Quantum RAM Based Neural Netoworks

A mathematical quantisation of a Random Access Memory (RAM) is proposed starting from its matrix representation. This quantum RAM (q-RAM) is employed as the neural unit of q-RAM-based Neural Networks, q-RbNN, which can be seen as the quantisation of the corresponding RAM-based ones. The models proposed here are direct realisable in quantum circuits, have a natural adaptation of the classical learning algorithms and physical feasibility of quantum learning in contrast to what has been proposed in the literature. 1 Quantum computation and Mathematical Quantisation Quantum computing [14] was originally proposed by Richard Feynman [7] in the 1980s and had its formalisation with David Deutsch which proposed the quantum Turing machine [4]. Quantum Computing has been popularised through the quantum circuit model [5] which is a quantisation [15] of the classical boolean circuit model of computation. Quantum computer also became a potential parallel device for improving the computational efficiency of neural networks [6]. The quantum information unit is the quantum bit or ”qubit”. A very intuitive view of the quantisation procedure is put forward by Nik Weaver in the Preface of his book Mathematical Quantization [15] with just a phrase which says it all: ”The fundamental idea of mathematical quantisation is sets are replaced with Hilbert spaces”. The quantisation of the boolean circuit logic starts by simply embedding the the classical bits {0, 1} in a convenient Hilbert space. The natural way of doing this is to represent them as (orthonormal) basis of a Complex Hilbert space. In this context these basis elements are called the computational-basis states.Linear combinations (from Linear Algebra [9]) of the basis spans the whole space whose elements, called states, are said to be in superposition. Any basis can be used (recall from Linear Algebra [9] that there usually are many!). But in Quantum Computing it is customary to use the most conventional and well known one: |0〉, |1〉 are a pair of orthonormal basis vectors representing each classical bit, or ”cbit”, as column vector, |0〉 = [1 0]T and |1〉 = [0 1]T. A general state of the system (a vector) can be written as: |ψ〉 = α |0〉+β |1〉, where α, β are complex coefficients (called probability amplitudes) constrained by the normalization condition: |α| + |β| = 1. This is the model of one qubit. Multiple qubits are obtained via tensor products. By linearity we just need to say how tensor behaves on the basis: |i〉 ⊗ |j〉 = |i〉 |j〉 = |ij〉 , where i, j ∈ {0, 1} If that were all the story, quantum computing would be just a trivial extension of classical computing. But Quantum Mechanics Principles [14] restrain the kind of permissible operations. Operations on qubits are carried out only by unitary operators (i.e. ∗Supported by MCT-CNPq and PRONEX/FACEPE. On sabbatical leave at the School of Computer Science, University of Birmingham ESANN'2009 proceedings, European Symposium on Artificial Neural Networks Advances in Computational Intelligence and Learning. Bruges (Belgium), 22-24 April 2009, d-side publi., ISBN 2-930307-09-9.