Quantum Mechanics and Neural Networks

TL;DR

This work formulates a neural-network framework for quantum mechanics in Euclidean time by representing quantum correlators as NN-parameter averages and leveraging the Kosambi–Karhunen–Loève decomposition to show MQMs admit NN descriptions. It identifies two practical routes to mimic unitarity through reflection positivity: a parameter-splitting construction and a Markov-process approach, linking RP to how time is encoded in NN architectures. The authors develop a deep NN-QM formalism that preserves RP across layers and demonstrate it with Ornstein–Uhlenbeck inputs, recovering core QM features such as the commutator relation, Heisenberg uncertainty, and spectral structure, while also revealing non-Gaussian, interacting behavior in deep architectures. These results provide a universal approximation viewpoint for MQMs via NN expansions and suggest pathways for Hamiltonian engineering and ML-driven discovery of quantum theories. The work lays groundwork for extending to higher dimensions and more general RP-constrained quantum field theories, with potential implications for quantum simulations and constructive QFT.

Abstract

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.

Figures (8)

• Figure 1: A Venn diagram illustrating the various definitions used in this section. Within the space of all stochastic processes (SP), a subset (NN-SP) admit a representation as a neural network. We have proven that every MQM, which is a stochastic process obeying \ref{['assumption_one']} and \ref{['assumption_two']}, has a neural network description, and thus $\mathrm{MQM} \subset \mathrm{NN}\text{-}\mathrm{SP}$. One might impose additional restrictions upon minimal quantum models, such as the Osterwalder-Schrader axioms (OS-QM) or another set of conditions defining a notion of quantum mechanics of one's choosing (QM'), which carve out different subsets of MQM.
• Figure 2: The correlation function $G^{(2)} ( t , 0 )$ obtained by simulating Ornstein-Uhlenbeck processes with $\theta, \sigma \in \{ 1, 2 \}$ for a total time $T = 5$ and step size $\Delta t = 0.01$. The initial position is drawn from the stationary distribution (\ref{['ou_stationary']}) and updates are performed using the Euler-Maruyama method. We carry out 200 epochs of simulations with 10,000 sample paths per epoch, compute the average correlator $G^{(2)} ( t , 0 )$ for each time $t$, and use the standard deviation across epochs for the error bars. The experimental results are shown as scatter plots with cool colors (green, cyan, purple, blue), while the corresponding analytical curves (\ref{['OU_two_point']}) for the two-point functions are drawn above the data points in warm colors (red, orange, yellow, and orange-red) and lie within all of the error bars. The marker size for the simulated results has been enlarged to enhance visibility; the error bars are plotted but are not visible because they are smaller than the corresponding markers.
• Figure 3: The values of the commutator $C(t)$, defined in (\ref{['Ct_defn']}), for Ornstein-Uhlenbeck processes with $\theta = 1$ and varying values of $\sigma$. We generate $10,000$ sample paths per epoch for 200 epochs, using the Euler-Maruyama method with step size $\Delta t = 0.01$ and total time $T = 5$, and plot the average value $\langle C ( t ) \rangle$ with error bars given by the standard deviation across epochs. The value of the commutator is constant in time and scales as the square of $\sigma$, as expected from the relation $\hbar = \sigma^2 m$ of equation (\ref{['parameter_map']}). This reproduces the commutator relation $[ \hat{x} , \hat{p} ] = \hbar$ for a quantum model in Euclidean signature. Although the displayed error bars appear bigger for the commutators with larger values of $\sigma$, the ratio of the standard deviation to the mean is approximately constant across experiments.
• Figure 4: We display the results of computing the first $10$ energy eigenvalues of the harmonic oscillator for various values of $\theta$, using the algorithm described above. For each choice of $\theta$, we carry out $20$ separate experiments, each of which generates $5,000,000$ sample paths to estimate the Hamiltonian. Error bars are included, computed using the standard deviation across these $20$ experiments, but are too small to be seen. Dotted lines indicate the true energy eigenvalues, $E_n = \theta n$, where we recall that the ground state energy has been shifted to zero. The experimental values are more accurate for low-lying energies at small $n$ and then accrue larger errors for higher $E_n$, although the magnitude and sign of the errors differs between the different choices of $\theta$.
• Figure 5: We compare the first three numerical energy eigenstates (scatter), obtained by diagonalizing the estimated Hamiltonian, to the true harmonic oscillator eigenstates (solid line). For concreteness, we take $\theta = \sigma = 1$. The Hamiltonian is computed by generating $50,000,000$ one-step sample paths and applying a similarity transformation to the estimated Fokker-Planck operator following the procedure described in the main text.

Theorems & Definitions (9)

• Theorem 2.1: Kosambi-Karhunen-Loève
• Proposition 3.1
• proof
• Theorem 3.1
• proof
• Proposition 4.1
• Proposition 4.2
• Theorem 4.1
• proof