This paper initiates the study of quantum computing within the constraints of using a polylogarithmic ($O(\log^k n), k\geq 1$) number of qubits and a polylogarithmic number of computation steps. The current research in the literature has focussed on using a polynomial number of qubits. A new mathematical model of computation called \emph{Quantum Neural Networks (QNNs)} is defined, building on Deutsch's model of quantum computational network. The model introduces a nonlinear and irreversible gate, similar to the speculative operator defined by Abrams and Lloyd. The precise dynamics of this operator are defined and while giving examples in which nonlinear Schr��dinger's equations are applied, we speculate on its possible implementation. The many practical problems associated with the current model of quantum computing are alleviated in the new model. It is shown that QNNs of logarithmic size and constant depth have the same computational power as threshold circuits, which are used for modeling neural networks. QNNs of polylogarithmic size and polylogarithmic depth can solve the problems in \NC, the class of problems with theoretically fast parallel solutions. Thus, the new model may indeed provide an approach for building scalable parallel computers.

28 pages, 11 figures, To appear in Journal of Computer and System Sciences