  Quantum simulations of lattice gauge theories are anticipated to directly
probe the real time dynamics of QCD, but scale unfavorably with the required
truncation of the gauge fields. Improved Hamiltonians are derived to correct
for the effects of gauge field truncations on the SU(3) Kogut-Susskind
Hamiltonian. It is shown in $1+1D$ that this enables low chromo-electric field
truncations to quantitatively reproduce features of the untruncated theory over
a range of couplings and quark masses. In $3+1D$, an improved Hamiltonian is
derived for lattice QCD with staggered massless fermions. It is shown in the
strong coupling limit that the spectrum qualitatively reproduces aspects of two
flavor QCD and simulations of a small system are performed on IBM's {\tt Perth}
quantum processor.
