We discuss a quantum version of an artificial deep neural network where the role of neurons is taken over by qubits and the role of weights is played by unitaries. The role of the non-linear activation function is taken over by subsequently tracing out layers (qubits) of the network. We study two examples and discuss the learning from a quantum information theoretic point of view. In detail, we show that the lower bound of the Heisenberg uncertainty relations is defining the change of the gradient descent in the learning process. We raise the question if the limit by Nature to two non-commuting observables, quantified in the Heisenberg uncertainty relations, is ruling the optimization of the quantum deep neural network. We find a negative answer.
We discuss a quantum version of an artificial deep neural network where the role of neurons is taken over by qubits and the role of weights is played by unitaries. The role of the non-linear activation function is taken over by subsequently tracing out layers (qubits) of the network. We study two examples and discuss the learning from a quantum information theoretic point of view. In detail, we show that the lower bound of the Heisenberg uncertainty relations is defining the change of the gradient descent in the learning process. We raise the question if the limit by Nature to two non-commuting observables, quantified in the Heisenberg uncertainty relations, is ruling the optimization of the quantum deep neural network. We find a negative answer.
The remarkable generalization capabilities of overparameterized neural networks remain one of the most profound mysteries in modern machine learning. Despite having sufficient capacity to simply memorize training data, these networks consistently achieve superior performance on unseen test examples. Traditional analyses focusing on weight distributions, loss landscapes, and optimization dynamics have provided partial insights but fail to capture the complex correlational structure that emerges between neurons during learning. This paper introduces a comprehensive quantum information framework for analyzing neural network representations, treating hidden layer activations as quantum states and characterizing them through density matrices, von Neumann entropy, purity, and related quantum information metrics. Through systematic experiments on synthetic classification tasks and algorithmic reasoning problems spanning over 3000 cumulative training epochs and 5000 Wang-Landau sampling iterations, we demonstrate that these quantum metrics reveal structure completely invisible to classical analysis. Our results show that stochastic gradient descent systematically reduces von Neumann entropy while increasing purity as networks learn, transitioning from highly mixed quantum states with entropy approximately 1.42 and purity approximately 0.29 to purer configurations with entropy approximately 1.19 and purity approximately 0.37 that achieve superior test accuracy of 99.5%. Using Wang-Landau sampling to explore the equilibrium entropy landscape, we discover that neural networks naturally favor high-entropy states with entropy approximately 1.42 and moderate accuracy of 62.0%, and that optimization actively purifies the quantum state to achieve task performance. On modular arithmetic tasks exhibiting the grokking phenomenon, we observe that von Neumann entropy increases dramatically from 1.55 to 3.29, representing a 112% increase, during an extended 2000-epoch memorization phase before any improvement in generalization, while purity decreases correspondingly from 0.444 to 0.042, a 90.5% decrease, approaching the maximally mixed limit of 0.01. This entropy accumulation phase appears necessary for eventual generalization, with the network building quantum correlations at a rate of approximately 0.00097 nats per epoch during the plateau. Weight entropy, in contrast, shows minimal change of only 11% increase over the same period. Correlation analysis reveals that von Neumann entropy and purity are almost perfectly anticorrelated with a correlation coefficient of 1 -0.99, while weight entropy shows weak correlation with quantum metrics with absolute correlation coefficients less than 0.3. The effective rank of the density matrix stabilizes near 2.7 to 3.0 for the 6-neuron classifier, indicating that the network effectively uses approximately three independent activation modes. These findings establish quantum information metrics as powerful tools for understanding neural network learning dynamics, provide the first quantitative evidence for the entropy accumulation interpretation of grokking, and suggest that neural networks undergo a quantum-inspired phase transition during learning. We discuss implications for generalization theory, optimization algorithm design, and the statistical physics of deep learning, and outline a comprehensive research agenda for extending these methods to larger architectures and more complex tasks.
The remarkable generalization capabilities of overparameterized neural networks remain one of the most profound mysteries in modern machine learning. Despite having sufficient capacity to simply memorize training data, these networks consistently achieve superior performance on unseen test examples. Traditional analyses focusing on weight distributions, loss landscapes, and optimization dynamics have provided partial insights but fail to capture the complex correlational structure that emerges between neurons during learning. This paper introduces a comprehensive quantum information framework for analyzing neural network representations, treating hidden layer activations as quantum states and characterizing them through density matrices, von Neumann entropy, purity, and related quantum information metrics. Through systematic experiments on synthetic classification tasks and algorithmic reasoning problems spanning over 3000 cumulative training epochs and 5000 Wang-Landau sampling iterations, we demonstrate that these quantum metrics reveal structure completely invisible to classical analysis. Our results show that stochastic gradient descent systematically reduces von Neumann entropy while increasing purity as networks learn, transitioning from highly mixed quantum states with entropy approximately 1.42 and purity approximately 0.29 to purer configurations with entropy approximately 1.19 and purity approximately 0.37 that achieve superior test accuracy of 99.5%. Using Wang-Landau sampling to explore the equilibrium entropy landscape, we discover that neural networks naturally favor high-entropy states with entropy approximately 1.42 and moderate accuracy of 62.0%, and that optimization actively purifies the quantum state to achieve task performance. On modular arithmetic tasks exhibiting the grokking phenomenon, we observe that von Neumann entropy increases dramatically from 1.55 to 3.29, representing a 112% increase, during an extended 2000-epoch memorization phase before any improvement in generalization, while purity decreases correspondingly from 0.444 to 0.042, a 90.5% decrease, approaching the maximally mixed limit of 0.01. This entropy accumulation phase appears necessary for eventual generalization, with the network building quantum correlations at a rate of approximately 0.00097 nats per epoch during the plateau. Weight entropy, in contrast, shows minimal change of only 11% increase over the same period. Correlation analysis reveals that von Neumann entropy and purity are almost perfectly anticorrelated with a correlation coefficient of 1-0.99, while weight entropy shows weak correlation with quantum metrics with absolute correlation coefficients less than 0.3. The effective rank of the density matrix stabilizes near 2.7 to 3.0 for the 6-neuron classifier, indicating that the network effectively uses approximately three independent activation modes. These findings establish quantum information metrics as powerful tools for understanding neural network learning dynamics, provide the first quantitative evidence for the entropy accumulation interpretation of grokking, and suggest that neural networks undergo a quantum-inspired phase transition during learning. We discuss implications for generalization theory, optimization algorithm design, and the statistical physics of deep learning, and outline a comprehensive research agenda for extending these methods to larger architectures and more complex tasks.
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.
A novel neural networks model with quantum weights is present.Firstly, based on the information processing modes of biology neuron and quantum computing theory, a quantum neuron model is presented, which is composed of weighting, aggregating, activating, and inspiriting.Secondly the quantum neural networks model based on quantum neuron is constructed in which the input and the output are real vectors; the linked weight and the activation value are qubits.On the basic of the gradient descent algorithm, a learning algorithm of this model is proposed.It is shown that this algorithm is super-linearly convergent under certain conditions and can increase the probability of getting the global optimal solution.Finally, the availability of the model is illustrated in both convergence speed and convergence rate by two application examples of pattern recognition and function approximation.
Abstract We demonstrate that any Euclidean-time quantum mechanical theory may be represented as a neural network (NN), ensured by the Kosambi–Karhunen–Loève theorem, mean-square path continuity, and finite two-point functions. The additional constraint of reflection positivity, which is related to unitarity, may be achieved by a number of mechanisms, such as imposing NN parameter space splitting or the Markov property. Non-differentiability of the networks is related to the appearance of non-trivial commutators. Neural networks acting on Markov processes are no longer Markov, but still reflection positive, which facilitates the definition of deep NN quantum systems. We illustrate these principles in several examples using numerical implementations, recovering classic quantum mechanical results such as Heisenberg uncertainty, non-trivial commutators, and the spectrum.