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In [4] we found a mathematical formulation of a quantum
neural network through a quantum dot molecule coupled to the substrate
lattice through optical phonons, and subject to a timevarying external
field. In this case, the nonlinearity is a consequence of the realtime propagation of the
system coupled to its environment. Dissipation is not considered
here, although the general model can incorporate this possibility [6,4].
Using discretized Feynman path integrals, authors found that the real time
evolution of the system can be put into a form which resembles the equations
for the virtual neuron activation levels of an artificial neural
network. The timeline discretization points serve as virtual neurons.
Through the Feynman path integral formulation of quantum mechanics we can
write the expression for the time evolution of the quantum mechanical state
of a system as:

(5) 
Expression (5) is equivalent to the following one:

(6) 
Here
is the input state of the
quantum system at time and
is the output state at time . In this equation, is the mass,
is Planck's constant, and is the potential energy. In the second line,
the paths are discretized: , with the number of discretization
points,
.
Next: Specifying the System
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Gilson Giraldi
20020702