  The linear matter power spectrum is an essential ingredient in all
theoretical models for interpreting large-scale-structure observables. Although
Boltzmann codes such as CLASS or CAMB are very efficient at computing the
linear spectrum, the analysis of data usually requires $10^4$-$10^6$
evaluations, which means this task can be the most computationally expensive
aspect of data analysis. Here, we address this problem by building a neural
network emulator that provides the linear theory (total and cold) matter power
spectrum in about one millisecond with 0.2% (0.5%) accuracy over redshifts $z
\le 3$ ($z \le 9$), and scales $10^{-4} \le k \, [h {\rm Mpc^{-1}}] < 50$. We
train this emulator with more than 200,000 measurements, spanning a broad
cosmological parameter space that includes massive neutrinos and dynamical dark
energy. We show that the parameter range and accuracy of our emulator is enough
to get unbiased cosmological constraints in the analysis of a Euclid-like weak
lensing survey. Complementing this emulator, we train 15 other emulators for
the cross-spectra of various linear fields in Eulerian space, as predicted by
2nd-order Lagrangian Perturbation theory, which can be used to accelerate
perturbative bias descriptions of galaxy clustering. Our emulators are
specially designed to be used in combination with emulators for the nonlinear
matter power spectrum and for baryonic effects, all of which are publicly
available at http://www.dipc.org/bacco.
