Efficient Approximation of Quantum Channel Fidelity Exploiting Symmetry

TL;DR

This paper presents a symmetry-exploiting method to efficiently approximate quantum channel fidelity, reducing computational complexity from exponential to polynomial in hierarchy level and input dimension.

Contribution

It introduces a technique to compute semidefinite programming bounds for quantum fidelity in polynomial time by leveraging symmetries, enabling scalable approximations.

Findings

SDP computation time is polynomial in hierarchy level and input dimension.

Approximate quantum fidelity within epsilon can be achieved efficiently.

Symmetry exploitation significantly reduces computational complexity.

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.