Communication complexity provides a natural framework for quantifying the classical resources required to reproduce quantum statistics. In the qubit prepare-and-measure scenario, two classical bits have been shown to be necessary and sufficient to simulate arbitrary qubit states and arbi- trary quantum measurements exactly. However, this result does not exclude the possibility that restricted families of measurements may admit accurate 1-bit classical approximations. We use a neural network procedure to demonstrate that a single bit can achieve high average accuracy for specific measurement families. A performance analysis of our neural network reveals that symmet- ric measurements with uniformly weighted elements, such as those forming regular polyhedra, are particularly amenable to this restricted communication. By analyzing the patterns learned by the neural network, we derive an analytical protocol that is extremely accurate for finite information- ally complete symmetric configurations and becomes exact in the limit of a continuous isotropic measurement.
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9 pages with the main text and 9 pages with Appendix
Communication complexity provides a natural framework for quantifying the classical resources required to reproduce quantum statistics. In the qubit prepare-and-measure scenario, two classical bits have been shown to be necessary and sufficient to simulate arbitrary qubit states and arbi- trary quantum measurements exactly. However, this result does not exclude the possibility that restricted families of measurements may admit accurate 1-bit classical approximations. We use a neural network procedure to demonstrate that a single bit can achieve high average accuracy for specific measurement families. A performance analysis of our neural network reveals that symmet- ric measurements with uniformly weighted elements, such as those forming regular polyhedra, are particularly amenable to this restricted communication. By analyzing the patterns learned by the neural network, we derive an analytical protocol that is extremely accurate for finite information- ally complete symmetric configurations and becomes exact in the limit of a continuous isotropic measurement.
Bell's theorem states that Local Hidden Variables (LHVs) cannot fully explain the statistics of measurements on some entangled quantum states. It is natural to ask how much supplementary classical communication would be needed to simulate them. We study two long-standing open questions in this field with neural network simulations and other tools. First, we present evidence that all projective measurements on partially entangled pure two-qubit states require only one bit of communication. We quantify the statistical distance between the exact quantum behaviour and the product of the trained network, or of a semianalytical model inspired by it. Second, while it is known on general grounds (and obvious) that one bit of communication cannot eventually reproduce all bipartite quantum correlation, explicit examples have proved evasive. Our search failed to find one for several bipartite Bell scenarios with up to 5 inputs and 4 outputs, highlighting the power of one bit of communication in reproducing quantum correlations.
Simulating quantum algorithms with classical resources generally requires exponential resources. However, heuristic classical approaches are often very efficient in approximately simulating special circuit structures, for example with limited entanglement, or based on one-dimensional geometries. Here we introduce a classical approach to the simulation of general quantum circuits based on neural-network quantum states (NQS) representations. Considering a set of universal quantum gates, we derive rules for exactly applying single-qubit and two-qubit Z rotations to NQS, whereas we provide a learning scheme to approximate the action of Hadamard gates. Results are shown for the Hadamard and Fourier transform of entangled initial states for systems sizes and total circuit depths exceeding what can be currently simulated with state-of-the-art brute-force techniques. The overall accuracy obtained by the neural-network states based on Restricted Boltzmann machines is satisfactory, and offers a classical route to simulating highly-entangled circuits. In the test cases considered, we find that our classical simulations are comparable to quantum simulations affected by an incoherent noise level in the hardware of about 10−3 per gate.
Network quantum steering plays a pivotal role in quantum information science, enabling robust certification of quantum correlations in scenarios with asymmetric trust assumptions among network parties. The intricate nature of quantum networks, however, poses significant challenges for the detection and quantification of steering. In this work, we develop a neural network-based method for measuring network quantum steerability, which can be generalized to arbitrary quantum networks and naturally applied to standard steering scenarios. Our method provides an effective framework for steerability analysis, demonstrating remarkable accuracy and efficiency in standard bipartite and multipartite steering scenarios. Numerical simulations involving isotropic states and noisy Greenberger-Horne-Zeilinger states yield results that are consistent with established findings in these respective scenarios. Furthermore, we demonstrate its utility in the bilocal network steering scenario, where an untrusted central party shares two-qubit isotropic states of different visibilities, $\ensuremath{\nu}$ and $\ensuremath{\omega}$, with trusted end-point parties and performs a single Bell state measurement. Through explicit construction of a network local hidden state model derived from numerical results and incorporation of the entanglement properties of network assemblages, we analytically demonstrate that the network steering thresholds are determined by the curve $\ensuremath{\nu}\ensuremath{\omega}=1/3$ under the corresponding configuration.
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.
We investigate the relationship between two distinct classical approaches to quantum systems: direct simulation from a classical description and sample-based learning from measurement data. While both tasks ultimately aim to reproduce Born-rule statistics, complexity-theoretic results suggest that simulability and learnability need not coincide in general. Here we study this relationship empirically using a fixed deep energy-based generative model trained on measurement samples from controlled families of quantum states. We independently tune two quantum resources associated with classical simulation cost: entanglement, through the bond dimension of random matrix product states, and non-stabilizerness, through the number of T gates in Clifford-dominated circuits. Learning difficulty is characterized using two probes of neural-network complexity: the largest Hessian eigenvalue at convergence and Random Subspace Optimization. For both quantum resources, increasing simulation hardness systematically correlates with sharper loss landscapes and degraded reconstruction performance under constrained capacity. Our results indicate that, within the regimes studied here, classical learnability tracks known simulation complexity measures, suggesting that neural-network training dynamics can provide an empirical probe of quantum computational hardness.
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.