A Quantum Dot Neural Network

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 time-varying external field. In this case, the nonlinearity is a consequence of the real-time 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:

$$\mid \psi \left( x_{f},T\right) \rangle =\int\limits_{\left... ...eft( x_{f},T\right) }D\left[ x\left( t\right) \right] \times$$

$$\exp \left( \frac{i}{%%TCIMACRO{\UNICODE[m]{0x127}} \rlap{... ...) \right] \right) \mid \psi \left( x_{0},0\right) \rangle .$$ (5)

Expression (5) is equivalent to the following one:

$$\begin{array}[t]{c} lim \\ N \rightarrow \infty \end{ar... ...325em}{.1ex}}h\Delta t}\right) ^{\left( N-1\right) /2}\times$$

$$\exp \left( \sum\limits_{j=0}^{N}\left[ \frac{im\Delta t}{2... ...ht) \right] \right) \mid \psi \left( x_{0},0\right) \rangle$$ (6)

Here $\mid \psi \left( x_{0},0\right) \rangle$ is the input state of the quantum system at time $t=0$ and $\mid \psi \left( x_{f},T\right) \rangle$ is the output state at time $t=T$. In this equation, $m$ is the mass, $2\pi \rlap{\protect\rule[1.1ex]{.325em}{.1ex}}h$ is Planck's constant, and $V$ is the potential energy. In the second line, the paths are discretized: $N\Delta t=T$, with the number of discretization points, $N\rightarrow \infty$.