Majorization and the time complexity of linear optical networks.

This work shows that the majorization of photon distributions is related to the runtime of classically simulating multimode passive linear optics, which explains one aspect of the boson sampling hardness. A Shur-concave quantity which we name the \emph{Boltzmann entropy of elementary quantum complexity} ($S_B^q$) is introduced to present some quantitative analysis of the relation between the majorization and the classical runtime for simulating linear optics. We compare $S_B^q$ with two quantities that are important criteria for understanding the computational cost of the photon scattering process, $\mathcal{T}$ (the runtime for the classical simulation of linear optics) and $\mathcal{E}$ (the additive error bound for an approximated amplitude estimator). First, for all the known algorithms for computing the permanents of matrices with repeated rows and columns, the runtime $\mathcal{T}$ becomes shorter as the input and output distribution vectors are more majorized. Second, the error bound $\mathcal{E}$ decreases as the majorization difference of input and output states increases. We expect that our current results would help in understanding the feature of linear optical networks from the perspective of quantum computation.