to converge to zero. The non-monotonic decrease is to be expected as we are varying the input states. The network now teleports any qubit state: picking 1000 states at random from the Haar measure (uniform distribution over the Bloch sphere) gives a cost function distribution with mean $5.0371\times 10^{-4}$ and standard deviation $1.7802\times 10^{-4}$, which is effectively zero.](teleportation_cost.png){height="5cm"}

Discussion {#discussion .unnumbered}
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Quantum vs. classical {#quantum-vs.-classical .unnumbered}
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Can these neural networks show some form of quantum supremacy? The comparison of classical and quantum neural networks is well-defined within our set-up, as the classical networks correspond to a particular parameter regime for the quantum networks. A key type of quantum supremacy is that the quantum network can take and process quantum inputs: it can for example process $\ket{+}$ and $\ket{-}$ differently. Thus, there are numerous quantum tasks it can do that the classical network cannot, including the two examples above. We anticipate that they will moreover, in some cases be able to process classical inputs faster, by turning them into superpositions—investigating this is a natural follow-on from this work.

We also mention that we term our above design a [*quantum*]{} neural network with [*classical learning parameters*]{}, as the parameters in 