Efficient Approximation of Quantum Channel Fidelity Exploiting Symmetry
This work makes the computation of optimal quantum channel fidelity scalable for fixed output dimension, addressing a key bottleneck in quantum information theory.
The authors exploit symmetries in semidefinite programs to compute quantum channel fidelity approximations in polynomial time, achieving ε-accuracy in poly(1/ε, input dimension) time.
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 $ε$ in $\mathrm{poly}(1/ε, \text{input dimension})$ time.