A note on tomography of states with low stabilizer rank

Abstract

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

arXiv comment7 pages