  Conventional methods of quantum simulation involve trade-offs that limit
their applicability to specific contexts where their use is optimal. In
particular, the interaction picture simulation has been found to provide
substantial asymptotic advantages for some Hamiltonians, but incurs prohibitive
constant factors and is incompatible with methods like qubitization. We provide
a framework that allows different simulation methods to be hybridized and
thereby improve performance for interaction picture simulations over known
algorithms. These approaches show asymptotic improvements over the individual
methods that comprise them and further make interaction picture simulation
methods practical in the near term. Physical applications of these hybridized
methods yield a gate complexity scaling as $\log^2 \Lambda$ in the electric
cutoff $\Lambda$ for the Schwinger Model and independent of the electron
density for collective neutrino oscillations, outperforming the scaling for all
current algorithms with these parameters. For the general problem of
Hamiltonian simulation subject to dynamical constraints, these methods yield a
query complexity independent of the penalty parameter $\lambda$ used to impose
an energy cost on time-evolution into an unphysical subspace.
