A note on tomography of states with low stabilizer rank

Abstract

In this note, we consider tomography of $n$-qubit quantum states $|\psi \rangle$ that are promised to have stabilizer rank $k$ i.e., admit a stabilizer decomposition of $|\psi\rangle = \sum_{i=1}^k c_i |\phi_i\rangle$ where each $|\phi_i\rangle$ is a stabilizer state. We give a tomography protocol for such states up to trace distance $\varepsilon>0$ running in $\textsf{poly}(n,2^k,1/\varepsilon)$ time and thus can handle $k=O(\log n)$ efficiently. This follows from first showing that the stabilizer extent $\xi(|\psi\rangle)$, which is the minimal $\ell_1$ norm over the coefficients $\{c_i\}$ of any stabilizer decomposition of $|\psi\rangle$, can be bounded as $\xi(|\psi\rangle) \leq 2^{O(k)}$ 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.