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.
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.
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.
This chapter offers a clear and accessible introduction to the field of quantum-inspired neural computing. This area combines ideas from quantum mechanics with traditional neural network models to create new possibilities in machine learning and optimization metaheuristic algorithms. Unlike actual quantum computing, quantum-inspired approaches work on classical computers but borrow key concepts like superposition, entanglement, and probabilistic behaviour to improve how neural networks learn and process information. The study begins by explaining why researchers are interested in using quantum principles in neural networks. It then introduces the basic mathematical ideas behind these concepts, such as quantum states and transformations, and shows how they can be used to design more powerful and flexible learning systems. It focuses on discussing current models, outlining the distinctions between quantum-inspired and true quantum systems, and evaluating the potential benefits and challenges of this approach. In addition, Quantum-inspired neural computing introduces a novel framework that bridges classical artificial intelligence with principles derived from quantum mechanics. By simulating quantum behaviours on classical hardware, it opens new avenues for developing more adaptive and efficient learning models. This emerging approach offers promising advantages in areas such as optimization, parallel processing, and high-dimensional data representation. Finally, despite its potential, the field faces notable limitations and challenges. These include the lack of limited theoretical formalization, and difficulties in scaling models for complex real-world applications. Moreover, distinguishing genuine quantum-inspired advantages from conservative algorithmic improvements remains an current research apprehension.
Artificial neural networks, usually just called neural networks, computing systems indefinitely inspired by the biological neural networks and they are extensive in both research as well as industry. It is critical to design quantum Neural Networks for complete quantum learning tasks. In this project, we suggest a computational neural network model based on principles of quantum mechanics which form a quantum feed-forward neural network proficient in universal quantum computation. This structure takes input from one layer of qubits and drives that input onto another layer of qubits. This layer of qubits evaluates this information and drives on the output to the next layer. Eventually, the path leads to the final layer of qubits. The layers do not have to be of the same breadth, meaning they need not have the same number of qubits as the layer before and/or after it. This assembly is trained on which path to take identical to classical ANN. The intended project can be compiled by the subsequent points provided here: 1. The expert training of the quantum neural network utilizing the fidelity as a cost function, providing both conventional and efficient quantum implementations. 2. Use of methods that enable quick optimization with reduced memory requirements. 3. Benchmarking our proposal for the quantum task of learning an unknown unitary and find extraordinary generality and a remarkable sturdiness to noisy training data.
Quantum neural networks (QNNs) represent a cutting-edge approach to machine learning and artificial intelligence, harnessing the principles of quantum mechanics to process and analyze data in fundamentally new ways. Unlike classical neural networks, which operate on classical bits and are limited by the laws of classical physics, QNNs use quantum bits (qubits) and exploit quantum phenomena such as superposition and entanglement to achieve unprecedented computational power and efficiency. This chapter explores the principles and applications of QNNs, including their potential for solving complex optimisation problems, pattern recognition tasks, and quantum simulation tasks. We discuss various QNN architectures, training algorithms, and applications, highlighting their advantages and challenges compared to classical neural networks. Additionally, we explore the potential impact of QNNs on fields such as quantum computing, quantum communication, and quantum sensing, as well as their implications for society and the economy. Overall, this chapter provides insights into the emerging field of quantum neural networks and sets the stage for further research and innovation in this exciting area.
Even the most sophisticated artificial neural networks are built by aggregating substantially identical units called neurons. A neuron receives multiple signals, internally combines them, and applies a non-linear function to the resulting weighted sum. Several attempts to generalize neurons to the quantum regime have been proposed, but all proposals collided with the difficulty of implementing non-linear activation functions, which is essential for classical neurons, due to the linear nature of quantum mechanics. Here we propose a solution to this roadblock in the form of a small quantum circuit that naturally simulates neurons with threshold activation. Our quantum circuit defines a building block, the "quantum neuron", that can reproduce a variety of classical neural network constructions while maintaining the ability to process superpositions of inputs and preserve quantum coherence and entanglement. In the construction of feedforward networks of quantum neurons, we provide numerical evidence that the network not only can learn a function when trained with superposition of inputs and the corresponding output, but that this training suffices to learn the function on all individual inputs separately. When arranged to mimic Hopfield networks, quantum neural networks exhibit properties of associative memory. Patterns are encoded using the simple Hebbian rule for the weights and we demonstrate attractor dynamics from corrupted inputs. Finally, the fact that our quantum model closely captures (traditional) neural network dynamics implies that the vast body of literature and results on neural networks becomes directly relevant in the context of quantum machine learning.
A random access memory, or RAM, is a device that, when interrogated, returns the content of a memory location in a memory array. A quantum RAM, or qRAM, allows one to access superpositions of memory sites, which may contain either quantum or classical information. RAMs and qRAMs with n-bit addresses can access 2n memory sites. Any design for a RAM or qRAM then requires O(2n) two-bit logic gates. At first sight this requirement might seem to make large scale quantum versions of such devices impractical, due to the difficulty of constructing and operating coherent devices with large numbers of quantum logic gates. Here we analyze two different RAM architectures (the conventional fanout and the ``bucket brigade'') and propose some proof-of-principle implementations, which show that, in principle, only O(n) two-qubit physical interactions need take place during each qRAM call. That is, although a qRAM needs O(2n) quantum logic gates, only O(n) need to be activated during a memory call. The resulting decrease in resources could give rise to the construction of large qRAMs that could operate without the need for extensive quantum error correction.