Sample-optimal single-copy quantum state tomography via 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 a 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 $\mathcal{O}(dr^2\log d/\epsilon^2)$ 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.

arXiv:2509.12703 [quant-ph]

(or arXiv:2509.12703v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2509.12703 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.