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The virial expansion provides a non-perturbative view into the thermodynamicsof quantum many-body systems in dilute regimes. While powerful, the expansionis challenging as calculating its coefficients at each order $n$ requiresanalyzing (if not solving) the quantum $n$-body problem. In this work, wepresent a comprehensive review of automated algebra methods, which we developedto calculate high-order virial coefficients. The methods are computational butnon-stochastic, thus avoiding statistical effects; they are also for the mostpart analytic, not numerical, and amenable to massively parallel computerarchitectures. We show formalism and results for coefficients characterizingthe thermodynamics (pressure, density, energy, static susceptibilities) ofhomogeneous and harmonically trapped systems, and explain how to generalizethem to other observables such as the momentum distribution, Tan's contact, andthe structure factor.

Toward an automated-algebra framework for high orders in the virial expansion of quantum matter