  Higher-fidelity entry simulations can be enabled by integrating finer
thermo-chemistry models into compressible flow physics. One such class of
models are State-to-State (StS) kinetics, which explicitly track species
populations among quantum energy levels. StS models can represent
thermo-chemical non-equilibrium effects that are hardly captured by standard
multi-temperature models. However, the associated increase in computational
cost is dramatic. For implicit solution techniques that rely on standard
block-sparse representations of the Jacobian, both the spatial complexity and
the temporal complexity grow quadratically with respect to the number of
quantum levels represented. We introduce a more efficient way to represent the
Jacobian arising in first-order implicit simulations for compressible flow
physics coupled with StS models. The key idea is to recognize that the density
of local blocks of the Jacobian comes from rank-one updates that can be managed
separately. This leads to a new Jacobian structure, consisting of a
fully-sparse matrix and block-wise rank-one updates, whose overall complexity
grows linearly with the number of quantum levels. This structure also brings
forth a potentially faster variation of the block-Jacobi preconditioning
algorithm by leveraging the Sherman-Morrison-Woodbury inversion formula.
