Specifying the System

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 doubly-degenerate ground state (in the absence of environmental potentials). These states can be thought of as the polarization $P$ of the molecule, equal to $\pm 1$, that is, the Pauli matrix operator $\sigma _{z}$. In Equation (5) this would be the value $x\left( t\right)$. In addition to adjusting or training $V\left( x\right)$, 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) electron-phonon coupling parameter of $0.08<<1$. Instead of taking $N\rightarrow \infty$ like in (6), we take $N$ to be finite (quasi-continuum). Equation (6) becomes:

$$\mid \psi \left( \sigma _{z}\left( N\Delta t\right) ,T\right) \rangle =$$

$$\sum_{\left\{ \sigma _{z}\left( j\Delta t\right) \right\} }... ...t) \sigma _{z}\left( j\Delta t\right) \right] \right) \times$$

$$I\left[ \sigma _{z}\left( \Delta t\right) \right] \mid \psi \left( \sigma _{z}\left( 0\right) ,0\right) \rangle$$ (7)

where the path integral over possible positions at each time, $x(t)$, has been written as a finite set of sums over states of the polarization, $%% \sigma _{z}$, at each time slice $j\Delta t$. Also, at each time slice the polarization can be either $+1$ or $-1$. The potential energy $V$ comes from a time-varying electric field, $\epsilon \left( t\right)$, and the kinetic energy term, in this two-state basis, now has the form $K\sigma _{x}\left( j\Delta t\right)$, where $\sigma _{x}$ is the Pauli matrix. Since $\sigma _{x}$ is off-diagonal 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 $K$ (the tunneling amplitude), is determined by the physics of the dot molecule: how easy it is for the electrons to tunnel from polarization state $+1$ to $%% -1$. The effect of the optical phonons is summarized by the influence functional $I\left[ \sigma _{z}\left( t\right) \right]$, given by:

$$I\left[ \sigma _{z}\left( t\right) \right] =\int \prod_{k}D... ....1ex}}h}\int\limits_{0}^{T}d\tau \sum\limits_{k}S_{k}\right)$$ (8)

where

$$S_{k}=\frac{m_{k}}{2}\stackrel{\cdot }{\alpha _{k}^{2}}\lef... ...pha _{k}\left( \tau \right) \sigma _{z}\left( \tau \right) ,$$

and $\alpha _{k}$ is the position variable of the $k^{th}$ harmonic oscillator (phonon), $m_{k}$ its mass, $\omega _{k}$ its frequency, and $%% \lambda _{k}$ 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:

$$I\left[ \sigma _{z}\left( t\right) \right] =\exp \left( \s... ...right) \sigma _{z}\left( j^{\prime }\Delta t\right) \right)$$ (9)

where $\chi \left( \tau ,\tau ^{\prime }\right) =\chi \left( \left\vert \tau -\tau ^{\prime }\right\vert \right) =\chi \left( \tau ^{\prime \prime }\right)$ is the influence phase, proportional to the response function of the bath. For the phonon bath,

$$\chi \left( \tau \right) =\sum\limits_{k}\frac{\lambda _{k}... ...ect\rule[1.1ex]{.325em}{.1ex}}h\omega _{k}}{2}\right) \times$$

$$\left[ \cosh \left( \frac{\beta %%TCIMACRO{\UNICODE[m]{0x12... ..._{k}}{2}\right) \sin \left( \omega _{k}\tau \right) \right]$$ (10)

where it was introduced also a (suitably low) temperature, given by $1/\beta$ in units of Boltzmanns constant. In [4] authors consider the obtained $N$ intermediate states to be the states of $N$ virtual quantum neurons, one at each time slice $j\Delta t$. The nonlinearity necessary for neural computation is inherent in the kinetic energy term, $\left( x_{j+1}-x_{j}\right)$, and in the exponential. Each of the $N$ neurons different possible states contributes to the final measured state; the amount it contributes, can be adjusted by changing the potential energy, $V\left( x\right)$. The trainable parameters set can be any of those that appear above ( $\lambda _{k},\omega _{k})$ or even the values of the electric field at each time slice $j$ $\left\{ \epsilon \left( j\Delta t\right) ;j=0,...,N\right\}$. 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].