  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\"{o}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.
