Detecting mixed-unitary quantum channels is NP-hard

Colin Do-Yan Lee and John Watrous

Institute for Quantum Computing and School of Computer Science, University of Waterloo, Canada

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Abstract

A quantum channel is said to be a $\textit{mixed-unitary}$ channel if it can be expressed as a convex combination of unitary channels. We prove that, given the Choi representation of a quantum channel $\Phi$, it is NP-hard with respect to polynomial-time Turing reductions to determine whether or not $\Phi$ is a mixed-unitary channel. This hardness result holds even under the assumption that $\Phi$ is not within an inverse-polynomial distance (in the dimension of the space upon which $\Phi$ acts) of the boundary of the mixed-unitary channels.

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Cited by

[1] Mark Girard, Debbie Leung, Jeremy Levick, Chi-Kwong Li, Vern Paulsen, Yiu Tung Poon, and John Watrous, "On the mixed-unitary rank of quantum channels", arXiv:2003.14405.

[2] Uma Girish, Ran Raz, and Wei Zhan, "Quantum Logspace Algorithm for Powering Matrices with Bounded Norm", arXiv:2006.04880.

[3] D. W. Kribs, J. Levick, K. Olfert, R. Pereira, and M. Rahaman, "Nullspaces of Entanglement Breaking Channels and Applications", arXiv:2007.15893.

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