  Finite-temperature phases of many-body quantum systems are fundamental to
phenomena ranging from condensed-matter physics to cosmology, yet they are
generally difficult to simulate. Using an ion trap quantum computer and
protocols motivated by the Quantum Approximate Optimization Algorithm (QAOA),
we generate nontrivial thermal quantum states of the transverse-field Ising
model (TFIM) by preparing thermofield double states at a variety of
temperatures. We also prepare the critical state of the TFIM at zero
temperature using quantum-classical hybrid optimization. The entanglement
structure of thermofield double and critical states plays a key role in the
study of black holes, and our work simulates such nontrivial structures on a
quantum computer. Moreover, we find that the variational quantum circuits
exhibit noise thresholds above which the lowest depth QAOA circuits provide the
best results.
