  We provide in this paper a concrete method for training a quantum neural
network to maximize the relevant information about a property that is
transmitted through the network. This is significant because it gives an
operationally well founded quantity to optimize when training autoencoders for
problems where the inputs and outputs are fully quantum. We provide a rigorous
algorithm for computing the value of the quantum information bottleneck
quantity within error $\epsilon$ that requires $O(\log^2(1/\epsilon) +
1/\delta^2)$ queries to a purification of the input density operator if its
spectrum is supported on $\{0\}~\bigcup ~[\delta,1-\delta]$ for $\delta>0$ and
the kernels of the relevant density matrices are disjoint. We further provide
algorithms for estimating the derivatives of the QIB function, showing that
quantum neural networks can be trained efficiently using the QIB quantity given
that the number of gradient steps required is polynomial.
