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The European Symposium on Artificial Neural Networks

Vector space weightless neural networks.

Wilson Rosa de Oliveira, Adenilton José da Silva, Teresa B. Ludermir

Published Jan 1, 2014• 3 citations
OpenAlex

Abstract

By embedding the boolean space Z2 as an orthonormal basis in a vec- tor space we can treat the RAM based neuron as a matrix (operator) acting on the vector space. We show how this model (inspired by our research on quantum neu- ral networks) is of sufficient generality as to have classical weighted (perceptron- like), classical weightless (RAM-based, PLN, etc), quantum weighted and quantum weightless neural models as particular cases. It is also indicated how one could use it to polynomially solve 3-SAT and briefly mention how could one train this novel model.

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Brazilian Symposium on Neural Networks (SBRN)•Oct 1, 2010•16 citations
By embedding the boolean space Z2 as an orthonormal basis in a vec- tor space we can treat the RAM based neuron as a matrix (operator) acting on the vector space. We show how this model (inspired by our research on quantum neu- ral networks) is of sufficient generality as to have classical weighted (perceptron- like), classical weightless (RAM-based, PLN, etc), quantum weighted and quantum weightless neural models as particular cases. It is also indicated how one could use it to polynomially solve 3-SAT and briefly mention how could one train this novel model.
Despite neural networks have super-Turing computing power, there is no known algorithm for obtaining a classical neural networks that solves NP-complete problems in polynomial time. However this paper shows that a quantum neural networks model coupled with a non-unitary operator can solve 3-SAT in polynomial time. The proposed method uses a network circuit to represent a Boolean logic function and a non-unitary operator to decide the satisfiability. The parameters of the network is set deterministically and manually, accordingly to the problem at hand with neither quantum nor classical learning.
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.
In this paper we investigate the computational power of the quantum weightless neural networks (q-WNN) and propose a novel quantum weightless neural node. The new quantum neuron is derived from the quantum probabilistic logic node (qPLN), which is a mathematical quantisation of the weightless neural node PLN. By a slight modification on the input lines of the qPLN node it is possible to define the quantum psi-neuron. We show in this paper that psi-neurons can be used to approximate any quantum circuit.
Quantum machine learning models based on parametrized quantum circuits, also called quantum neural networks (QNNs), are considered to be among the most promising candidates for applications on near-term quantum devices. Here we explore the expressivity and inductive bias of QNNs by exploiting an exact mapping from QNNs with inputs xxx to classical perceptrons acting on x⊗xx \otimes xx⊗x (generalised to complex inputs). The simplicity of the perceptron architecture allows us to provide clear examples of the shortcomings of current QNN models, and the many barriers they face to becoming useful general-purpose learning algorithms. For example, a QNN with amplitude encoding cannot express the Boolean parity function for n≥3n\geq 3n≥3, which is but one of an exponential number of data structures that such a QNN is unable to express. Mapping a QNN to a classical perceptron simplifies training, allowing us to systematically study the inductive biases of other, more expressive embeddings on Boolean data. Several popular embeddings primarily produce an inductive bias towards functions with low class balance, reducing their generalisation performance compared to deep neural network architectures which exhibit much richer inductive biases. We explore two alternate strategies that move beyond standard QNNs. In the first, we use a QNN to help generate a classical DNN-inspired kernel. In the second we draw an analogy to the hierarchical structure of deep neural networks and construct a layered non-linear QNN that is provably fully expressive on Boolean data, while also exhibiting a richer inductive bias than simple QNNs. Finally, we discuss characteristics of the QNN literature that may obscure how hard it is to achieve quantum advantage over deep learning algorithms on classical data.
Quantum analogues of the (classical) Logical Neural Networks (LNN) models are proposed in (q-LNN for short). We shall here further develop and investigate the q-LNN composed of the quantum analogue of the Probabilistic Logic Node (PLN) and the multiple-valued PLN (MPLN) variations, dubbed q-PLN and q-MPLN respectively. Besides a clearer mathematical description, we present a computationally efficient and simply described quantum learning algorithm in contrast to what has been proposed to the quantum weighted version.
The success of quantum computation is most commonly associated with speed up of classical algorithms, as the Shor's factoring algorithm and the Grover's search algorithm. But it should also be related with exponential storage capacity such as the super dense coding. In this work we use a probabilistic quantum memory proposed by Trugen berger, where one can store 2n\mathbf 2^n2n patterns with only n\mathbf nn quantum bits (qbits). We define a new model of a quantum weightless neural node with this memory in a similar fashion as to the classical Random Access Memory (RAM) node is used in classical weightless neural networks. Some advantages of the proposed model are that the memory of the node does not grow exponentially with the number of inputs and that the node can generalise.