Predicting quantum channels over general product distributions

Sitan Chen, Jaume de Dios Pont, Jun-Ting Hsieh, Hsin-Yuan Huang, Jane Lange, Jerry Li
Proceedings of Thirty Eighth Conference on Learning Theory, PMLR 291:986-1007, 2025.

Abstract

We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an $n$-qubit channel $\mathcal{E}$ and an observable $\mathcal{O}$, we aim to learn the mapping \begin{equation*} \rho \mapsto \Tr(\mathcal{O} \mathcal{E}[\rho]) \end{equation*} to within a small error for most $\rho$ sampled from a distribution $\mathcal{D}$. Previously, Huang et al. proved a surprising result that even if $\mathcal{E}$ is arbitrary, this task can be solved in time roughly $n^{O(\log(1/\epsilon))}$, where $\epsilon$ is the target prediction error. However, their guarantee applied only to input distributions $\mathcal{D}$ invariant under all single-qubit Clifford gates, and their algorithm fails for important cases such as general product distributions over product states $\rho$. In this work, we propose a new approach that achieves accurate prediction over essentially any product distribution $\mathcal{D}$, provided it is not “classical” in which case there is a trivial exponential lower bound. Our method employs a “biased Pauli analysis,” analogous to classical biased Fourier analysis. Implementing this approach requires overcoming several challenges unique to the quantum setting, including the lack of a basis with appropriate orthogonality properties. The techniques we develop to address these issues may have broader applications in quantum information.

Cite this Paper

BibTeX
@InProceedings{pmlr-v291-chen25c, title = {Predicting quantum channels over general product distributions}, author = {Chen, Sitan and {de Dios Pont}, Jaume and Hsieh, Jun-Ting and Huang, Hsin-Yuan and Lange, Jane and Li, Jerry}, booktitle = {Proceedings of Thirty Eighth Conference on Learning Theory}, pages = {986--1007}, 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/chen25c/chen25c.pdf}, url = {https://proceedings.mlr.press/v291/chen25c.html}, abstract = {We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an $n$-qubit channel $\mathcal{E}$ and an observable $\mathcal{O}$, we aim to learn the mapping \begin{equation*} \rho \mapsto \Tr(\mathcal{O} \mathcal{E}[\rho]) \end{equation*} to within a small error for most $\rho$ sampled from a distribution $\mathcal{D}$. Previously, Huang et al. proved a surprising result that even if $\mathcal{E}$ is arbitrary, this task can be solved in time roughly $n^{O(\log(1/\epsilon))}$, where $\epsilon$ is the target prediction error. However, their guarantee applied only to input distributions $\mathcal{D}$ invariant under all single-qubit Clifford gates, and their algorithm fails for important cases such as general product distributions over product states $\rho$. In this work, we propose a new approach that achieves accurate prediction over essentially any product distribution $\mathcal{D}$, provided it is not “classical” in which case there is a trivial exponential lower bound. Our method employs a “biased Pauli analysis,” analogous to classical biased Fourier analysis. Implementing this approach requires overcoming several challenges unique to the quantum setting, including the lack of a basis with appropriate orthogonality properties. The techniques we develop to address these issues may have broader applications in quantum information. } }
Endnote
%0 Conference Paper %T Predicting quantum channels over general product distributions %A Sitan Chen %A Jaume de Dios Pont %A Jun-Ting Hsieh %A Hsin-Yuan Huang %A Jane Lange %A Jerry Li %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-chen25c %I PMLR %P 986--1007 %U https://proceedings.mlr.press/v291/chen25c.html %V 291 %X We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an $n$-qubit channel $\mathcal{E}$ and an observable $\mathcal{O}$, we aim to learn the mapping \begin{equation*} \rho \mapsto \Tr(\mathcal{O} \mathcal{E}[\rho]) \end{equation*} to within a small error for most $\rho$ sampled from a distribution $\mathcal{D}$. Previously, Huang et al. proved a surprising result that even if $\mathcal{E}$ is arbitrary, this task can be solved in time roughly $n^{O(\log(1/\epsilon))}$, where $\epsilon$ is the target prediction error. However, their guarantee applied only to input distributions $\mathcal{D}$ invariant under all single-qubit Clifford gates, and their algorithm fails for important cases such as general product distributions over product states $\rho$. In this work, we propose a new approach that achieves accurate prediction over essentially any product distribution $\mathcal{D}$, provided it is not “classical” in which case there is a trivial exponential lower bound. Our method employs a “biased Pauli analysis,” analogous to classical biased Fourier analysis. Implementing this approach requires overcoming several challenges unique to the quantum setting, including the lack of a basis with appropriate orthogonality properties. The techniques we develop to address these issues may have broader applications in quantum information.
APA
Chen, S., de Dios Pont, J., Hsieh, J., Huang, H., Lange, J. & Li, J.. (2025). Predicting quantum channels over general product distributions. Proceedings of Thirty Eighth Conference on Learning Theory, in Proceedings of Machine Learning Research 291:986-1007 Available from https://proceedings.mlr.press/v291/chen25c.html.

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