  Nuclear magnetic resonance spectroscopy is one of the few remaining areas of
physical chemistry for which polynomially scaling simulation methods have not
so far been available. Here, we report such a method and illustrate its
performance by simulating common 2D and 3D liquid state NMR experiments
(including accurate description of spin relaxation processes) on isotopically
enriched human ubiquitin - a protein containing over a thousand nuclear spins
forming an irregular polycyclic three-dimensional coupling lattice. The
algorithm uses careful tailoring of the density operator space to only include
nuclear spin states that are populated to a significant extent. The reduced
state space is generated by analyzing spin connectivity and decoherence
properties: rapidly relaxing states as well as correlations between
topologically remote spins are dropped from the basis set. In the examples
provided, the resulting reduction in the quantum mechanical simulation time is
by many orders of magnitude.
