Improved sample upper and lower bounds for trace estimation of quantum state powers

Kean Chen, Qisheng Wang
Proceedings of Thirty Eighth Conference on Learning Theory, PMLR 291:1008-1028, 2025.

Abstract

As often emerges in various basic quantum properties such as entropy, the trace of quantum state powers $\operatorname{tr}(\rho^q)$ has attracted a lot of attention. The recent work of Liu and Wang (SODA 2025) showed that $\operatorname{tr}(\rho^q)$ can be estimated to within additive error $\varepsilon$ with a dimension-independent sample complexity of $\widetilde O(1/\varepsilon^{3+\frac{2}{q-1}})$ for any constant $q > 1$, where only an $\Omega(1/\varepsilon)$ lower bound was given. In this paper, we significantly improve the sample complexity of estimating $\operatorname{tr}(\rho^q)$ in both the upper and lower bounds. In particular: - For $q > 2$, we settle the sample complexity with matching upper and lower bounds $\widetilde \Theta(1/\varepsilon^2)$. - For $1 < q < 2$, we provide an upper bound $\widetilde O(1/\varepsilon^{\frac{2}{q-1}})$, with a lower bound $\Omega(1/\varepsilon^{\max\{\frac{1}{q-1}, 2\}})$ for dimension-independent estimators, implying there is only room for a quadratic improvement. Our upper bounds are obtained by (non-plug-in) quantum estimators based on weak Schur sampling, in sharp contrast to the prior approach based on quantum singular value transformation and samplizer.

Cite this Paper

BibTeX
@InProceedings{pmlr-v291-chen25d, title = {Improved sample upper and lower bounds for trace estimation of quantum state powers}, author = {Chen, Kean and Wang, Qisheng}, booktitle = {Proceedings of Thirty Eighth Conference on Learning Theory}, pages = {1008--1028}, year = {2025}, editor = {Haghtalab, Nika and Moitra, Ankur}, volume = {291}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--04 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v291/main/assets/chen25d/chen25d.pdf}, url = {https://proceedings.mlr.press/v291/chen25d.html}, abstract = {As often emerges in various basic quantum properties such as entropy, the trace of quantum state powers $\operatorname{tr}(\rho^q)$ has attracted a lot of attention. The recent work of Liu and Wang (SODA 2025) showed that $\operatorname{tr}(\rho^q)$ can be estimated to within additive error $\varepsilon$ with a dimension-independent sample complexity of $\widetilde O(1/\varepsilon^{3+\frac{2}{q-1}})$ for any constant $q > 1$, where only an $\Omega(1/\varepsilon)$ lower bound was given. In this paper, we significantly improve the sample complexity of estimating $\operatorname{tr}(\rho^q)$ in both the upper and lower bounds. In particular: - For $q > 2$, we settle the sample complexity with matching upper and lower bounds $\widetilde \Theta(1/\varepsilon^2)$. - For $1 < q < 2$, we provide an upper bound $\widetilde O(1/\varepsilon^{\frac{2}{q-1}})$, with a lower bound $\Omega(1/\varepsilon^{\max\{\frac{1}{q-1}, 2\}})$ for dimension-independent estimators, implying there is only room for a quadratic improvement. Our upper bounds are obtained by (non-plug-in) quantum estimators based on weak Schur sampling, in sharp contrast to the prior approach based on quantum singular value transformation and samplizer.} }
Endnote
%0 Conference Paper %T Improved sample upper and lower bounds for trace estimation of quantum state powers %A Kean Chen %A Qisheng Wang %B Proceedings of Thirty Eighth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2025 %E Nika Haghtalab %E Ankur Moitra %F pmlr-v291-chen25d %I PMLR %P 1008--1028 %U https://proceedings.mlr.press/v291/chen25d.html %V 291 %X As often emerges in various basic quantum properties such as entropy, the trace of quantum state powers $\operatorname{tr}(\rho^q)$ has attracted a lot of attention. The recent work of Liu and Wang (SODA 2025) showed that $\operatorname{tr}(\rho^q)$ can be estimated to within additive error $\varepsilon$ with a dimension-independent sample complexity of $\widetilde O(1/\varepsilon^{3+\frac{2}{q-1}})$ for any constant $q > 1$, where only an $\Omega(1/\varepsilon)$ lower bound was given. In this paper, we significantly improve the sample complexity of estimating $\operatorname{tr}(\rho^q)$ in both the upper and lower bounds. In particular: - For $q > 2$, we settle the sample complexity with matching upper and lower bounds $\widetilde \Theta(1/\varepsilon^2)$. - For $1 < q < 2$, we provide an upper bound $\widetilde O(1/\varepsilon^{\frac{2}{q-1}})$, with a lower bound $\Omega(1/\varepsilon^{\max\{\frac{1}{q-1}, 2\}})$ for dimension-independent estimators, implying there is only room for a quadratic improvement. Our upper bounds are obtained by (non-plug-in) quantum estimators based on weak Schur sampling, in sharp contrast to the prior approach based on quantum singular value transformation and samplizer.
APA
Chen, K. & Wang, Q.. (2025). Improved sample upper and lower bounds for trace estimation of quantum state powers. Proceedings of Thirty Eighth Conference on Learning Theory, in Proceedings of Machine Learning Research 291:1008-1028 Available from https://proceedings.mlr.press/v291/chen25d.html.

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