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Natural Science

Towards a Real Quantum Neuron

Wei Hu

Published Jan 1, 2018• 28 citations
Publisher

Abstract

Google’s AlphaGo represents the impressive performance of deep learning and the backbone of deep learning is the workhorse of highly versatile neural networks. Each network is made up of layers of interconnected neurons and the nonlinear activation function inside each neuron is one of the key factors that account for the unprecedented achievement of deep learning. Learning how to create quantum neural networks has been a long time pursuit since 1990’s from many researchers, unfortunately without much success. The main challenge is to know how to design a nonlinear activation function inside the quantum neuron, because the laws in quantum mechanics require the operations on quantum neurons be unitary and linear. A recent discovery uses a special quantum circuit technique called repeat-until-success to make a nonlinear activation function inside a quantum neuron, which is the hard part of creating this neuron. However, the activation function used in that work is based on the periodic tangent function. Because of this periodicity, the input to this function has to be restricted to the range of [0, π/2), which is a serious constraint for its applications in real world problems. The function’s periodicity also makes its neurons not suited for being trained with gradient descent as its derivatives oscillate. The purpose of our study is to propose a new nonlinear activation function that is not periodic so it can take any real numbers and its neurons can be trained with efficient gradient descent. Our quantum neuron offers the full benefit as a quantum entity to support superposition, entanglement, interference, while also enjoys the full benefit as a classical entity to take any real numbers as its input and can be trained with gradient descent. The performance of the quantum neurons with our new activation function is analyzed on IBM’s 5Q quantum computer and IBM’s quantum simulator.

DOI
10.4236/ns.2018.103011

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Google’s AlphaGo represents the impressive performance of deep learning and the backbone of deep learning is the workhorse of highly versatile neural networks. Each network is made up of layers of interconnected neurons and the nonlinear activation function inside each neuron is one of the key factors that account for the unprecedented achievement of deep learning. Learning how to create quantum neural networks has been a long time pursuit since 1990’s from many researchers, unfortunately without much success. The main challenge is to know how to design a nonlinear activation function inside the quantum neuron, because the laws in quantum mechanics require the operations on quantum neurons be unitary and linear. A recent discovery uses a special quantum circuit technique called repeat-until-success to make a nonlinear activation function inside a quantum neuron, which is the hard part of creating this neuron. However, the activation function used in that work is based on the periodic tangent function. Because of this periodicity, the input to this function has to be restricted to the range of [0, π/2), which is a serious constraint for its applications in real world problems. The function’s periodicity also makes its neurons not suited for being trained with gradient descent as its derivatives oscillate. The purpose of our study is to propose a new nonlinear activation function that is not periodic so it can take any real numbers and its neurons can be trained with efficient gradient descent. Our quantum neuron offers the full benefit as a quantum entity to support superposition, entanglement, interference, while also enjoys the full benefit as a classical entity to take any real numbers as its input and can be trained with gradient descent. The performance of the quantum neurons with our new activation function is analyzed on IBM’s 5Q quantum computer and IBM’s quantum simulator.
Abstract Artificial neural networks are becoming an integral part of digital solutions to complex problems. However, employing neural networks on quantum processors faces challenges related to the implementation of non-linear functions using quantum circuits. In this paper, we use repeat-until-success circuits enabled by real-time control-flow feedback to realize quantum neurons with non-linear activation functions. These neurons constitute elementary building blocks that can be arranged in a variety of layouts to carry out deep learning tasks quantum coherently. As an example, we construct a minimal feedforward quantum neural network capable of learning all 2-to-1-bit Boolean functions by optimization of network activation parameters within the supervised-learning paradigm. This model is shown to perform non-linear classification and effectively learns from multiple copies of a single training state consisting of the maximal superposition of all inputs.
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.
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.
Due to the linearity of quantum mechanics, it remains a challenge to design quantum generative machine learning models that embed nonlinear activations into the evolution of the state vector. However, some of the most successful classical generative models, such as those based on neural networks, involve highly nonlinear dynamics for quality training. In this paper we explore the effect of these dynamics in quantum generative modeling by introducing a model that adds nonlinear activations via a neural network structure onto the standard Born machine framework---the quantum neuron Born machine (QNBM). To achieve this, we utilize a previously introduced quantum neuron subroutine, which is a repeat-until-success circuit with midcircuit measurements and classical control. After introducing the QNBM, we investigate how its performance depends on network size, by training a three-layer QNBM with four output neurons and various input and hidden layer sizes. We then compare our nonlinear QNBM to the linear quantum circuit Born machine (QCBM). We allocate similar time and memory resources to each model such that the only major difference is the qubit overhead required by the QNBM. With gradient-based training, we show that while both models can easily learn a trivial uniform probability distribution, on a more challenging class of distributions, the QNBM achieves an almost three-times-smaller error rate than a QCBM with a similar number of tunable parameters. We therefore provide evidence that suggests that nonlinearity is a useful resource in quantum generative models, and we put forth the QNBM as a model with good generative performance and potential for quantum advantage.
In this paper, we propose a simple neural net that requires only O(n log_2k) number of qubits and O(nk) quantum gates: Here, N is the number of input parameters, and k is the number of weights applied to these parameters in the proposed neural net. We describe the network in terms of a quantum circuit, and then draw its equivalent classical neural net which involves O(k^n) nodes in the hidden layer. Then, we show that the network uses a periodic activation function of cosine values of the linear combinations of the inputs and weights. The backpropagation is described through the gradient descent, and then iris and breast cancer datasets are used for the simulations. The numerical results indicate the network can be used in machine learning problems and it may provide exponential speedup over the same structured classical neural net.
The current generation of quantum computers calls for quantum algorithms that require a limited number of quantum gates and are resilient to noises. A suitable design strategy is variational circuits where parameters of circuits are determined through training, an approach that conforms with characteristics of machine learning applications. In this paper, we propose a low-depth and implementation-efficient non-linear activation function for quantum neural networks (QNNs). The building block of a quantum circuit is quantum gate which is a unitary operation. Thus, building a non-linear component out of quantum gates is challenging. While the majority of prior works used measurement as the source of non-linearity, this method has limited ability in classifying datasets. We propose a quantum circuit for the popular Rectified Linear Unit (ReLU) activation function. Our proposed circuit is based on low-cost quantum gates that can be synthesized into primitive gates in contemporary quantum computers. We exploit QNNs that rely on quantum rotation to define decision boundaries for classification problems. In addition, we use controlled quantum gates to detect correlation in data through entanglement of qubits. Our evaluations reveal that QNNs equipped with our proposed quantum ReLU perform well on standard benchmark datasets while requiring dramatically fewer number of epochs for training compared with classical neural networks. In addition, we run QNNs with different number of quantum layers on an IBM quantum computer and show that our proposed circuits are practical and generate meaningful results on real quantum computers.