Quantum Channel Certification with Incoherent Measurements

Omar Fawzi, Nicolas Flammarion, Aurélien Garivier, Aadil Oufkir
Proceedings of Thirty Sixth Conference on Learning Theory, PMLR 195:1822-1884, 2023.

Abstract

In the problem of quantum channel certification, we have black box access to a quantum process and would like to decide if this process matches some predefined specification or is \eps-far from this specification. The objective is to achieve this task while minimizing the number of times the black box is used. Note that the state certification problem is a special case where the black box has no input. Here, we focus on two relevant extreme cases. The first one is when the predefined specification is a unitary channel, e.g., a gate in a quantum circuit. In this case, we show that testing whether the black box is described by a fixed unitary or \eps-far from it in the trace norm requires Θ(d/\eps2) uses of the black box. The second setting we consider is when the predefined specification is a completely depolarizing channels with input dimension \din and output dimension \dout. In this case, we prove that, in the non-adaptive setting, \TildeΘ(\din2\dout1.5/\eps2) uses of the channel are necessary and sufficient to verify whether it is equal to the depolarizing channel or \eps-far from it in the diamond norm. Finally, we prove a lower bound of Ω(\din2\dout/\eps2) for this problem in the adaptive setting. Note that the special case \din=1 corresponds to the well-studied quantum identity testing problem.

Cite this Paper

BibTeX
@InProceedings{pmlr-v195-fawzi23a, title = {Quantum Channel Certification with Incoherent Measurements}, author = {Fawzi, Omar and Flammarion, Nicolas and Garivier, Aur{\'e}lien and Oufkir, Aadil}, booktitle = {Proceedings of Thirty Sixth Conference on Learning Theory}, pages = {1822--1884}, year = {2023}, editor = {Neu, Gergely and Rosasco, Lorenzo}, volume = {195}, series = {Proceedings of Machine Learning Research}, month = {12--15 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v195/fawzi23a/fawzi23a.pdf}, url = {https://proceedings.mlr.press/v195/fawzi23a.html}, abstract = {In the problem of quantum channel certification, we have black box access to a quantum process and would like to decide if this process matches some predefined specification or is $\eps$-far from this specification. The objective is to achieve this task while minimizing the number of times the black box is used. Note that the state certification problem is a special case where the black box has no input. Here, we focus on two relevant extreme cases. The first one is when the predefined specification is a unitary channel, e.g., a gate in a quantum circuit. In this case, we show that testing whether the black box is described by a fixed unitary or $\eps$-far from it in the trace norm requires $\Theta(d/\eps^2)$ uses of the black box. The second setting we consider is when the predefined specification is a completely depolarizing channels with input dimension $\din$ and output dimension $\dout$. In this case, we prove that, in the non-adaptive setting, $\Tilde{\Theta}(\din^2\dout^{1.5}/\eps^2)$ uses of the channel are necessary and sufficient to verify whether it is equal to the depolarizing channel or $\eps$-far from it in the diamond norm. Finally, we prove a lower bound of $\Omega(\din^2\dout/\eps^2)$ for this problem in the adaptive setting. Note that the special case $\din = 1$ corresponds to the well-studied quantum identity testing problem.} }
Endnote
%0 Conference Paper %T Quantum Channel Certification with Incoherent Measurements %A Omar Fawzi %A Nicolas Flammarion %A Aurélien Garivier %A Aadil Oufkir %B Proceedings of Thirty Sixth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2023 %E Gergely Neu %E Lorenzo Rosasco %F pmlr-v195-fawzi23a %I PMLR %P 1822--1884 %U https://proceedings.mlr.press/v195/fawzi23a.html %V 195 %X In the problem of quantum channel certification, we have black box access to a quantum process and would like to decide if this process matches some predefined specification or is $\eps$-far from this specification. The objective is to achieve this task while minimizing the number of times the black box is used. Note that the state certification problem is a special case where the black box has no input. Here, we focus on two relevant extreme cases. The first one is when the predefined specification is a unitary channel, e.g., a gate in a quantum circuit. In this case, we show that testing whether the black box is described by a fixed unitary or $\eps$-far from it in the trace norm requires $\Theta(d/\eps^2)$ uses of the black box. The second setting we consider is when the predefined specification is a completely depolarizing channels with input dimension $\din$ and output dimension $\dout$. In this case, we prove that, in the non-adaptive setting, $\Tilde{\Theta}(\din^2\dout^{1.5}/\eps^2)$ uses of the channel are necessary and sufficient to verify whether it is equal to the depolarizing channel or $\eps$-far from it in the diamond norm. Finally, we prove a lower bound of $\Omega(\din^2\dout/\eps^2)$ for this problem in the adaptive setting. Note that the special case $\din = 1$ corresponds to the well-studied quantum identity testing problem.
APA
Fawzi, O., Flammarion, N., Garivier, A. & Oufkir, A.. (2023). Quantum Channel Certification with Incoherent Measurements. Proceedings of Thirty Sixth Conference on Learning Theory, in Proceedings of Machine Learning Research 195:1822-1884 Available from https://proceedings.mlr.press/v195/fawzi23a.html.

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