poly}}(n,2^k,1\mathrm{/}\epsilon )$ time and thus can handle $k=O(\log n)$ efficiently. This follows from first showing that the stabilizer extent $\xi (\mathrm{\mid }\psi ⟩)$, which is the minimal ${\mathrm{\ell }}_1$ norm over the coefficients $\{c_i\}$ of any stabilizer decomposition of $\mathrm{\mid }\psi ⟩$, can be bounded as $\xi (\mathrm{\mid }\psi ⟩)\le 2^{O(k)}$ and improving upon the previous best bound of $k^{O(k)}$. The protocol is then obtained by utilizing a tomography protocol for states with bounded stabilizer extent.