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Quantum computing models for artificial neural networks
Stefano Mangini, Francesco Tacchino, Dario Gerace, Daniele Bajoni, Chiara Macchiavello
TL;DR
Neural networks have driven AI progress, while their increasing computational demands motivate interest in quantum approaches. This review surveys proposals for implementing neural-network functions on quantum architectures and strategies for addressing their challenges. It reports theoretical indications of greater QNN expressivity and training efficiency, while noting that clear routes to quantum learning advantage remain under study.
Problem
Whether quantum computers can provide an advantage for learning models remains an open question, with only a few clear indications so far.
Source: §IV · p. 5Method
The article reviews proposals for quantum neural networks, strategies for addressing their challenges, and prospects for quantum machine learning.
Source: §I · p. 1Results
Effective-dimension analysis indicates that quantum neural networks can be more expressive and efficient during training than classical counterparts.
Source: §1 · p. 6Takeaways & Limitations
No clear separation has yet been found for quantum learning on data that are inherently quantum.
Source: §3 · p. 6Trainability results often rely on strong ansatz assumptions, so QNNs require case-by-case analysis for precise conclusions.
Source: §1 · p. 5
Abstract
from arXiv · showhide
Neural networks are computing models that have been leading progress in Machine Learning (ML) and Artificial Intelligence (AI) applications. In parallel, the first small scale quantum computing devices have become available in recent years, paving the way for the development of a new paradigm in information processing. Here we give an overview of the most recent proposals aimed at bringing together these ongoing revolutions, and particularly at implementing the key functionalities of artificial neural networks on quantum architectures. We highlight the exciting perspectives in this context and discuss the potential role of near term quantum hardware in the quest for quantum machine learning advantage.
I. INTRODUCTION
Neural networks have driven AI progress, but their growing computational demands have sharpened interest in quantum computing. Because quantum computation is linear and does not permit arbitrary-precision copying, classical neural-network algorithms cannot simply be transferred to quantum platforms; the article reviews proposed approaches and prospects.
I. INTRODUCTION
- Feedforward neural networks repeatedly apply affine transformations and element-wise nonlinear activation functions, with weights and biases adjusted during training.Source: §I · p. 1
- Neural networks learn by adjusting weights to fit target tasks, while generalization concerns prediction on data not used for training.Source: §I · p. 1
- Growing classical neural-network data, feature, and parameter sizes have driven sharply rising energy and computational demands, motivating interest in quantum computing.Source: §I · p. 1
- Quantum computation is intrinsically linear and cannot copy information with arbitrary precision, so classical neural-network algorithms cannot simply be ported to quantum platforms.Source: §I · p. 1
- The article reviews approaches to quantum artificial neural networks, strategies for overcoming implementation difficulties, and future prospects.Source: §I · p. 1
- Figure 1 contrasts classical feedforward networks with quantum variational circuits, perceptrons, kernel methods, convolutional networks, and dissipative networks.Source: Figure 1 · p. 2
II. QUANTUM NEURAL NETWORKS
Quantum neural networks are parametrized quantum circuits whose repeated, self-similar layers resemble classical neural-network architectures. The section introduces their circuit framework, explains why data re-uploading can increase expressivity, and surveys implementation choices and the shift toward parametrized circuits.
II. QUANTUM NEURAL NETWORKS
- Data encoding loads inputs into a quantum circuit, followed by a variational ansatz and measurements that inform classical parameter updates.Source: §II · p. 2
- A quantum neural network is a parametrized quantum circuit with repeated self-similar layers, combining variational gates with fixed, often entangling operations.Source: §II · p. 3
- A naive QNN structure may represent only a limited range of input functions, often mostly sine functions; repeated data re-uploading can increase its expressivity.Source: §II · p. 3
- Unlike earlier approaches that sought quantum speedups for linear algebra, current research has shifted toward parametrized circuits as quantum counterparts to classical learning models.Source: §II · p. 3
- QNN implementations retain flexibility in choosing input encoding and variational circuits, with different architectures developed for different purposes.Source: §II · p. 3
1. Quantum Perceptron Models
Quantum perceptron research has addressed how quantum systems can reproduce classical neuron behavior despite their linear unitary evolution. Proposals have demonstrated simple image-recognition tasks, and connected quantum neurons have been implemented in proof-of-principle experiments.
1. Quantum Perceptron Models
- Quantum perceptron proposals have carried out simple pattern-recognition tasks on binary and grayscale images.Source: §1 · p. 3
- Early quantum perceptron work sought to reproduce classical perceptron nonlinearities using systems governed by linear unitary evolution.Source: §1 · p. 3
- Models connecting multiple quantum neurons have been realized in proof-of-principle experiments on superconducting devices.Source: §1 · p. 3
2. Kernel methods
Quantum kernel methods map data into quantum states and use state inner products as similarities for classification or regression. Their potential advantage lies in constructing kernels that are difficult to compute classically.
2. Kernel methods
- Quantum kernel methods map each input to a quantum state and use inner products in an exponentially large Hilbert space as similarity measures.Source: §2 · p. 3
- The resulting kernel matrix can be used with classical methods such as support vector machines for classification or regression.Source: §2 · p. 3
- Quantum support vector machines may offer a classification advantage by constructing kernel functions that are hard to compute classically.Source: §2 · p. 3
3. Generative Models
The section surveys quantum approaches to generative modeling, including quantum versions of classical neural models and models based on quantum probabilities.
3. Generative Models
- A generative model learns to reproduce an unknown probability distribution from a training set.Source: §3 · pp. 3–4
- Quantum Boltzmann Machines replace classical nodes with qubits and the energy function with a quantum Hamiltonian for generative learning.Source: §3 · p. 4
- Quantum GANs use QNNs as generator and discriminator, while Born machines generate data according to Born’s rule.Source: §3 · p. 4
4. Quantum Convolutional Neural Networks
The section describes quantum convolutional and dissipative neural networks, alongside the central challenge of training learning models efficiently.
4. Quantum Convolutional Neural Networks
- Quantum convolutional networks alternate parametrized unitary operations on qubit subsets with pooling by measurement, repeating until few qubits remain.They use logarithmically many parameters in the number of qubits, and are described as useful for quantum phase recognition and quantum error correction.Source: §4 · p. 4
5. Quantum Dissipative Neural Networks
- Dissipative QNNs progressively discard qubit layers after interaction and can learn general quantum transformations, enabling universal quantum computation.Their backpropagation-like training algorithm offers a possible route to quantum analogues of deep networks for quantum data.Source: §5 · p. 4
III. TRAINABILITY OF QNN
- Efficient training is pivotal for practical learning models; historically, limited computational power and training routines kept AI methods largely academic.Source: §III · p. 4
1. Barren plateaus
Barren plateaus can make QNN optimization difficult, but their occurrence and severity depend on cost functions, circuit structure, and noise.
1. Barren plateaus
- Under a unitary 2-design ansatz, cost gradients have mean zero and variance approximately 2^-n, yielding an essentially flat landscape except near minima.A unitary 2-design is sufficiently random that sampling matches the Haar distribution up to the second moment.Source: §1 · p. 4
- Global costs hinder trainability at any circuit depth, whereas local costs allow efficient trainability at depths of order O(log n).Evaluating costs on a small visible-qubit subset while tracing out hidden qubits may also produce a plateau.Source: §1 · pp. 4–5
- Entanglement can spread information across the network and contribute to plateaus, while hardware noise can induce them as well.Source: §1 · p. 5
- Trainability results depend on ansatz assumptions: dissipative QNNs can have plateaus with global unitaries, while convolutional QNNs seem immune.More expressive ansatzes generally have flatter landscapes; VQA training is argued to be NP-hard, though similar worst-case conclusions do not preclude classical NN applications.Source: §1 · p. 5
2. Avoiding plateaus
The section reviews strategies for mitigating barren plateaus and explains why quantum optimization often relies on classical routines, with several quantum-specific alternatives.
2. Avoiding plateaus
- Proposed plateau-mitigation strategies include layerwise training, identity-initialized parameter subsets, inter-layer parameter correlations, and classical recurrent-NN initialization heuristics.Source: §2 · p. 5
- Local cost functions and suitable entanglement scaling between hidden and visible units are important choices for avoiding barren plateaus.Source: §2 · p. 5
3. Optimization routines
- Quantum measurements prevent straightforward backpropagation through intermediate states, so training typically uses classical gradient-based or gradient-free optimizers.Quantum-tailored alternatives include the parameter shift rule, Quantum Natural Gradient, and closed update formulas.Source: §3 · p. 5
- Fully coherent quantum parameter updates remain out of reach for near-term hardware limited in qubit count and circuit depth.Source: §3 · p. 5
IV. QUANTUM ADVANTAGE
Whether quantum models can achieve a learning advantage remains an open question, despite initial results suggesting possible differences in capacity and training efficiency. The section reviews proposed ways to assess model power and the role of data in quantum advantage.
IV. QUANTUM ADVANTAGE
- The prospect of quantum learning advantage remains under study, with only a few clear indications of how to achieve it.Quantum computers are expected to outperform classical counterparts in some applications, particularly sampling from complex probability distributions.Source: §IV · p. 5
1. Power of Quantum Neural Networks
- “Expressivity” refers both to function approximation and distance from an exact 2-design, motivating the search for a unifying measure of QNN power.The two meanings respectively concern approximating any input function and spanning quantum states uniformly [53].Source: §1 · p. 5
- A universal capacity metric based on maximum information stored in model parameters was defined from the classical VC-dimension concept [67].The VC dimension measures the maximum number of independent classifications a learning model can implement.Source: §1 · p. 5
- Quantum models may be more expressive and efficient to train than classical counterparts under an effective-dimension measure, but this evidence comes from two specific QNN instances.The proposed advantage is linked to a more evenly spread Fisher Information spectrum; the examples use trivial data encoding and the encoding scheme in.Source: §1 · p. 6
2. The role of data
The role and encoding of data matter to assessments of quantum learning advantage. Geometric kernel tests and results involving quantum data suggest possible advantages in particular settings, but do not establish a general separation.
2. The role of data
- Projected quantum kernels reduced encoded-data effective dimension while retaining quantum correlations, and outperformed tested classical models in prediction error on engineered datasets.The geometric test uses training data and the model-induced kernel, independent of the target function; a large geometric distance indicates there exists a dataset on which the quantum model can outperform the classical one.Source: §2 · p. 6
- Careless encoding can spread data points too far apart, making the kernel matrix approximate the identity and contributing to poor quantum-model performance.When the geometric quantity is similar for classical and quantum learning, classical models perform similarly or better.Source: §2 · p. 6
3. Quantum data for quantum neural networks
- No clear quantum–classical separation has been found for quantum data; classical learners match quantum models in sample complexity and average prediction error, though worst-case advantage remains possible.Identifying and generating interesting quantum datasets remains an open problem.Source: §3 · p. 6
4. Quantum-inspired methods
Quantum-inspired classical procedures, including tensor networks, are relevant competitors and tools for evaluating quantum neural network proposals. The outlook identifies trainability, data, classical comparison, and practical hardware demonstrations as continuing priorities.
4. Quantum-inspired methods
- Tensor networks can simulate quantum computers well in specific regimes and provide a classical framework for some quantum-neural-network proposals.The paper anticipates competition between quantum processors and advanced classical methods, including neural network quantum states.Source: §4 · p. 6
V. OUTLOOKS
- Quantum machine learning models are presented as promising, while practical demonstrations of relevance remain a near-future focus amid rapidly evolving research.The authors expect further results on directions in the quest for quantum computational advantage in AI applications.Source: §V · pp. 6–7
- Trainability, suitable dataset identification and treatment, and systematic comparison with classical counterparts remain key theoretical and practical challenges.The outlook also anticipates attention to experimental demonstrations of practical relevance as hardware progresses.Source: §V · p. 7