Training the Quantum Network

We now set up a simulation of the quantum neural network. We specify as inputs the initial $\left( t=0\right)$ polarizations of each of two quantum dot molecules, spatially far enough from each other that they do not interact directly, but sharing the same substrate. The system output $\left( \mid \psi \left( \sigma _{z}\left( N\Delta t\right) ,T\right) \rangle \right)$ represents a combination of the basic states of polarization, say $\mid +\rangle$ and $\mid -\rangle$. To define a training rule we have to define a scalar function of the system output whose value is thresholded to decide if the desired behavior has been reproduced (a quantum logic gate, for example). In [4] the polarization of the first molecule at the final time is arbitrarily taken. Thus, the probability amplitude for the first molecules final state to be equal to the $\mid +\rangle$ state is computed (give by $\left\vert \langle +\mid \psi \left( \sigma _{z}\left( N\Delta t\right) ,T\right) \rangle \right\vert ^{2}$) and the signal of the following expression considered:

$$Out=\left\vert \langle +\mid \psi \left( \sigma _{z}\left( N\Delta t\right) ,T\right) \rangle \right\vert ^{2}-Desired$$ (11)

if $Out>0$ the network is considered to be trained. To achieve this goal, an Error Function is defined and a gradient descent algorithm was used for training:

$$Error=\frac{1}{4}Out^{2};\quad \lambda _{k}^{new}=\lambda _{k}^{old}-\eta \frac{\partial Error}{\partial \lambda _{k}},$$ (12)

where

$$\frac{\partial Error}{\partial \lambda _{k}}=\frac{1}{2} Ou... ...lambda _{k}}\langle +\mid \psi \rangle ^{*}\right] \right]$$ (13)

We shall emphasize that it is possible to train purely quantum gates such as a phase shift, because the network is quantum mechanical.