We propose a solution to the matrix dimension problem in quantum mechanical simulations of MRI experiments on complex molecules. This problem is very old; it arises when Kronecker products of spin operators and spatial dynamics generators are taken – the resulting matrices are far too large. However, spin and spatial operators individually have manageable dimensions, and the action by their Kronecker products on any vector may be computed without opening those products; this procedure is now implemented in Spinach.
MRI simulations of complex metabolites in 3D with diffusion, flow, kinetics, and quantum mechanical treatment of spin relaxation, are now possible, as well as simulations of spatially distributed ultrafast, pure shift, diffusion, and flow driven NMR experiments. This level of generality hinges on:
The ability to treat classical degrees of freedom (diffusion,
hydrodynamics, radiofrequency and microwave phases, stochastic
tumbling, etc.) at the same conceptual level as spin degrees of
The ability to survive enormous Kronecker products. In realistic
systems (ten spins in 3D), the direct products
of spin and spatial dynamics generators have the
dimension in excess of 1012 .
Code parallelisation over cluster architectures, including the
possibility of using a GPU on each node of the cluster.
This report is about solving all of this, and on where the dark art of simulating a time-domain magnetic resonance experiment stands at the moment. Two recent innovations are the abandonment of Liouville equation in favour of Fokker-Planck equation  as the core formalism, and the use of tensor structured objects that never open Kronecker products .
 I. Kuprov, “Fokker-Planck formalism in magnetic resonance simulations”, Journal of Magnetic Resonance, 2016, 270, 124-135, and references therein.
 A.J. Allami, M.G. Concilio, P. Lally, I. Kuprov, “Quantum mechanical MRI simulations: solving the matrix dimension problem”, Science Advances, in press.