Atomic Nuclei Reveal Limits of Neural Network Quantum Simulations

James W. T. Keeble and colleagues at TIFPA Trento Institute constructed neural quantum state representations of atomic nuclei, which possess strong entanglement and deviate from easily representable ‘stabilizer’ states. These representations reveal that states exhibiting greater non-stabilizerness prove more challenging for the network to learn, indicating this property sharply impacts the efficiency of compression and representation using restricted Boltzmann machines. The findings are key for optimising network architectures and improving the capacity to model highly entangled quantum systems. Quantum complexity limits neural network modelling of atomic nuclei Accuracy in representing medium-mass atomic nuclei dropped to below 13% error for the most complex cases, a substantial improvement over prior methods unable to model such highly entangled systems. This represents a significant advancement, as traditional methods, such as exact diagonalization and coupled-cluster theory, become computationally intractable for nuclei beyond a certain mass number due to the exponential scaling of the Hilbert space. The representational capability linked directly to ‘non-stabilizerness’, a measure of quantum complexity, with states exhibiting greater non-stabilizerness consistently proving more difficult for restricted Boltzmann machines (RBMs) to learn. Non-stabilizerness quantifies the extent to which a quantum state cannot be efficiently described by a stabilizer code, a type of quantum error-correcting code. Stabilizer states are relatively easy for classical computers to simulate, while non-stabilizer states require exponentially more resources. Previously hindering accurate simulations of nuclei exhibiting significant entanglement, a clear threshold now exists where conventional neural network approaches struggle. Researchers at the University of Surrey and the University of Edinburgh tailored a second-quantized formulation of neural quantum states for nuclear physics, sidestepping computational limitations encountered in earlier studies and enabling calculations within a manageable parameter space. This formulation leverages the fermionic nature of nucleons within the nucleus, employing creation and annihilation operators to describe their behaviour. Earlier attempts often relied on first-quantized approaches, which suffer from significant computational overhead when dealing with many-body systems. Detailed analysis revealed a strong correlation between accuracy and ‘non-stabilizerness’, while simple measures of entanglement, such as entanglement entropy, failed to show similar correlations, suggesting it is a primary factor limiting the ability of these networks to compress and represent highly entangled states. Entanglement entropy, while indicative of overall entanglement, doesn’t fully capture the specific type of entanglement that hinders neural network representation. The researchers found that non-stabilizerness provides a more nuanced and predictive measure of the difficulty faced by the RBM. Simulations using this approach allowed for modelling with approximately 10% of the parameters needed by traditional methods for the largest nucleus studied, 28Silicon. The use of restricted Boltzmann machines (RBMs) is central to this approach. RBMs are a type of generative stochastic artificial neural network capable of learning complex probability distributions. In this context, the RBM learns to represent the ground state wavefunction of the nucleus. The network’s parameters are adjusted through a training process, minimising the difference between the RBM-generated wavefunction and the true ground state obtained from more accurate, but computationally expensive, methods. The success of this method hinges on the RBM’s ability to efficiently encode the correlations present within the nuclear wavefunction. Although this new approach successfully maps the complex behaviour of atomic nuclei using neural networks, a key limitation remains regarding the computational demands of modelling increasingly complex systems. The accuracy of these networks is sharply impacted by ‘non-factorability’, essentially the degree to which a nucleus’s behaviour deviates from simple, predictable patterns. A highly factorable system would allow for the wavefunction to be expressed as a product of single-particle states, greatly simplifying the calculation. However, real nuclei exhibit strong correlations between nucleons, leading to significant non-factorability. Addressing these challenges will require substantially more computational power and refined network architectures, potentially necessitating exploration of alternative neural network designs or hybrid computational strategies. The significance of this work extends beyond merely achieving a reduction in computational cost. It provides fundamental insights into the relationship between quantum complexity and machine learning. Understanding which properties of quantum states are most challenging for neural networks to represent is crucial for developing more effective algorithms for quantum simulation.

This research establishes a valuable baseline for utilising artificial intelligence in nuclear physics, acknowledging the considerable challenge of representing highly complex nuclei. A direct link now exists between the complexity of quantum states and the ability of restricted Boltzmann machines (RBMs) to accurately represent them. Modelling medium-mass atomic nuclei showed that higher ‘non-stabilizerness’ proved systematically more difficult to learn, offering an important insight for refining these networks. This finding highlights it as a key factor governing how efficiently RBMs can compress and represent entangled quantum systems, and further development will begin to unlock even more complex simulations. The ability to accurately model larger and more complex nuclei could lead to a better understanding of nuclear structure, nuclear reactions, and the synthesis of heavy elements, potentially extending the reach of neural networks into previously inaccessible regimes of nuclear physics. Future research will likely focus on exploring different network architectures, such as variational quantum circuits, and developing more efficient training algorithms to overcome the limitations imposed by non-stabilizerness and non-factorability. The research demonstrated that restricted Boltzmann machines (RBMs) struggle to accurately learn representations of quantum states with higher ‘non-stabilizerness’. This matters because it identifies a key property of quantum systems that impacts how efficiently neural networks can compress and represent complex information. The findings suggest that non-stabilizerness is a primary factor limiting the performance of RBMs when modelling entangled systems like medium-mass atomic nuclei. Authors propose that future work will explore more sophisticated network architectures to address these limitations and improve representational efficiency. 👉 More information 🗞 Neural Quantum States in Non-Stabilizer Regimes: Benchmarks with Atomic Nuclei 🧠 ArXiv: https://arxiv.org/abs/2603.28646 Tags: