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The formulation above is general. A QNN approach comes out when the system is
that of a quantum dot molecule with five dots arranged as the pips on a
playing card. The dots are close enough to each other that
tunneling is possible between any two neighbors. Two electrons are fed into
the molecule, which then has a doublydegenerate ground state (in the
absence of environmental potentials). These states can be thought of as the
polarization of the molecule, equal to , that is, the Pauli
matrix operator . In Equation (5) this would be the value
.
In addition to adjusting or training
, we
can obtain an
additional trainable nonlinearity by coupling the system to its
environment. The environment is modeled by a set of Gaussians, that is, the
environment has a quadratic Hamiltonian, or, equivalently, a normal
distribution; if the set is taken to be infinite, any desired influence
including dissipation can be produced. In this model this would be
represented by the coupling between the electronic state of the dot
molecules and the lattice through optical phonons. Physically the coupling
would have to be weak enough to be represented accurately as linear; for
example, GaAs substrate satisfies this, with a (unitless) electronphonon
coupling parameter of .
Instead of taking
like in (6), we take
to be finite (quasicontinuum). Equation (6) becomes:

(7) 
where the path integral over possible positions at each time, , has
been written as a finite set of sums over states of the polarization,
, at each time slice . Also, at each time slice the
polarization can be either or . The potential energy comes from
a timevarying electric field,
, and the kinetic
energy term, in this twostate basis, now has the form
, where is the Pauli matrix. Since is offdiagonal in the polarization basis, this term contains the
(nonlinear) coupling between the states of the quantum dot molecule at
successive time slices. The size of this term, given by the parameter
(the tunneling amplitude), is determined by the physics of the dot molecule:
how easy it is for the electrons to tunnel from polarization state to . The effect of the optical phonons is summarized by the influence
functional
, given by:

(8) 
where
and is the position variable of the harmonic
oscillator (phonon), its mass, its frequency, and
its coupling strength to the system. The advantage of a
linearly coupled harmonic bath is that the path integrals over the phonons
can be performed immediately, giving us the nonlinear functional:

(9) 
where
is the influence phase, proportional to the response function of the bath.
For the phonon bath,

(10) 
where it was introduced also a (suitably low) temperature, given by in units of Boltzmanns constant.
In [4] authors consider the obtained
intermediate states
to be the states of virtual quantum neurons, one at each time
slice . The nonlinearity necessary for neural computation is
inherent in the kinetic energy term,
, and in
the exponential. Each of the neurons different possible states
contributes to the final measured state; the amount it contributes, can be
adjusted by changing
the potential energy,
.
The trainable parameters set can be any of those that appear above (
or even the values of the electric field at each time
slice
.
Combinations of these sets are also possible to be trained. It is important
to emphasize that any of these parameters can be physically controlled
[4].
Subsections
Next: Training the Quantum Network
Up: Quantum Computation and Nonlinearity
Previous: A Quantum Dot Neural
Gilson Giraldi
20020702