  Recent advances in automated algebra for dilute Fermi gases in the virial
expansion, where coarse temporal lattices were found advantageous, motivate the
study of more general computational schemes that could be applied to arbitrary
densities, beyond the dilute limit where the virial expansion is physically
reasonable. We propose here such an approach by developing what we call the
Quantum Thermodynamics Computational Engine (QTCE). In QTCE, the imaginary-time
direction is discretized and the interaction is accounted for via a quantum
cumulant expansion, where the coefficients are expressed in terms of
noninteracting expectation values. The aim of QTCE is to enable the systematic
resolution of interaction effects at fixed temporal discretization, as in
lattice Monte Carlo calculations, but here in an algebraic rather than
numerical fashion. Using this approach, in combination with numerical
integration techniques (both known and alternative ones proposed here), we
explore the thermodynamics of spin-1/2 fermions, focusing on the unitary limit
in 3 spatial dimensions, but also exploring the effects of continuously varying
the spatial dimension below 3. We find that, remarkably, extremely coarse
temporal lattices, when suitably renormalized using known results from the
virial expansion, yield stable partial sums for QTCE's cumulant expansion which
are qualitatively and quantitatively correct in wide regions, compared with
known experimental results.
