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This paper initiates the study of quantum computing within the constraints of using a polylogarithmic (O(^k n), k≥ 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 Quantum Neural Networks (QNNs) is defined, building on Deutsch's model of quantum computational network. The model introduces a nonlinear and irreversible gate, similar toarXiv:quant-ph/0201144v1 fatcat:jkd6gsmqyrhthcdgpz54b7baq4