We investigate the tomography problem for $k$‑sparse quantum states. Unlike classical distribution learning, where tight sample and time bounds in terms of support size are well understood, prior work offered no non‑trivial guarantees for sparse quantum states. This paper presents the first near‑optimal algorithm for learning $n$‑qubit $k$‑sparse pure states. With high probability it achieves fidelity at least $1-\varepsilon$ using only $$\tilde{O}\left(\frac{k}{\varepsilon}\right)$$ copies of the state and runs in $$\tilde{O}\left(\frac{kn}{\varepsilon}\right)$$ time. Both bounds are optimal up to polylogarithmic factors. By employing a random purification channel, we extend the technique to $k$‑sparse rank‑$r$ mixed states, obtaining a sample complexity of $$\tilde{O}\left(\frac{kr}{\varepsilon}\right)$$, again near optimal. An important open question remains: for $r>1$, can we achieve a time complexity that nearly matches the sample complexity?
Review: The work bridges a gap between classical sparse distribution learning and quantum state tomography, delivering sample‑optimal and almost time‑optimal procedures. It opens a clear path toward efficient tomography of high‑dimensional quantum systems and highlights the challenge of further reducing runtime for mixed‑state scenarios.