Abstract The emerging technology of quantum neural networks (QNNs) offers a quantum advantage over classical artificial neural networks (ANNs) in terms of speed or efficiency of information processing tasks. It is well established that nonlinear mapping between input and output is an indispensable feature of classical ANNs, while in a QNN the roles of nonlinearity are not yet fully understood. As one tends to think of QNNs as physical systems, it is natural to think of nonlinear mapping originating from a physical nonlinearity of the system, such as Kerr nonlinearity. Here we investigate the effect of Kerr nonlinearity on a bosonic QNN in the context of both classical (simulating an XOR gate) and quantum (generating Schrödinger cat states) tasks. Aside offering a mechanism of nonlinear input-output mapping, Kerr nonlinearity reduces the effect of noise or losses, which are particularly important to consider in the quantum setting. We note that nonlinear mapping may also be introduced through a nonlinear input-output encoding rather than a physical nonlinearity: for example, an output intensity is already a nonlinear function of input amplitude. While in such cases Kerr nonlinearity is not strictly necessary, it still increases the performance in the face of noise or losses.
Abstract The emerging technology of quantum neural networks (QNNs) offers a quantum advantage over classical artificial neural networks (ANNs) in terms of speed or efficiency of information processing tasks. It is well established that nonlinear mapping between input and output is an indispensable feature of classical ANNs, while in a QNN the roles of nonlinearity are not yet fully understood. As one tends to think of QNNs as physical systems, it is natural to think of nonlinear mapping originating from a physical nonlinearity of the system, such as Kerr nonlinearity. Here we investigate the effect of Kerr nonlinearity on a bosonic QNN in the context of both classical (simulating an XOR gate) and quantum (generating Schrödinger cat states) tasks. Aside offering a mechanism of nonlinear input-output mapping, Kerr nonlinearity reduces the effect of noise or losses, which are particularly important to consider in the quantum setting. We note that nonlinear mapping may also be introduced through a nonlinear input-output encoding rather than a physical nonlinearity: for example, an output intensity is already a nonlinear function of input amplitude. While in such cases Kerr nonlinearity is not strictly necessary, it still increases the performance in the face of noise or losses.
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 power and expressivity of deep classical neural networks can be attributed to nonlinear input-output relations. Such nonlinearities are at the heart of many computational tasks, such as data classification and pattern recognition. Quantum neural networks, on the other hand, are necessarily linear as they process information via unitary operations. Here we show that effective nonlinearities can be implemented in these platforms by exploiting the relationship between information processing and many-body quantum dynamics. The crucial point is that quantum many-body systems can show emergent collective behavior in the vicinity of phase transitions, which leads to an effectively nonlinear dynamics in the thermodynamic limit. In the context of quantum neural networks, which are necessarily finite, this translates into metastability with transient nonergodic behavior. By using a quantum neural network the architecture of which is inspired by dissipative many-body quantum spin models, we show that this mechanism indeed allows one to realize nonlinear data classification, despite the underlying dynamics being local and linear. Our proof-of-principle study may pave the way for the systematic construction of quantum neural networks with emergent nonlinear properties.
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.
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 x to classical perceptrons acting on x⊗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≥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 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.
We report on a gate-based variational quantum classifier implemented with single photons and probabilistic gates, to emulate the standard quantum circuit model framework. We evaluate the expressive power of two deployable quantum neural networks (QNNs) by computing their effective dimension, a capacity measure grounded in a proven generalization-error bound, and compare them with classical artificial neural networks (ANNs) of equivalent trainable-parameter count. Supervised binary classification tasks are used to benchmark performance across photonic and superconducting QNNs, both of which exhibit superior converged (lower) cross-entropy loss and (higher) prediction accuracy relative to matched-parameter ANNs. For a nonlinearly separable task, our photonic QNN with a single pair of trainable parameters successfully converged (loss 0.04 and accuracy 100%), whereas the equivalent ANN failed to learn the decision boundary, saturating at random-guessing performance. We simulate photonic quantum circuits, training them on the XOR problem and a two-class Iris subset using gradient-free optimization, and assess their robustness to sampling errors under realistic noise processes including photon loss and phase-shifter imperfections. Circuits with comparatively high effective dimension were deployed remotely on a six-qubit photonic quantum processor, achieving classification accuracies of up to 100% in both online and offline learning settings. Notably, even the simplest QNN deployed, with just two trainable parameters, successfully solved tasks that classically require ANNs with at least quadruple the number of parameters, suggesting an emergent algorithmic advantage. Overall, these results demonstrate a clear proof-of-principle that gate-based QNNs can be realized and trained effectively on current photonic hardware, providing proof of algorithmic advantage on a gate-based photonic QNN.