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We investigate a quantum gauge theory at finite temperature and density usinga variational algorithm for near-term quantum devices. We adapt $\beta$-VQE toevaluate thermal and quantum expectation values and study the phase diagram formassless Schwinger model along with the temperature and density. By comperingthe exact variational free energy, we find the variational algorithm work for$T>0$ and $\mu>0$ for the Schwinger model. No significant volume dependence ofthe variational free energy is observed in $\mu/g \in[0, 1.4]$. We calculatethe chiral condensate and take the continuum extrapolation. As a result, weobtain qualitative picture of the phase diagram for massless Schwinger model.

Schwinger model at finite temperature and density with beta VQE