Instance-optimal high-precision shadow tomography with few-copy measurements: A metrological approach

Senrui Chen, Weiyuan Gong, Sisi Zhou
Proceedings of Thirty Ninth Conference on Learning Theory, PMLR 336:1115-1185, 2026.

Abstract

We give the first instance-optimal sample complexity bounds for shadow tomography using few-copy measurements in the high-precision regime. More concretely, we study the problem of learning expectation values of a given set of observables of an unknown quantum state to precision $\epsilon$ in $L_p$-norm, using (possibly adaptive) measurements that act on one or a few copies at a time, and we are interested in the regime that $\epsilon$ is below some concrete and potentially dimension-dependent threshold. In this setup, we prove the necessary and sufficient number of copies, for any given set of observables, is characterized by a simple optimization formula involving a quadratic form of the inverse Fisher information matrix up to a logarithmic factor. Our results establish a rigorous correspondence between quantum learning and quantum metrology.

Cite this Paper

BibTeX
@InProceedings{pmlr-v336-chen26c, title = {Instance-optimal high-precision shadow tomography with few-copy measurements: A metrological approach}, author = {Chen, Senrui and Gong, Weiyuan and Zhou, Sisi}, booktitle = {Proceedings of Thirty Ninth Conference on Learning Theory}, pages = {1115--1185}, year = {2026}, editor = {Hanneke, Steve and Lattimore, Tor}, volume = {336}, series = {Proceedings of Machine Learning Research}, month = {29 Jun--03 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v336/main/assets/chen26c/chen26c.pdf}, url = {https://proceedings.mlr.press/v336/chen26c.html}, abstract = {We give the first instance-optimal sample complexity bounds for shadow tomography using few-copy measurements in the high-precision regime. More concretely, we study the problem of learning expectation values of a given set of observables of an unknown quantum state to precision $\epsilon$ in $L_p$-norm, using (possibly adaptive) measurements that act on one or a few copies at a time, and we are interested in the regime that $\epsilon$ is below some concrete and potentially dimension-dependent threshold. In this setup, we prove the necessary and sufficient number of copies, for any given set of observables, is characterized by a simple optimization formula involving a quadratic form of the inverse Fisher information matrix up to a logarithmic factor. Our results establish a rigorous correspondence between quantum learning and quantum metrology. } }
Endnote
%0 Conference Paper %T Instance-optimal high-precision shadow tomography with few-copy measurements: A metrological approach %A Senrui Chen %A Weiyuan Gong %A Sisi Zhou %B Proceedings of Thirty Ninth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2026 %E Steve Hanneke %E Tor Lattimore %F pmlr-v336-chen26c %I PMLR %P 1115--1185 %U https://proceedings.mlr.press/v336/chen26c.html %V 336 %X We give the first instance-optimal sample complexity bounds for shadow tomography using few-copy measurements in the high-precision regime. More concretely, we study the problem of learning expectation values of a given set of observables of an unknown quantum state to precision $\epsilon$ in $L_p$-norm, using (possibly adaptive) measurements that act on one or a few copies at a time, and we are interested in the regime that $\epsilon$ is below some concrete and potentially dimension-dependent threshold. In this setup, we prove the necessary and sufficient number of copies, for any given set of observables, is characterized by a simple optimization formula involving a quadratic form of the inverse Fisher information matrix up to a logarithmic factor. Our results establish a rigorous correspondence between quantum learning and quantum metrology.
APA
Chen, S., Gong, W. & Zhou, S.. (2026). Instance-optimal high-precision shadow tomography with few-copy measurements: A metrological approach. Proceedings of Thirty Ninth Conference on Learning Theory, in Proceedings of Machine Learning Research 336:1115-1185 Available from https://proceedings.mlr.press/v336/chen26c.html.

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