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 Schrödinger'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.
The role of interference and entanglement in quantum neural processing is discussed. It is argued that on contrast to the quantum computing the problem of the use of exponential resources as the payment for the absense of entanglement does not exist for quantum neural processing. This is because of corresponding systems, as any modern classical artificial neural systems, do not realize functions precisely, but approximate them by training on small sets of examples. It can permit to implement quantum neural systems optically, because in this case there is no need in exponential resources of opt…
The early definition of a quantum neural network as a new field that combines the classical neurocomputing with quantum computing was rather vague and satisfactory in the 2000s. The widespread in 2020 modern definition of a quantum neural network as a model or machine learning algorithm that combines the functions of quantum computing with artificial neural networks deprives quantum neural networks of their fundamental importance. We argue that the concept of a quantum neural network should be defined in terms of its most general function as a tool for representing the amplitude of an arbitrar…
The Hopfield neural networks and the holographic neural networks are models which were successfully simulated on conventional computers. Starting with these models, an analogous fundamental quantum information processing system is developed in this article. Neuro-quantum interaction can regulate the "collapse"-readout of quantum computation results. This paper is a comprehensive introduction into associative processing and memory-storage in quantum-physical framework.
It is suggested that a quantum neural network (QNN), a type of artificial neural network, can be built using the principles of quantum information processing. The input and output qubits in the QNN can be implemented by optical modes with different polarization, the weights of the QNN can be implemented by optical beam splitters and phase shifters
Machine Learning provides powerful tools for a variety of applications, including disease diagnosis through medical image classification. In recent years, quantum machine learning techniques have been put forward as a way to potentially enhance performance in machine learning applications, both through quantum algorithms for linear algebra and quantum neural networks. In this work, we study two different quantum neural network techniques for medical image classification: first by employing quantum circuits in training of classical neural networks, and second, by designing and training quantum …
With the overwhelming success in the field of quantum information in the last decades, the "quest" for a Quantum Neural Network (QNN) model began in order to combine quantum computing with the striking properties of neural computing. This article presents a systematic approach to QNN research, which so far consists of a conglomeration of ideas and proposals. It outlines the challenge of combining the nonlinear, dissipative dynamics of neural computing and the linear, unitary dynamics of quantum computing. It establishes requirements for a meaningful QNN and reviews existing literature against …
Artificial neural networks have achieved great success in many fields ranging from image recognition to video understanding. However, its high requirements for computing and memory resources have limited further development on processing big data with high dimensions. In recent years, advances in quantum computing show that building neural networks on quantum processors is a potential solution to this problem. In this paper, we propose a novel neural network model named Quantum Convolutional Neural Network (QCNN), aiming at utilizing the computing power of quantum systems to accelerate classic…
We revisit the application of neural networks techniques to quantum state tomography. We confirm that the positivity constraint can be successfully implemented with trained networks that convert outputs from standard feed-forward neural networks to valid descriptions of quantum states. Any standard neural-network architecture can be adapted with our method. Our results open possibilities to use state-of-the-art deep-learning methods for quantum state reconstruction under various types of noise.
Quantum machine learning techniques have been proposed as a way to potentially enhance performance in machine learning applications. In this paper, we introduce two new quantum methods for neural networks. The first one is a quantum orthogonal neural network, which is based on a quantum pyramidal circuit as the building block for implementing orthogonal matrix multiplication. We provide an efficient way for training such orthogonal neural networks; novel algorithms are detailed for both classical and quantum hardware, where both are proven to scale asymptotically better than previously known t…
This study concerns with the dynamics of a quantum neural network unit in order to examine the suitability of simple neural computing tasks. More specifically, we examine the dynamics of an interacting spin model chosen as a candidate of a quantum perceptron for closed and open quantum systems. We adopt a collisional model enables examining both Markovian and non-Markovian dynamics of the proposed quantum system. We show that our quantum neural network (QNN) unit has a stable output quantum state in contact with an environment carrying information content. By the performed numerical simulation…
Artificial neural network, consisting of many neurons in different layers, is an important method to simulate humain brain. Usually, one neuron has two operations: one is linear, the other is nonlinear. The linear operation is inner product and the nonlinear operation is represented by an activation function. In this work, we introduce a kind of quantum neuron whose inputs and outputs are quantum states. The inner product and activation operator of the quantum neurons can be realized by quantum circuits. Based on the quantum neuron, we propose a model of quantum neural network in which the wei…
X, The Moonshot Factory（X，登月工厂）；Google Research（谷歌研究院）
We introduce Quantum Graph Neural Networks (QGNN), a new class of quantum neural network ansatze which are tailored to represent quantum processes which have a graph structure, and are particularly suitable to be executed on distributed quantum systems over a quantum network. Along with this general class of ansatze, we introduce further specialized architectures, namely, Quantum Graph Recurrent Neural Networks (QGRNN) and Quantum Graph Convolutional Neural Networks (QGCNN). We provide four example applications of QGNNs: learning Hamiltonian dynamics of quantum systems, learning how to create …
Quantum computing enables quantum neural networks (QNNs) to have great potentials to surpass artificial neural networks (ANNs). The powerful generalization of neural networks is attributed to nonlinear activation functions. Although various models related to QNNs have been developed, they are facing the challenge of merging the nonlinear, dissipative dynamics of neural computing into the linear, unitary quantum system. In this paper, we establish different quantum circuits to approximate nonlinear functions and then propose a generalizable framework to realize any nonlinear quantum neuron. We …
University of British Columbia（不列颠哥伦比亚大学）；Stewart Blusson Quantum Matter Institute（斯图尔特·布鲁森量子物质研究所）；University of Cambridge（剑桥大学）
Recurrent neural networks play an important role in both research and industry. With the advent of quantum machine learning, the quantisation of recurrent neural networks has become recently relevant. We propose fully quantum recurrent neural networks, based on dissipative quantum neural networks, capable of learning general causal quantum automata. A quantum training algorithm is proposed and classical simulations for the case of product outputs with the fidelity as cost function are carried out. We thereby demonstrate the potential of these algorithms to learn complex quantum processes with …
We study the representational power of Boltzmann machines (a type of neural network) in quantum many-body systems. We prove that any (local) tensor network state has a (local) neural network representation. The construction is almost optimal in the sense that the number of parameters in the neural network representation is almost linear in the number of nonzero parameters in the tensor network representation. Despite the difficulty of representing (gapped) chiral topological states with local tensor networks, we construct a quasi-local neural network representation for a chiral $p$-wave superc…
This paper proposes a quantum-classical algorithm to evaluate and select classical artificial neural networks architectures. The proposed algorithm is based on a probabilistic quantum memory and the possibility to train artificial neural networks in superposition. We obtain an exponential quantum speedup in the evaluation of neural networks. We also verify experimentally through a reduced experimental analysis that the proposed algorithm can be used to select near-optimal neural networks.
In recent years, with rapid progress in the development of quantum technologies, quantum machine learning has attracted a lot of interest. In particular, a family of hybrid quantum-classical neural networks, consisting of classical and quantum elements, has been massively explored for the purpose of improving the performance of classical neural networks. In this paper, we propose a novel hybrid quantum-classical algorithm called quantum dilated convolutional neural networks (QDCNNs). Our method extends the concept of dilated convolution, which has been widely applied in modern deep learning al…
This paper highlights a practical research of the possibility of forming quantum circuits for training neural networks. The demonstrated quantum circuits were based on the principles of Grover's Search Algorithm. The perceptron was chosen as the architecture for the example neural network. The multilayer perceptron is a popular neural network architecture due to its scalability and applicability for solving a wide range of problems.
In this work, we propose a quantum neural network named quantum perceptron over a field (QPF). Quantum computers are not yet a reality and the models and algorithms proposed in this work cannot be simulated in actual (or classical) computers. QPF is a direct generalization of a classical perceptron and solves some drawbacks found in previous models of quantum perceptrons. We also present a learning algorithm named Superposition based Architecture Learning algorithm (SAL) that optimizes the neural network weights and architectures. SAL searches for the best architecture in a finite set of neura…
George Mason University（乔治梅森大学）；University of Notre Dame（圣母大学）；University of Connecticut（康涅狄格大学）
With the constant increase of the number of quantum bits (qubits) in the actual quantum computers, implementing and accelerating the prevalent deep learning on quantum computers are becoming possible. Along with this trend, there emerge quantum neural architectures based on different designs of quantum neurons. A fundamental question in quantum deep learning arises: what is the best quantum neural architecture? Inspired by the design of neural architectures for classical computing which typically employs multiple types of neurons, this paper makes the very first attempt to mix quantum neuron d…
Korea Advanced Institute of Science and Technology（韩国高等科学技术研究院）；Keio University（庆应义塾大学）
In two pervious papers \cite{dndiep3}, \cite{dndiep4}, the first author constructed the least square quantum neural networks (LS-QNN), and ploynomial interpolation quantum neural networks ( PI-QNN), parametrico-stattistical QNN like: leanr regrassion quantum neural networks (LR-QNN), polynomial regression quantum neural networks (PR-QNN), chi-squared quantum neural netowrks ($χ^2$-QNN). We observed that the method works also in the cases by using nonparametric statistics. In this paper we analyze and implement the nonparametric tests on QNN such as: linear nonparametric regression quantum neur…
We develop a new quantum neural network layer designed to run efficiently on a quantum computer but that can be simulated on a classical computer when restricted in the way it entangles input states. We first ask how a classical neural network architecture, both fully connected or convolutional, can be executed on a quantum computer using quantum phase estimation. We then deform the classical layer into a quantum design which entangles activations and weights into quantum superpositions. While the full model would need the exponential speedups delivered by a quantum computer, a restricted clas…
Quantum deep learning is a research field for the use of quantum computing techniques for training deep neural networks. The research topics and directions of deep learning and quantum computing have been separated for long time, however by discovering that quantum circuits can act like artificial neural networks, quantum deep learning research is widely adopted. This paper explains the backgrounds and basic principles of quantum deep learning and also introduces major achievements. After that, this paper discusses the challenges of quantum deep learning research in multiple perspectives. Last…
To address Quantum Artificial Neural Networks as quantum dynamical computing systems, a formalization of quantum artificial neural networks as dynamical systems is developed, expanding the concept of unitary map to the neural computation setting and introducing a quantum computing field theory on the network. The formalism is illustrated in a simulation of a quantum recurrent neural network and the resulting field dynamics is researched upon, showing emergent neural waves with excitation and relaxation cycles at the level of the quantum neural activity field, as well as edge of chaos signature…
Thang Long University（升龙大学）；Institute of Mathematics, Vietnam National Academy of Science and Technology（越南国家科学技术院数学研究所）
Some basic quantum neural networks were analyzed and constructed in the recent work of the author \cite{dndiep3}, published in International Journal of Theoretical Physics (2020). In particular the Least Quare Problem (LSP) and the Linear Regression Problem (LRP) was discussed. In this second paper we continue to analyze and construct the least square quantum neural network (LS-QNN), the polynomial interpolation quantum neural network (PI-QNN), the polynomial regression quantum neural network (PR-QNN) and chi-squared quantum neural network ($χ^2$-QNN). We use the corresponding solution or test…
Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering（霍里亚·胡卢贝伊国家物理与核工程研发研究所）
Neural networks are being used to improve the probing of the state spaces of many particle systems as approximations to wavefunctions and in order to avoid the recurring sign problem of quantum monte-carlo. One may ask whether the usual classical neural networks have some actual hidden quantum properties that make them such suitable tools for a highly coupled quantum problem. I discuss here what makes a system quantum and to what extent we can interpret a neural network as having quantum remnants.
Our work intends to show that: (1) Quantum Neural Networks (QNN) can be mapped onto spinnetworks, with the consequence that the level of analysis of their operation can be carried out on the side of Topological Quantum Field Theories (TQFT); (2) Deep Neural Networks (DNN) are a subcase of QNN, in the sense that they emerge as the semiclassical limit of QNN; (3) A number of Machine Learning (ML) key-concepts can be rephrased by using the terminology of TQFT. Our framework provides as well a working hypothesis for understanding the generalization behavior of DNN, relating it to the topological f…
In many cases, Neural networks can be mapped into tensor networks with an exponentially large bond dimension. Here, we compare different sub-classes of neural network states, with their mapped tensor network counterpart for studying the ground state of short-range Hamiltonians. We show that when mapping a neural network, the resulting tensor network is highly constrained and thus the neural network states do in general not deliver the naive expected drastic improvement against the state-of-the-art tensor network methods. We explicitly show this result in two paradigmatic examples, the 1D ferro…
University of Cambridge（剑桥大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）
We examine the usefulness of applying neural networks as a variational state ansatz for many-body quantum systems in the context of quantum information-processing tasks. In the neural network state ansatz, the complex amplitude function of a quantum state is computed by a neural network. The resulting multipartite entanglement structure captured by this ansatz has proven rich enough to describe the ground states and unitary dynamics of various physical systems of interest. In the present paper, we initiate the study of neural network states in quantum information-processing tasks. We demonstra…
