Efficient Approximation of Quantum Channel Fidelity Exploiting Symmetry

• URL: http://arxiv.org/abs/2308.15884v3
• Date: Thu, 21 Mar 2024 04:01:35 GMT
• Title: Efficient Approximation of Quantum Channel Fidelity Exploiting Symmetry
• Authors: Yeow Meng Chee, Hoang Ta, Van Khu Vu,
• Abstract summary: We show that for a fixed output dimension of a quantum channel, we can compute the SDP in time with respect to the level of the hierarchy and input dimension. As a consequence of our result, the optimal fidelity can be approximated with an accuracy of $epsilon$ in $mathrmpoly (1/epsilon, textinput dimension)$ time.
• Score: 14.524074846672526
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of $\epsilon$ in $\mathrm{poly}(1/\epsilon, \text{input dimension})$ time.

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