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

Abstract

Quantum state tomography (QST) is a central task in quantum information, and its efficiency is commonly characterized by sample complexity. Although collective measurements on multiple copies achieve optimal performance, they are difficult to implement on near-term devices, motivating the study of single-copy approaches. Here, we introduce an ancilla-free single-copy QST protocol based on logarithmic-depth local circuits on an $n$-qubit system. For rank-$r$ states in dimension $d=2^n$, our protocol achieves trace-norm error $\epsilon$ using $\mathcal{O}(d{r}^2\log d/{\epsilon }^2)$ copies, matching the single-copy lower bound up to a logarithmic factor. For full-rank mixed states, it removes this logarithmic overhead and achieves the optimal scaling $\mathcal{O}(d^3/{\epsilon }^2)$, with nearly optimal classical runtime for explicit matrix output. These results show that sample-optimal QST can be realized using experimentally accessible shallow-depth measurements.