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Nonlocal quantum fluids emerge as dark-matter models and tools for quantumsimulations and technologies. However, strongly nonlinear regimes, like thoseinvolving multi-dimensional self-localized solitary waves, are marginallyexplored for what concerns quantum features. We study the dynamics of 3D+1solitons in the second-quantized nonlocal nonlinear Schroedinger-Newtonequation. We theoretically investigate the quantum diffusion of the solitoncenter of mass and other parameters, varying the interaction length. 3D+1simulations of the Ito partial differential equations arising from the positiveP-representation of the density matrix validate the theoretical analysis. Thenumerical results unveil the onset of non-Gaussian statistics of the soliton,which may signal quantum-gravitational effects and be a resource for quantumcomputing. The non-Gaussianity arises from the interplay between the solitonparameter quantum diffusion and stable invariant propagation. The fluctuationsand the non-Gaussianity are universal effects expected for any nonlocality anddimensionality.

Random walk and non-Gaussianity of the 3D second-quantized Schr\"odinger-Newton nonlocal soliton