Sample-optimal single-copy quantum state tomography via shallow depth measurements

Quantum state tomography (QST) is one of the fundamental problems in quantum information. Among various metrics, sample complexity is widely used to evaluate QST algorithms. While multi-copy measurements are known to achieve optimal sample complexity, they are challenging to implement on near-term quantum devices. In practice, single-copy measurements with shallow-depth circuits are more feasible. Although a near-optimal QST algorithm under single-qubit measurements has recently been proposed, its sample complexity does not match the known lower bound for single-copy measurements. Here, we make two contributions by employing circuits with depth $\mathcal{O}(\log n)$ on an $n$-qubit system. First, QST for rank-$r$ $d$-dimensional state $\rho$ can be achieved with sample complexity $\mathcal{O}\text{\hspace{0.17em}⁣}\left(\frac{d{r}^2\ln d}{{\epsilon }^2}\right)$ to error $\epsilon$ in trace distance, which is near-optimal up to a $\ln d$ factor compared to the known lower bound $\mathrm{\Omega }\left(\frac{d{r}^2}{{\epsilon }^2}\right)$. Second, for the general case of $r=d$, we can remove the $\ln d$ factor, yielding an optimal sample complexity of $\mathcal{O}\text{\hspace{0.17em}⁣}\left(\frac{d^3}{{\epsilon }^2}\right)$.