Quantum Mechanics and Neural Networks
This work introduces a novel framework for representing quantum mechanics as neural networks, potentially enabling new computational approaches in quantum physics.
The authors demonstrated that any Euclidean-time quantum mechanical theory can be represented as a neural network, using principles like the Kosambi-Karhunen-Loève theorem and reflection positivity, and illustrated this with numerical examples recovering quantum results such as Heisenberg uncertainty and non-trivial commutators.
We demonstrate that any Euclidean-time quantum mechanical theory may be represented as a neural network, 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 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.