  The recently developed variational discrete action theory (VDAT) provides a
systematic variational approach to the ground state of the quantum many-body
problem, where the quality of the solution is controlled by an integer
$\mathcal{N}$, and increasing $\mathcal{N}$ monotonically approaches the exact
solution. VDAT can be exactly evaluated in the $d=\infty$ multi-orbital Hubbard
model using the self-consistent canonical discrete action theory (SCDA), which
requires a self-consistency condition for the integer time Green's functions.
Previous work demonstrates that $\mathcal{N}=3$ accurately captures
multi-orbital Mott/Hund physics at a cost similar to the Gutzwiller
approximation. Here we employ a gauge constraint to automatically satisfy the
self-consistency condition of the SCDA at $\mathcal{N}=3$, yielding an even
more efficient algorithm with enhanced numerical stability. We derive closed
form expressions of the gauge constrained algorithm for the multi-orbital
Hubbard model with general density-density interactions, allowing VDAT at
$\mathcal{N}=3$ to be straightforwardly applied to the seven orbital Hubbard
model. We present results and a performance analysis using $\mathcal{N}=2$ and
$\mathcal{N}=3$ for the $\textrm{SU}(2\textrm{N}_{\textrm{orb}})$ Hubbard model
in $d=\infty$ with $\textrm{N}_{\textrm{orb}}=2-8$, and compare to numerically
exact dynamical mean-field theory solutions where available. The developments
in this work will greatly facilitate the application of VDAT at $\mathcal{N}=3$
to strongly correlated electron materials.
