The ability of artificial neural networks (ANNs) to adapt to input data and perform generalizations is intimately connected to the use of nonlinear activation and propagation functions. Quantum versions of ANN have been proposed to take advantage of the possible supremacy of quantum over classical computing. To date, all proposals faced the difficulty of implementing nonlinear activation functions since quantum operators are linear. This brief presents an architecture to simulate the computation of an arbitrary nonlinear function as a quantum circuit. This computation is performed on the phase of an adequately designed quantum state, and quantum phase estimation recovers the result, given a fixed precision, in a circuit with linear complexity in function of ANN input size.
The ability of artificial neural networks (ANNs) to adapt to input data and perform generalizations is intimately connected to the use of nonlinear activation and propagation functions. Quantum versions of ANN have been proposed to take advantage of the possible supremacy of quantum over classical computing. To date, all proposals faced the difficulty of implementing nonlinear activation functions since quantum operators are linear. This brief presents an architecture to simulate the computation of an arbitrary nonlinear function as a quantum circuit. This computation is performed on the phase of an adequately designed quantum state, and quantum phase estimation recovers the result, given a fixed precision, in a circuit with linear complexity in function of ANN input size.
Quantum computing enables quantum neural networks (QNNs) to have great potential to surpass artificial neural networks. 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 present two quantum neuron examples based on the proposed framework. The quantum resources required to construct a single quantum neuron are polynomial in function of the input size. Finally, both IBM Quantum Experience results and numerical simulations illustrate the effectiveness of the proposed framework.
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.
An artificial neural network, consisting of many neurons in different layers, is an important method to simulate the human brain. Usually, one neuron has two operations: one is linear, the other is nonlinear. The linear operation is the 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 a quantum neural network in which the weights between neurons are all quantum states. We also construct a quantum circuit to realize this quantum neural network model. A learning algorithm is proposed meanwhile. We show the validity of the learning algorithm theoretically and demonstrate the potential of the quantum neural network numerically.
A non-linear activation function is an integral component of neural network algorithms used for various tasks such as data classification and pattern recognition. In neu-romorphic/emerging-hardware-based implementations of neural network algorithms, the non-linear activation function is often implemented through dedicated analog electronics. This enables faster execution of the activation function during training and inference of neural networks compared to conventional digital implementation. Here, with a similar motivation, we propose a novel non-linear activation function that can be used in a neural network for data classification. Our activation function can be implemented by taking advantage of the inherent nonlinearity in qubit preparation and SU(2) operation in quantum mechanics. These operations are directly implementable on quantum hardware through single-qubit quantum gates as we show here. In addition, the SU(2) parameters are adjustable here making the activation function adaptable; we adjust the parameters through classical feedback like in a variational algorithm in quantum machine learning. Using our proposed quantum activation function, we report accurate classification using popular machine learning data sets like Fisher's Iris, Wisconsin's Breast Cancer (WBC), Abalone, and MNIST on three different platforms: simulations on a classical computer, simulations on a quantum simulation framework like Qiskit, and experimental implementation on quantum hardware (IBM-Q). Then we use a Bloch-sphere-based approach to intuitively explain how our proposed quantum activation function, with its adaptability, helps in data classification.
Modeling nonlinear activation functions on quantum computers is vital for quantum neurons employed in fully quantum neural networks, however, remains a challenging task. We introduce an amplitude-based implementation for approximating nonlinearity in the form of the unit step function on a quantum computer. Our approach expands upon repeat-until-success protocols, suggesting a modification that requires a single measurement only. We describe two distinct circuit types which receive their input either directly from a classical computer or as a quantum state when embedded in a more advanced quantum algorithm. All quantum circuits are theoretically evaluated using numerical simulation and executed on Noisy Intermediate-Scale Quantum hardware. We demonstrate that reliable data with high precision can be obtained from our quantum circuits involving up to eight qubits and up to 25 CX-gate applications, enabled by state-of-the-art hardware-optimization techniques and measurement error mitigation.
Abstract The field of artificial neural networks is expected to strongly benefit from recent developments of quantum computers. In particular, quantum machine learning, a class of quantum algorithms which exploit qubits for creating trainable neural networks, will provide more power to solve problems such as pattern recognition, clustering and machine learning in general. The building block of feed-forward neural networks consists of one layer of neurons connected to an output neuron that is activated according to an arbitrary activation function. The corresponding learning algorithm goes under the name of Rosenblatt perceptron. Quantum perceptrons with specific activation functions are known, but a general method to realize arbitrary activation functions on a quantum computer is still lacking. Here, we fill this gap with a quantum algorithm which is capable to approximate any analytic activation functions to any given order of its power series. Unlike previous proposals providing irreversible measurement–based and simplified activation functions, here we show how to approximate any analytic function to any required accuracy without the need to measure the states encoding the information. Thanks to the generality of this construction, any feed-forward neural network may acquire the universal approximation properties according to Hornik’s theorem. Our results recast the science of artificial neural networks in the architecture of gate-model quantum computers.