Quantum Mechanics and Neural Networks (2504.05462v1)

Abstract: We demonstrate that any Euclidean-time quantum mechanical theory may be represented as a neural network, ensured by the Kosambi-Karhunen-Lo`eve 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 neural network 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 neural network 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.

• The paper demonstrates that a wide class of Euclidean-time quantum mechanical systems can be represented using neural networks, supported by the KKL theorem.
• Mechanisms such as parameter space splitting and leveraging Markov processes ensure reflection positivity, enabling unitary quantum system representation.
• Numerical examples recover known quantum results, suggesting practical applications for simulating quantum systems and engineering Hamiltonians.

Insights on Quantum Mechanics and Neural Networks

The paper "Quantum Mechanics and Neural Networks" by Christian Ferko and James Halverson explores a promising intersection of quantum mechanics (QM) and neural networks (NNs), providing an innovative framework for representing Euclidean-time quantum mechanical theories using neural network architectures. The authors assert that any quantum mechanical system defined in Euclidean time can be mapped onto a neural network, justified by the Kosambi-Karhunen-Loéve (KKL) theorem, alongside properties unique to quantum mechanics, such as reflection positivity—a property linked with unitarity.

Key Contributions

1. Universality of Neural Networks in Quantum Mechanics: The authors establish that a wide class of quantum mechanical systems, adhering to mean-square continuity and possessing finite two-point functions, can be represented using neural networks. This claim is substantiated by the KKL theorem, which provides a spectral decomposition framework suitable for neural networks interpretation.
2. Mechanisms Ensuring Unitarity and Reflection Positivity: Two mechanisms are identified for achieving reflection positivity—parameter space splitting and leveraging the Markov property. Reflection positivity is crucial in confirming that a Euclidean-time system can be transitioned into a Lorentzian quantum system with unitary characteristics.

• Parameter Space Splitting: This method involves decomposing the parameter set such that it naturally generates reflection-positive processes, albeit at the cost of translation invariance unless engineered for all times.
• Markov Processes: The authors demonstrate that neural networks acting on Markov processes, while no longer retaining Markov property, maintain reflection positivity, providing a potent construction method for deep neural network quantum systems.

1. Numerical Implementations and Quantum Mechanical Properties: The paper further illustrates these principles through numerical examples, recovering known quantum mechanical results like the Heisenberg uncertainty principle and the spectral decomposition of quantum states.
2. Deep Neural Network Quantum Mechanics (Deep NN-QM): The conception of Deep NN-QM suggests that neural networks can define complex Euclidean quantum systems, which are inherently reflection positive. Practical implementations show how the Ornstein-Uhlenbeck process, an analogue to the quantum harmonic oscillator, and various neural network architectures can portray rich dynamics akin to quantum mechanical paradigms.

Implications and Future Directions

• Practical Applications in Quantum Theory: This representation of quantum mechanics using neural network transforms offers a robust platform for simulating quantum systems computationally. The realizability of reflection-positive models through neural networks paves the way for computational techniques in quantum field theories and might inspire novel algorithms in quantum simulations.
• Engineering Hamiltonians with Desired Properties: By leveraging neural networks and machine learning, it includes the possibility of engineering Hamiltonians for specific quantum behaviors, optimizing NN architectures to emulate desired quantum scenarios through a learning paradigm.
• Generalization to Higher Dimensions: Although the focus is primarily on quantum mechanics as a one-dimensional field theory, the theoretical foundation provided hints towards extending the framework to higher-dimensional field theories. This potential extension would require considering distributions rather than ordinary functions in the field of field theories.

The research marks a significant step in quantum computing and neural networks, offering theoretical insights that could bridge the classical-quantum computational divide. As an exploration of the deep connections between machine learning and fundamental physics, it invites further theoretical and practical investigations into the cross-pollination of these domains.