Learning Distributions over Quantum Measurement Outcomes

Weiyuan Gong, Scott Aaronson
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:11598-11613, 2023.

Abstract

Shadow tomography for quantum states provides a sample efficient approach for predicting the measurement outcomes of quantum systems. However, these shadow tomography procedures yield poor bounds if there are more than two outcomes per measurement. In this paper, we consider a general problem of learning properties from quantum states: given an unknown $d$-dimensional quantum state $\rho$ and $M$ unknown quantum measurements $\mathcal{M}_1,...,\mathcal{M}_M$ with $K\geq 2$ outcomes, estimating the probability distribution for applying $\mathcal{M}_i$ on $\rho$ to within total variation distance $\epsilon$. Compared to the special case when $K=2$, we have to learn unknown distributions instead of values. Here, we propose an online shadow tomography procedure that solves this problem with high success probability requiring $\tilde{O}(K\log^2M\log d/\epsilon^4)$ copies of $\rho$. We further prove an information-theoretic lower bound showing that at least $\Omega(\min\{d^2,K+\log M\}/\epsilon^2)$ copies of $\rho$ are required to solve this problem with high success probability. Our shadow tomography procedure requires sample complexity with only logarithmic dependence on $M$ and $d$ and is sample-optimal concerning the dependence on $K$.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-gong23a, title = {Learning Distributions over Quantum Measurement Outcomes}, author = {Gong, Weiyuan and Aaronson, Scott}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {11598--11613}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/gong23a/gong23a.pdf}, url = {https://proceedings.mlr.press/v202/gong23a.html}, abstract = {Shadow tomography for quantum states provides a sample efficient approach for predicting the measurement outcomes of quantum systems. However, these shadow tomography procedures yield poor bounds if there are more than two outcomes per measurement. In this paper, we consider a general problem of learning properties from quantum states: given an unknown $d$-dimensional quantum state $\rho$ and $M$ unknown quantum measurements $\mathcal{M}_1,...,\mathcal{M}_M$ with $K\geq 2$ outcomes, estimating the probability distribution for applying $\mathcal{M}_i$ on $\rho$ to within total variation distance $\epsilon$. Compared to the special case when $K=2$, we have to learn unknown distributions instead of values. Here, we propose an online shadow tomography procedure that solves this problem with high success probability requiring $\tilde{O}(K\log^2M\log d/\epsilon^4)$ copies of $\rho$. We further prove an information-theoretic lower bound showing that at least $\Omega(\min\{d^2,K+\log M\}/\epsilon^2)$ copies of $\rho$ are required to solve this problem with high success probability. Our shadow tomography procedure requires sample complexity with only logarithmic dependence on $M$ and $d$ and is sample-optimal concerning the dependence on $K$.} }
Endnote
%0 Conference Paper %T Learning Distributions over Quantum Measurement Outcomes %A Weiyuan Gong %A Scott Aaronson %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-gong23a %I PMLR %P 11598--11613 %U https://proceedings.mlr.press/v202/gong23a.html %V 202 %X Shadow tomography for quantum states provides a sample efficient approach for predicting the measurement outcomes of quantum systems. However, these shadow tomography procedures yield poor bounds if there are more than two outcomes per measurement. In this paper, we consider a general problem of learning properties from quantum states: given an unknown $d$-dimensional quantum state $\rho$ and $M$ unknown quantum measurements $\mathcal{M}_1,...,\mathcal{M}_M$ with $K\geq 2$ outcomes, estimating the probability distribution for applying $\mathcal{M}_i$ on $\rho$ to within total variation distance $\epsilon$. Compared to the special case when $K=2$, we have to learn unknown distributions instead of values. Here, we propose an online shadow tomography procedure that solves this problem with high success probability requiring $\tilde{O}(K\log^2M\log d/\epsilon^4)$ copies of $\rho$. We further prove an information-theoretic lower bound showing that at least $\Omega(\min\{d^2,K+\log M\}/\epsilon^2)$ copies of $\rho$ are required to solve this problem with high success probability. Our shadow tomography procedure requires sample complexity with only logarithmic dependence on $M$ and $d$ and is sample-optimal concerning the dependence on $K$.
APA
Gong, W. & Aaronson, S.. (2023). Learning Distributions over Quantum Measurement Outcomes. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:11598-11613 Available from https://proceedings.mlr.press/v202/gong23a.html.

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