TEPID-ADAPT: Adaptive variational method for simultaneous preparation of low-temperature Gibbs and low-lying eigenstates

Preparing Gibbs states, which describe systems in equilibrium at finite temperature, is of great importance, particularly at low temperatures. In this work, we propose a new method -- TEPID-ADAPT -- that prepares the thermal Gibbs state of a given Hamiltonian at low temperatures using a variational method that is partially adaptive and uses a purification with a minimal number of ancillary qubits. We also present an alternative implementation without ancillary qubits. A key technical innovation here is to use a mixed-state ansatz where the entropy can be efficiently calculated, with no computational overhead. Our algorithm uses a truncated, parametrized eigenspectrum of the Hamiltonian. Beyond preparing Gibbs states, this approach also straightforwardly gives us access to the truncated low-energy eigenspectrum of the Hamiltonian, making it also a method that prepares excited states simultaneously. As a result of this, we are also able to prepare thermal states at any lower temperature of the same Hamiltonian without further optimization.