  Preparation of Gibbs distributions is an important task for quantum
computation. It is a necessary first step in some types of quantum simulations
and further is essential for quantum algorithms such as quantum Boltzmann
training. Despite this, most methods for preparing thermal states are
impractical to implement on near-term quantum computers because of the memory
overheads required. Here we present a variational approach to preparing Gibbs
states that is based on minimizing the free energy of a quantum system. The key
insight that makes this practical is the use of Fourier series approximations
to the logarithm that allows the entropy component of the free-energy to be
estimated through a sequence of simpler measurements that can be combined
together using classical post processing. We further show that this approach is
efficient for generating high-temperature Gibbs states, within constant error,
if the initial guess for the variational parameters for the programmable
quantum circuit are sufficiently close to a global optima. Finally, we examine
the procedure numerically and show the viability of our approach for five-qubit
Hamiltonians using Trotterized adiabatic state preparation as an ansatz.
