Detecting mixed-unitary quantum channels is NP-hard
Institute for Quantum Computing and School of Computer Science, University of Waterloo, Canada
Published: | 2020-04-16, volume 4, page 253 |
Eprint: | arXiv:1902.03164v3 |
Doi: | https://doi.org/10.22331/q-2020-04-16-253 |
Citation: | Quantum 4, 253 (2020). |
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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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[2] Otfried Gühne, Yuanyuan Mao, and Xiao-Dong Yu, "Geometry of Faithful Entanglement", Physical Review Letters 126 14, 140503 (2021).
[3] 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.
[4] Cristhiano Duarte, Barbara Amaral, Marcelo Terra Cunha, and Matthew Leifer, "Investigating Coarse-Grainings and Emergent Quantum Dynamics with Four Mathematical Perspectives", arXiv:2011.10349.
[5] Uma Girish, Ran Raz, and Wei Zhan, "Quantum Logspace Algorithm for Powering Matrices with Bounded Norm", arXiv:2006.04880.
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