Quantum Stochastic Neural Maps and Quantum Neural Networks

Abstract

Quantum neural computation employs quantum computational circuits that consist of chains of conditional unitary operators that follow the network’s neural links leading to entanglement between the local neuron-level computation and the network. When such quantum computational circuits are applied iteratively, instead of an input and output density, we have a sequence of density operators, with the transition from one density to another resulting from a form of unitary quantum map. It has been shown that these unitary quantum neural maps lead to complex emergent dynamics at the level of relevant quantum averages. The present work expands the research on quantum neural maps combining it with quantum stochastic processes theory, introducing the concept of a quantum stochastic neural map, resulting from a coupling to an external noise source. The unitary and stochastic maps are implemented for a quantum recurrent neural network, showing evidence of complex emergent dynamics, including, in the case of the stochastic map, fractal attractors that leave a signature at the level of the energy versus mutual information plots, as well as local (neuron-level) entropy resilient dynamics, where each neuron, as an open computing unit, exhibits dissipation but does not reach full entanglement-related decoherence.

Keywords: Quantum Neural Maps, Quantum Stochastic Processes, Quantum Neural Computation, Open Quantum Computing Systems, Recurrent Neural Networks, Decoherence, Fractal Attractors