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

Abstract: 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 $\mathcal{O}!\left(\tfrac{dr^2 \ln d}{\epsilon^2}\right)$ to error $\epsilon$ in trace distance, which is near-optimal up to a $\ln d$ factor compared to the known lower bound $\Omega\left(\frac{dr^2}{\epsilon^2}\right)$. Second, for the general case of $r = d$, we can remove the $\ln d$ factor, yielding an optimal sample complexity of $\mathcal{O}!\left(\frac{d^3}{\epsilon^2}\right)$.